81 Newton s Law of Universal Gravitation


 Jeffrey Small
 2 years ago
 Views:
Transcription
1 81 Newton s Law of Univesal Gavitation One of the most famous stoies of all time is the stoy of Isaac Newton sitting unde an apple tee and being hit on the head by a falling apple. It was this event, so the stoy goes, that led Newton to ealize that the same foce that bought the apple down on his head was also esponsible fo keeping the Moon in its obit aound the ath, and fo keeping all the planets of the sola system, including ou own planet ath, in obit aound the Sun. This foce is the foce of gavity. It is had to ovestate the impact of Newton s wok on gavity. Pio to Newton, it was widely thought that thee was one set of physical laws that explained how things woked on ath (explaining why apples fall down, fo instance), and a completely diffeent set of physical laws that explained the motion of the stas in the heavens. Amed with the insight that events on ath, as well as the behavio of stas, can be explained by a elatively simple equation (see the box below), humankind awoke to the undestanding that ou fates ae not detemined by the whims of gods, but depend, in fact, on the way we inteact with the ath, and in the way the ath inteacts with the Moon and the Sun. This simple, yet poweful idea, that we have some contol ove ou own lives, helped tigge a eal enlightenment in many aeas of ats and sciences. The foce of gavity does not equie the inteacting objects to be in contact with one anothe. The foce of gavity is an attactive foce that is popotional to the poduct of the masses of the inteacting objects, and invesely popotional to the squae of the distance between them. A gavitational inteaction involves the attactive foce that any object with mass exets on any othe object with mass. The geneal equation to detemine the gavitational foce an object of mass M exets on an object of mass m when the distance between thei centesofmass is is: v GmM FG = ˆ (quation 8.1: Newton s Law of Univesal Gavitation) 11 whee G = N m / kg is known as the univesal gavitational constant. The magnitude of the foce is equal to GmM / while the diection is given by ˆ, which means that the foce is attactive, diected back towad the object exeting the foce. v v At the suface of the ath, should we use FG = mg o Newton s Law of Univesal Gavitation instead? Why is g equal to 9.8 N/kg at the suface of the ath, anyway? The two equations must be equivalent to one anothe, at least at the suface of the ath, because they epesent the same gavitational inteaction. If we set the expessions equal to one anothe we get: GmM GM mg = which gives g =. At the suface of the ath M is the mass of the ath, M = kg, and is the 6 adius of the ath, R = m. So, the magnitude of g at the ath s suface is: g 11 4 GM ( N m /kg )( kg) = = = 9.83 N/kg. R 6 ( m) Fo any object at the suface of the ath, when we use Newton s Law of Univesal Gavitation, the factos G, M, and R ae all constants, so, until this point in the book, we have simply been eplacing the constant value of GM / R by g = 9.8 N/kg. 4 Chapte 8 Gavity Page 1
2 XAMPL 8.1 A twodimensional situation Thee balls, of mass m, m, and 3m, ae placed at the cones of a squae measuing L on each side, as shown in Figue 8.5. Assume this set of thee balls is not inteacting with anything else in the univese. What is the magnitude and diection of the net gavitational foce on the ball of mass m? Figue 8.1: Thee balls placed at the cones of a squae. SOLUTION Let s begin by attaching foce vectos to the ball of mass m. In Figue 8.6 each vecto is colocoded based on the object exeting the foce. The length of each vecto is popotional to the magnitude of the foce it epesents. Figue 8.: Attaching foce vectos to the ball of mass m. We can find the two individual foces acting on the ball of mass m using Newton s Law of Univesal Gavitation. Let s define +x to the ight and +y up. v Gm( m) Fom the ball of mass m: F1 = to the ight. L v Gm(3 m) Fom the ball of mass 3m: 31 at 45 o F = below the xaxis. L + L Finding the net foce is a vectoaddition poblem. In the xdiection, we get: In the ydiection, we get: v v v F F F Gm 3Gm o 3 Gm 1x = 1x + 31x =+ + cos 45 = + v v v F F F L L L 3Gm o 3 Gm 1y = 1y + 31y = 0 sin45 = L L.. The Pythagoean theoem gives the magnitude of the net foce on the ball of mass m: Gm Gm 1 = 1x + 1y = = 3.4 F F F F1 y 3 The angle is given by: tan θ = = =. F1 x So, the angle is 19.1 below the xaxis. Figue 8.3: The tiangle epesenting the vecto addition poblem being solved above. Related ndofchapte xecises: 16, ssential Question 8.1: The Sun has a much lage mass than the ath. Which object exets a lage gavitational foce on the othe, the Sun o the ath? L L. Chapte 8 Gavity Page
3 Answe to ssential Question 8.1: Newton s thid law tells us that the gavitational foce the Sun exets on the ath is equal in magnitude (and opposite in diection) to the gavitational foce the ath exets on the Sun. This follows fom quation 8.1, because, whethe we look at the foce exeted by the Sun o the ath, the factos going into the equation ae the same. 8 The Pinciple of Supeposition XPLORATION 8. Thee objects in a line Thee balls, of mass m, m, and 3m, ae equally spaced along a line. The spacing between the balls is. We can aange the balls in thee diffeent ways, as shown in Figue 8.. In each case the balls ae in an isolated egion of space vey fa fom anything else. Figue 8.: Thee diffeent aangements of thee balls of mass m, m, and 3m placed on a line with a distance between neighboing balls. Step 1 How many foces does each ball expeience in each case? ach ball expeiences two gavitational foces, one fom each of the othe balls. We can neglect any othe inteactions. Step Conside Case 1. Is the foce that the ball of mass m exets on the ball of mass 3m affected by the fact that the ball of mass m lies between the othe two balls? Inteestingly, no. To find the net foce on any object, we simply add the individual foces acting on an object as vectos. This is known as the pinciple of supeposition, and it applies to many diffeent physical situations. In case 1, fo instance, we find the foce the ball of mass m applies to the ball of mass 3m as if the ball of mass m is not pesent. The net foce on the ball of mass 3m is the vecto addition of that foce and the foce on the 3m ball fom the ball of mass m. Step 3 In which case does the ball of mass m expeience the lagestmagnitude net foce? Ague qualitatively. Let s attach aows to the ball of mass m, as in Figue 8.3, to epesent the two foces the ball expeiences in each case. The length of each aow is popotional to the foce. Figue 8.3: Attaching foce vectos to the ball of mass m. The vectos ae colocoded based on the colo of the object exeting the foce. The length of each vecto is dawn in units of Gm /. Chapte 8 Gavity Page 3
4 In case 1, the two foces patly cancel, and, in case, the foces add but give a smalle net foce than that in case 3. Thus, the ball of mass m expeiences the lagestmagnitude net foce in Case 3. Step 4 Calculate the foce expeienced by the ball of mass m in each case. To do this, we will make extensive use of Newton s Univesal Law of Gavitation. Let s define ight to be the positive diection, and use the notation F v 1 fo the foce that the ball of mass m expeiences fom the ball of mass m. In each case: v v v F = F + F, net 1 3 Case 1: Case : Case 3: v v v Gm( m) G( m)(3 m) Gm 6Gm 4Gm F, net = F1 + F3 = + = + =+ v v v Gm( m) G( m)(3 m) Gm 3Gm 7Gm F, net = F1 + F3 =+ + =+ + =+ ( ) v v v Gm( m) G( m)(3 m) Gm 6Gm 13Gm F, net = F1 + F3 = = = ( ) This appoach confims that the ball of mass m expeiences the lagestmagnitude net foce in case 3. Step 5  Rank the thee cases, fom lagest to smallest, based on the magnitude of the net foce exeted on the ball in the middle of the set of thee balls. Let s extend ou pictoial method by attaching foce vectos to each ball in each case, as in Figue 8.4. Figue 8.4: Attaching foce vectos to the balls in each case. The foce vectos ae colocoded accoding to the ball applying the foce. The length of each vecto is dawn in units of Gm /. Again, when consideing the net foce on the middle ball, we need to add the individual foces as vectos. Refeing to Figue 8.4, anking the cases based on the magnitude of the net foce exeted on the middle ball gives Case 1 > Case 3 > Case. Key idea about the pinciple of supeposition: The net foce acting on an object can be found using the pinciple of supeposition, adding all the individual foces togethe as vectos and emembeing that each individual foce is unaffected by the pesence of othe foces. Related nd of Chapte xecises: 15 and 7. ssential Question 8.: In the xploation above, which ball expeiences the lagestmagnitude net foce in (i) Case 1 (ii) Case (iii) Case 3? Chapte 8 Gavity Page 4
5 Answe to ssential Question 8.: We could detemine the net foce on each object quantitatively, but Figue 8.4 shows that the object expeiencing the lagestmagnitude net foce is the object of mass 3m in cases 1 and, and the object of mass m in case 3. In geneal, in the case of thee objects of diffeent mass aanged in a line the object expeiencing the lagest net foce will be one of the objects at the end of the line, the one with the lage mass. The object in the middle will not have the lagest net foce because the two foces it expeiences ae in opposite diections. 83 Gavitational Field Let s discuss the concept of a gavitational field, which is epesented by g v. So fa, we have efeed to g v as the acceleation due to gavity, but a moe appopiate name is the stength of the local gavitational field. A field is something that has a magnitude and diection at all points in space. One way to define the gavitational field at a paticula point is in tems of the gavitational foce that an object of mass m would expeience if it wee placed at that point: v v F g = G. (quation 8.: Gavitational field) m The units fo gavitational field ae N/kg, o m/s. A special case is the gavitational field outside an object of mass M, such as the ath, that is poduced by that object: v GM g = ˆ, (quation 8.3: Gavitational field fom a point mass) whee is the distance fom the cente of the object to the point. The magnitude of the field is GM /, while the diection is given by ˆ, which means that the field is diected back towad the object poducing the field. One way to think about a gavitational field is the following: it is a measue of how an object, o a set of objects, with mass influences the space aound it. Visualizing the gavitational field It can be useful to daw a pictue that epesents the gavitational field nea an object, o a set of objects, so we can see at a glance what the field in the egion is like. In geneal thee ae two ways to do this, by using eithe field lines o field vectos. The fieldline epesentation is shown in Figue 8.7. If Figue 8.7 (a) epesents the field at the suface of the ath, Figue 8.7 (b) could epesent the field at the suface of anothe planet whee g is twice as lage as it is at the suface of the ath. In both these cases we have a unifom field, because the field lines ae equally spaced and paallel. In Figue 8.7 (c) we have zoomed out fa fom a planet to get a wide pespective on how the planet affects the space aound it, while in Figue 8.7 (d) we have done the same thing fo a diffeent planet with half the mass, but the same adius, as the planet in (c). Chapte 8 Gavity Page 5
6 Figue 8.7: Fieldline diagams fo vaious situations. Diagams a and b epesent unifom gavitational fields, with the field in b two times lage than that in a. Diagams c and d epesent nonunifom fields, such as the fields nea a planet. The field at the suface of the planet in c is two times lage than that at the suface of the planet in d. Question: How is the diection of the gavitational field at a paticula point shown on a fieldline diagam? What indicates the elative stength of the gavitational field at a paticula point on the fieldline diagam? Answe: ach field line has a diection maked on it with an aow that shows the diection of the gavitational field at all points along the field line. The elative stength of the gavitational field is indicated by the density of the field lines (i.e., by how close the lines ae). The moe lines thee ae in a given aea the lage the field. A second method of epesenting a field is to use field vectos. A field vecto diagam has the nice featue of einfocing the idea that evey point in space has a gavitational field associated with it, because a gid made up of equally spaced dots is supeimposed on the pictue and a vecto is attached to each of these gid points. All the vectos ae the same length. The situations epesented by the fieldline pattens in Figue 8.7 ae now edawn in Figue 8.8 using the fieldvecto epesentation. Figue 8.8: Fieldvecto diagams fo vaious situations. In figues a and b the field is unifom and diected down. The field vectos ae dake in figue b, eflecting the fact that the field has a lage magnitude in figue b than in figue a. Figues c and d epesent nonunifom fields, such as those found nea a planet. Again, the fact that each field vecto in figue c is dake than its countepat in figue d tells us that the field at any point in figue c has a lage magnitude than the field at an equivalent point in figue d. Related nd of Chapte xecises: 18, 36. ssential Question 8.3: How is the diection of the gavitational field at a paticula point shown on a fieldvecto diagam? What indicates the elative stength of the gavitational field at a paticula point on the fieldvecto diagam? Chapte 8 Gavity Page 6
7 Answe to ssential Question 8.3: The diection of the gavitational field at a paticula point is epesented by the diection of the field vecto at that point (o the ones nea it if the point does not coespond exactly to the location of a field vecto). The elative stength of the field is indicated by the dakness of the aow. The lage the field s magnitude, the dake the aow. 84 Gavitational Potential negy The expession we have been using fo gavitational potential enegy up to this point, UG = mgh, applies when the gavitational field is unifom. In geneal, the equation fo gavitational potential enegy is: GmM UG =. (quation 8.4: Gavitational potential enegy, in geneal) This gives the enegy associated with the gavitational inteaction between two objects, of mass m and M, sepaated by a distance. The minus sign tells us the objects attact one anothe. Conside the diffeences between the mgh equation fo gavitational potential enegy and the moe geneal fom. Fist, when using quation 8.4 we ae no longe fee to define the potential enegy to be zeo at some convenient point. Instead, the gavitational potential enegy is zeo when the two objects ae infinitely fa apat. Second, when using quation 8.4 we find that the gavitational potential enegy is always negative, which is cetainly not what we found with mgh. That should not woy us, howeve, because what is citical is how potential enegy changes as objects move with espect to one anothe. If you dop you pen and it falls to the floo, fo instance, both foms of the gavitational potential enegy equation give consistent esults fo the change in the pen s gavitational potential enegy. quation 8.4 also einfoces the idea that, when two objects ae inteacting via gavity, neithe object has its own gavitational potential enegy. Instead, gavitational potential enegy is associated with the inteaction between the objects. XPLORATION 8.4 Calculate the total potential enegy in a system Thee balls, of mass m, m, and 3m, ae placed in a line, as shown in Figue What is the total gavitational potential enegy of this system? Figue 8.10: Thee equally spaced balls placed in a line. To detemine the total potential enegy of the system, conside the numbe of inteacting pais. In this case thee ae thee ways to pai up the objects, so thee ae thee tems to add togethe to find the total potential enegy. Because enegy is a scala, we do not have to woy about diection. Using a subscipt of 1 fo the ball of mass m, fo the ball of mass m, and 3 fo the ball of mass 3m, we get: Gm(3 m) G( m)(3 m) Gm( m) 10Gm UTotal = U13 + U3 + U1 = =. Key ideas fo gavitational potential enegy: Potential enegy is a scala. The total gavitational potential enegy of a system of objects can be found by adding up the enegy associated with each inteacting pai of objects. Related ndofchapte xecises: 5, 9, 40. Chapte 8 Gavity Page 7
8 XAMPL 8.4 Applying consevation ideas A ball of mass 1.0 kg and a ball of mass 3.0 kg ae initially sepaated by 4.0 m in a egion of space in which they inteact only with one anothe. When the balls ae eleased fom est, they acceleate towad one anothe. When they ae sepaated by.0 m, how fast is each ball going? SOLUTION Figue 8.11: The initial situation shows the balls at est. The foce of gavity causes them to acceleate towad one anothe. Figue 8.11 shows the balls at the beginning and when they ae sepaated by.0 m. Analyzing foces, we find that the foce on each ball inceases as the distance between the balls deceases. This makes it difficult to apply a foce analysis. negy consevation is a simple appoach. Ou enegy equation is: U + K + W = U + K. i i nc f f In this case, thee ae no nonconsevative foces acting, and in the initial state the kinetic enegy is zeo because both objects ae at est. This gives Ui = U f + K f. The final kinetic enegy epesents the kinetic enegy of the system, the sum of the kinetic enegies of the two objects. Let s solve this geneally, using a mass of m and a final speed of v1 fo the 1.0 kg ball, and a mass of 3m and a final speed of v fo the 3.0 kg ball. The enegy equation becomes: Gm(3 m) Gm(3 m) 1 1 = + mv1 + (3 m ) v. 4.0 m.0 m 3Gm 3Gm 1 3 Canceling factos of m gives: = + v1 + v. 4.0 m.0 m 3 Multiplying though by, and combining tems, gives: Gm + = v v..0 m Because thee is no net extenal foce, the system s momentum is conseved. Thee is no initial momentum. Fo the net momentum to emain zeo, the two momenta must always be equalandopposite. Defining ight to be positive, momentum consevation gives: 0=+ mv1 3mv, which we can simplify to v1 = 3v. Substituting this into the expession we obtained fom applying enegy consevation: 3Gm + = (3 v) + 3v = 1v.0 m Gm Gm This gives v =, and v1 = 3v = m 8.0 m 6 6 Using m = 1.0 kg, we get v =.9 10 m / s and v 1 = m / s. Related ndofchapte xecises: Poblems ssential Question 8.4: Retun to the pevious xample. If you epeat the expeiment with balls of mass.0 kg and 6.0 kg instead, would the final speeds change? If so, how? Chapte 8 Gavity Page 8
9 Answe to ssential Question 8.4: If we double each mass, the analysis above still woks. Plugging m =.0 kg into ou speed equations shows that the speeds incease by a facto of. 85 xample Poblems XAMPL 8.5A Whee is the field zeo? Locations whee the net gavitational field is zeo ae special, because an object placed whee the field is zeo expeiences no net gavitational foce. Let s place a ball of mass m at the oigin, and place a second ball of mass 9m on the xaxis at x = +4a. Find all the locations nea the balls whee the net gavitational field associated with these balls is zeo. SOLUTION A diagam of the situation is shown in Figue 8.9. Let s now appoach the poblem conceptually. At evey point nea the balls thee ae two gavitational fields, one fom each ball. The net field is zeo only whee the two fields ae equalandopposite. These fields ae in exactly opposite diections only at locations on the xaxis between the balls. If we get too close to the fist ball it dominates, and if we get too close to the second ball it dominates; thee is just one location between the balls whee the fields exactly balance. Figue 8.9: The two balls in xample 8.5A. An equivalent appoach is to use foces. Imagine having a thid ball (we geneally call this a test mass) and placing it nea the othe two balls. The thid ball expeiences two foces, one fom each of the oiginal balls, and these foces have to exactly balance. This happens at one location between the oiginal two balls. Whethe we think about fields o foces, the appoach is equivalent. The special place whee the net field is zeo is close to the ball with the smalle mass. To make up fo a facto of 9, epesenting the atio of the two masses, we need to have a facto of 3 (which gets squaed to 9) in the distances. In othe wods, we need to be thee times futhe fom the ball with a mass of 9m than we ae fom the ball of mass m fo the fields to be of equal magnitude. This occus at x = +a. We can also get this answe using a quantitative appoach. Using the subscipt 1 fo the ball of mass m, and fo the ball of mass 9m, we can expess the net field as: g v = g v + g v = net 1 0. Define ight to be positive. If the point we e looking fo is between the balls a distance x fom the ball of mass m, it is (4a x) fom the ball of mass 9m. Using the definition of g v gives: Gm G(9 m) + = 0. x (4 a x) Canceling factos of G and m, and eaanging gives: 1 = 9. x (4 a x) Cossmultiplying leads to: (4 a x) = 9 x. We could use the quadatic equation to solve fo x, but let s instead take the squae oot of both sides of the equation. When we take a squae oot the esult can be eithe plus o minus: 4a x=± 3 x. Chapte 8 Gavity Page 9
10 Using the positive sign, we get 4a= + 4x, so x = + a. This is the coect solution, lying between the balls and close to the ball with the smalle mass. Because it is thee times fathe fom the ball of mass 9m than the ball of mass m, and because the distance is squaed in the equation fo field, this exactly balances the facto of 9 in the masses. Using a minus sign gives a second solution, x = a. This location is thee times fathe (6a) fom the ball of mass 9m than fom the ball of mass m (a). Thus at x = a the two fields have the same magnitude, but they point in the same diection so they add athe than canceling. Related ndofchapte xecises: 13, 14, 0. XAMPL 8.5B scape fom ath When you thow a ball up into the ai, it comes back down. How fast would you have to launch a ball so that it neve came back down, but instead it escaped fom the ath? The minimum speed equied to do this is known as the escape speed. SOLUTION A diagam is shown in Figue 8.1. Let s assume the ball stats at the suface of the ath and that we can neglect ai esistance (this would be fine if we wee escaping fom the Moon, but it is a poo assumption if we e escaping fom ath  let s not woy about that, howeve). We ll also assume the ath is the only object in the Univese. So, this is an inteesting calculation but the esult will only be a ough appoximation of eality. Let s apply the enegy consevation equation: U + K + W = U + K. i i nc f f Figue 8.1: negy ba gaphs ae shown in addition to the pictues showing the initial and final situations. We e neglecting any wok done by nonconsevative foces, so W nc = 0. The final gavitational potential enegy is negligible, because the distance between the ball and ath is vey lage (we can assume it to be infinite). What about the final kinetic enegy? Because we e looking fo the minimum initial speed let s use the minimum possible speed of the ball when it is vey fa fom ath, which we can assume to be zeo. This leads to an equation in which eveything on the ighthand side is zeo: Ui + Ki = 0. GmM 1 + mvescape = 0. R The mass of the ball does not matte, because it cancels out. This gives: 11 4 GM ( N m / kg )( kg) vescape = = = 11. km / s. 6 R m This is athe fast, and explains why objects we thow up in the ai come down again! Related ndofchapte xecises: 41, 4. ssential Question 8.5: Let s say we wee on a diffeent planet that had the same mass as ath but twice ath s adius. How would the escape speed compae to that on ath? Chapte 8 Gavity Page 10
11 Answe to ssential Question 8.5: Since v escape GM doubling the adius educes the escape speed by a facto of. =, keeping the mass the same while R 86 Obits Imagine that we have an object of mass m in a cicula obit aound an object of mass M. An example could be a satellite obiting the ath. What is the total enegy associated with this object in its cicula obit? The total enegy is the sum of the potential enegy plus the kinetic enegy: GmM 1 = U + K = + mv. This is a lovely equation, but it doesn t tell us much. Let s conside foces to see if we can shed moe light on what s going on. Fo the object of mass m to expeience unifom cicula motion about the lage mass it must expeience a net foce diected towad the cente of the cicle (i.e., towad the object of mass M). This is the gavitational foce exeted by the object of mass M. Applying Newton s Second Law gives: v v mv Σ F = ma =, diected towad the cente. GmM mv GmM =, which tells us that mv =. Substituting this esult into the enegy expession gives: GmM GmM GmM = + =. This esult is geneally tue fo the case of a lighte object taveling in a cicula obit aound a moe massive object. We can make a few obsevations about this. Fist, the magnitude of the total enegy equals the kinetic enegy; the kinetic enegy has half the magnitude of the gavitational potential enegy; and the total enegy is half of the gavitational potential enegy. All this is tue when the obit is cicula. Second, the total enegy is negative, which is tue fo a bound system (a system in which the components emain togethe). Systems in which the total enegy is positive tend to fly apat. What happens when an object has a velocity othe than that necessay to tavel in a cicula obit? One way to think of this is to stat the obiting object off at the same place, with a velocity diected pependicula to the line connecting the two objects, and simply vay the speed. If the speed necessay to maintain a cicula obit is denoted by v, let s conside what happens if the speed is 0% less than v ; 0% lage than v ; the special case of v ; and 1.5v. The obits followed by the object in these cases ae shown in Figue Unless the object s initial speed is too small, causing it to eventually collide with the moe massive object, an initial speed that is less than v will poduce an elliptical obit whee the initial point tuns out to be the futhest the object eve gets fom the moe massive object. The initial point is special because at that point the object s velocity is pependicula to the gavitational foce the object expeiences. Chapte 8 Gavity Page 11
12 If the initial speed is lage than v the esult depends on how much lage it is. When the initial speed is v that is the escape speed, and is thus a special case. The shape of the obit is paabolic, and this path maks the bounday between the elliptical paths in which the object emains in obit and the highespeed hypebolic paths in which the object escapes fom the gavitational pull of the massive object. Figue 8.13: The obits esulting fom stating at a paticula spot, the ightmost point on each obit, with initial velocities diected the same way (up in the figue) but with diffeent initial speeds. The dak blue obit epesents the almostcicula obit of the ath, whee the distances on each axis ae in units of metes and the Sun is not shown but is located at the intesection of the axes. If the ath s speed wee suddenly educed by 0% the ath would instead follow the light puple obit, coming athe close to the Sun. If instead the ath s speed wee inceased by 0% the esulting elliptical obit would take us quite a long way fom the Sun befoe coming back again. Inceasing the ath s speed to times its cuent speed (an incease of a little moe than 40%) the ath would be moving at the escape speed and we would follow the light blue paabolic obit to infinity (and beyond). Any initial speed lage than this would esult in a hypebolic obit to infinity, such as that shown in the dak puple. Note that the speeds given in the key to the ight of the gaph epesent initial speeds, the speed the ath would have at the ightmost point in the obit to follow the coesponding path. Related ndofchapte xecises: 47, 59, and 60. ssential Question 8.6: Is linea momentum conseved fo any of these obits? If so, which? Chapte 8 Gavity Page 1
Revision Guide for Chapter 11
Revision Guide fo Chapte 11 Contents Student s Checklist Revision Notes Momentum... 4 Newton's laws of motion... 4 Gavitational field... 5 Gavitational potential... 6 Motion in a cicle... 7 Summay Diagams
More informationEpisode 401: Newton s law of universal gravitation
Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce
More informationFXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.
Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing
More informationChapter 13 Gravitation. Problems: 1, 4, 5, 7, 18, 19, 25, 29, 31, 33, 43
Chapte 13 Gavitation Poblems: 1, 4, 5, 7, 18, 19, 5, 9, 31, 33, 43 Evey object in the univese attacts evey othe object. This is called gavitation. We e use to dealing with falling bodies nea the Eath.
More informationmv2. Equating the two gives 4! 2. The angular velocity is the angle swept per GM (2! )2 4! 2 " 2 = GM . Combining the results we get !
Chapte. he net foce on the satellite is F = G Mm and this plays the ole of the centipetal foce on the satellite i.e. mv mv. Equating the two gives = G Mm i.e. v = G M. Fo cicula motion we have that v =!
More information2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,
3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects
More informationGRAVITATIONAL FIELD: CHAPTER 11. The groundwork for Newton s great contribution to understanding gravity was laid by three majors players:
CHAPT 11 TH GAVITATIONAL FILD (GAVITY) GAVITATIONAL FILD: The goundwok fo Newton s geat contibution to undestanding gavity was laid by thee majos playes: Newton s Law of Gavitation o gavitational and inetial
More informationChapter 13 Gravitation
Chapte 13 Gavitation Newton, who extended the concept of inetia to all bodies, ealized that the moon is acceleating and is theefoe subject to a centipetal foce. He guessed that the foce that keeps the
More informationSamples of conceptual and analytical/numerical questions from chap 21, C&J, 7E
CHAPTER 1 Magnetism CONCEPTUAL QUESTIONS Cutnell & Johnson 7E 3. ssm A chaged paticle, passing though a cetain egion of space, has a velocity whose magnitude and diection emain constant, (a) If it is known
More informationThe force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges
The foce between electic chages Coulomb s Law Two chaged objects, of chage q and Q, sepaated by a distance, exet a foce on one anothe. The magnitude of this foce is given by: kqq Coulomb s Law: F whee
More informationA) 2 B) 2 C) 2 2 D) 4 E) 8
Page 1 of 8 CTGavity1. m M Two spheical masses m and M ae a distance apat. The distance between thei centes is halved (deceased by a facto of 2). What happens to the magnitude of the foce of gavity between
More informationGravitation. AP Physics C
Gavitation AP Physics C Newton s Law of Gavitation What causes YOU to be pulled down? THE EARTH.o moe specifically the EARTH S MASS. Anything that has MASS has a gavitational pull towads it. F α Mm g What
More informationThe Role of Gravity in Orbital Motion
! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State
More informationPhysics 235 Chapter 5. Chapter 5 Gravitation
Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus
More informationVoltage ( = Electric Potential )
V1 Voltage ( = Electic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is
More informationLab 5: Circular Motion
Lab 5: Cicula motion Physics 193 Fall 2006 Lab 5: Cicula Motion I. Intoduction The lab today involves the analysis of objects that ae moving in a cicle. Newton s second law as applied to cicula motion
More informationVoltage ( = Electric Potential )
V1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage
More informationResources. Circular Motion: From Motor Racing to Satellites. Uniform Circular Motion. Sir Isaac Newton 3/24/10. Dr Jeff McCallum School of Physics
3/4/0 Resouces Cicula Motion: Fom Moto Racing to Satellites D Jeff McCallum School of Physics http://www.gapsystem.og/~histoy/mathematicians/ Newton.html http://www.fga.com http://www.clke.com/clipat
More information2008 QuarterFinal Exam Solutions
2008 Quatefinal Exam  Solutions 1 2008 QuateFinal Exam Solutions 1 A chaged paticle with chage q and mass m stats with an initial kinetic enegy K at the middle of a unifomly chaged spheical egion of
More informationF G r. Don't confuse G with g: "Big G" and "little g" are totally different things.
G1 Gavity Newton's Univesal Law of Gavitation (fist stated by Newton): any two masses m 1 and m exet an attactive gavitational foce on each othe accoding to m m G 1 This applies to all masses, not just
More informationCh. 8 Universal Gravitation. Part 1: Kepler s Laws. Johannes Kepler. Tycho Brahe. Brahe. Objectives: Section 8.1 Motion in the Heavens and on Earth
Ch. 8 Univesal Gavitation Pat 1: Keple s Laws Objectives: Section 8.1 Motion in the Heavens and on Eath Objectives Relate Keple s laws of planetay motion to Newton s law of univesal gavitation. Calculate
More information1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2
Chapte 5 Example The helium atom has 2 electonic enegy levels: E 3p = 23.1 ev and E 2s = 20.6 ev whee the gound state is E = 0. If an electon makes a tansition fom 3p to 2s, what is the wavelength of the
More informationMagnetic Field and Magnetic Forces. Young and Freedman Chapter 27
Magnetic Field and Magnetic Foces Young and Feedman Chapte 27 Intoduction Reiew  electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field
More informationExam I. Spring 2004 Serway & Jewett, Chapters 15. Fill in the bubble for the correct answer on the answer sheet. next to the number.
Agin/Meye PART I: QUALITATIVE Exam I Sping 2004 Seway & Jewett, Chaptes 15 Assigned Seat Numbe Fill in the bubble fo the coect answe on the answe sheet. next to the numbe. NO PARTIAL CREDIT: SUBMIT ONE
More informationChapter 3: Vectors and Coordinate Systems
Coodinate Systems Chapte 3: Vectos and Coodinate Systems Used to descibe the position of a point in space Coodinate system consists of a fied efeence point called the oigin specific aes with scales and
More informationMultiple choice questions [60 points]
1 Multiple choice questions [60 points] Answe all o the ollowing questions. Read each question caeully. Fill the coect bubble on you scanton sheet. Each question has exactly one coect answe. All questions
More informationPHYSICS 111 HOMEWORK SOLUTION #13. May 1, 2013
PHYSICS 111 HOMEWORK SOLUTION #13 May 1, 2013 0.1 In intoductoy physics laboatoies, a typical Cavendish balance fo measuing the gavitational constant G uses lead sphees with masses of 2.10 kg and 21.0
More informationPY1052 Problem Set 3 Autumn 2004 Solutions
PY1052 Poblem Set 3 Autumn 2004 Solutions C F = 8 N F = 25 N 1 2 A A (1) A foce F 1 = 8 N is exeted hoizontally on block A, which has a mass of 4.5 kg. The coefficient of static fiction between A and the
More informationPY1052 Problem Set 8 Autumn 2004 Solutions
PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ighthand end. If H 6.0 m and h 2.0 m, what
More information14. Gravitation Universal Law of Gravitation (Newton):
14. Gavitation 1 Univesal Law of Gavitation (ewton): The attactive foce between two paticles: F = G m 1m 2 2 whee G = 6.67 10 11 m 2 / kg 2 is the univesal gavitational constant. F m 2 m 1 F Paticle #1
More information12. Rolling, Torque, and Angular Momentum
12. olling, Toque, and Angula Momentum 1 olling Motion: A motion that is a combination of otational and tanslational motion, e.g. a wheel olling down the oad. Will only conside olling with out slipping.
More informationGeneral Physics (PHY 2130)
Geneal Physics (PHY 130) Lectue 11 Rotational kinematics and unifom cicula motion Angula displacement Angula speed and acceleation http://www.physics.wayne.edu/~apetov/phy130/ Lightning Review Last lectue:
More informationChapter 26  Electric Field. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University
Chapte 6 lectic Field A PowePoint Pesentation by Paul. Tippens, Pofesso of Physics Southen Polytechnic State Univesity 7 Objectives: Afte finishing this unit you should be able to: Define the electic field
More informationPHYSICS 111 HOMEWORK SOLUTION #5. March 3, 2013
PHYSICS 111 HOMEWORK SOLUTION #5 Mach 3, 2013 0.1 You 3.80kg physics book is placed next to you on the hoizontal seat of you ca. The coefficient of static fiction between the book and the seat is 0.650,
More informationExam 3: Equation Summary
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics Physics 8.1 TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t= Exam 3: Equation Summay total = Impulse: I F( t ) = p Toque: τ = S S,P
More informationSolutions to Homework Set #5 Phys2414 Fall 2005
Solution Set #5 1 Solutions to Homewok Set #5 Phys414 Fall 005 Note: The numbes in the boxes coespond to those that ae geneated by WebAssign. The numbes on you individual assignment will vay. Any calculated
More informationChapter 23: Gauss s Law
Chapte 3: Gauss s Law Homewok: Read Chapte 3 Questions, 5, 1 Poblems 1, 5, 3 Gauss s Law Gauss s Law is the fist of the fou Maxwell Equations which summaize all of electomagnetic theoy. Gauss s Law gives
More informationDetermining solar characteristics using planetary data
Detemining sola chaacteistics using planetay data Intoduction The Sun is a G type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this inestigation
More informationSo we ll start with Angular Measure. Consider a particle moving in a circular path. (p. 220, Figure 7.1)
Lectue 17 Cicula Motion (Chapte 7) Angula Measue Angula Speed and Velocity Angula Acceleation We ve aleady dealt with cicula motion somewhat. Recall we leaned about centipetal acceleation: when you swing
More informationProblem Set 6: Solutions
UNIVESITY OF ALABAMA Depatment of Physics and Astonomy PH 164 / LeClai Fall 28 Poblem Set 6: Solutions 1. Seway 29.55 Potons having a kinetic enegy of 5. MeV ae moving in the positive x diection and ente
More informationHour Exam No.1. p 1 v. p = e 0 + v^b. Note that the probe is moving in the direction of the unit vector ^b so the velocity vector is just ~v = v^b and
Hou Exam No. Please attempt all of the following poblems befoe the due date. All poblems count the same even though some ae moe complex than othes. Assume that c units ae used thoughout. Poblem A photon
More informationPhysics 107 HOMEWORK ASSIGNMENT #14
Physics 107 HOMEWORK ASSIGNMENT #14 Cutnell & Johnson, 7 th edition Chapte 17: Poblem 44, 60 Chapte 18: Poblems 14, 18, 8 **44 A tube, open at only one end, is cut into two shote (nonequal) lengths. The
More informationGravity. A. Law of Gravity. Gravity. Physics: Mechanics. A. The Law of Gravity. Dr. Bill Pezzaglia. B. Gravitational Field. C.
Physics: Mechanics 1 Gavity D. Bill Pezzaglia A. The Law of Gavity Gavity B. Gavitational Field C. Tides Updated: 01Jul09 A. Law of Gavity 3 1a. Invese Squae Law 4 1. Invese Squae Law. Newton s 4 th law
More informationChapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere.
Chapte.3 What is the magnitude of a point chage whose electic field 5 cm away has the magnitude of.n/c. E E 5.56 1 11 C.5 An atom of plutonium39 has a nuclea adius of 6.64 fm and atomic numbe Z94. Assuming
More informationFigure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!
1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the
More informationLab #7: Energy Conservation
Lab #7: Enegy Consevation Photo by Kallin http://www.bungeezone.com/pics/kallin.shtml Reading Assignment: Chapte 7 Sections 1,, 3, 5, 6 Chapte 8 Sections 14 Intoduction: Pehaps one of the most unusual
More informationGravitation and Kepler s Laws Newton s Law of Universal Gravitation in vectorial. Gm 1 m 2. r 2
F Gm Gavitation and Keple s Laws Newton s Law of Univesal Gavitation in vectoial fom: F 12 21 Gm 1 m 2 12 2 ˆ 12 whee the hat (ˆ) denotes a unit vecto as usual. Gavity obeys the supeposition pinciple,
More informationChapter F. Magnetism. Blinn College  Physics Terry Honan
Chapte F Magnetism Blinn College  Physics 46  Tey Honan F.  Magnetic Dipoles and Magnetic Fields Electomagnetic Duality Thee ae two types of "magnetic chage" o poles, Noth poles N and South poles S.
More informationPhysics 111 Fall 2007 Electrostatic Forces and the Electric Field  Solutions
Physics 111 Fall 007 Electostatic Foces an the Electic Fiel  Solutions 1. Two point chages, 5 µc an 8 µc ae 1. m apat. Whee shoul a thi chage, equal to 5 µc, be place to make the electic fiel at the
More informationUNIT 21: ELECTRICAL AND GRAVITATIONAL POTENTIAL Approximate time two 100minute sessions
Name St.No.  Date(YY/MM/DD) / / Section Goup# UNIT 21: ELECTRICAL AND GRAVITATIONAL POTENTIAL Appoximate time two 100minute sessions OBJECTIVES I began to think of gavity extending to the ob of the moon,
More informationSolution Derivations for Capa #8
Solution Deivations fo Capa #8 1) A ass spectoete applies a voltage of 2.00 kv to acceleate a singly chaged ion (+e). A 0.400 T field then bends the ion into a cicula path of adius 0.305. What is the ass
More informationCopyright 2008 Pearson Education, Inc., publishing as Pearson AddisonWesley.
Chapte 5. Foce and Motion In this chapte we study causes of motion: Why does the windsufe blast acoss the wate in the way he does? The combined foces of the wind, wate, and gavity acceleate him accoding
More informationDisplacement, Velocity And Acceleration
Displacement, Velocity And Acceleation Vectos and Scalas Position Vectos Displacement Speed and Velocity Acceleation Complete Motion Diagams Outline Scala vs. Vecto Scalas vs. vectos Scala : a eal numbe,
More informationDeflection of Electrons by Electric and Magnetic Fields
Physics 233 Expeiment 42 Deflection of Electons by Electic and Magnetic Fields Refeences Loain, P. and D.R. Coson, Electomagnetism, Pinciples and Applications, 2nd ed., W.H. Feeman, 199. Intoduction An
More informationVector Calculus: Are you ready? Vectors in 2D and 3D Space: Review
Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.7. find the vecto defined
More information(a) The centripetal acceleration of a point on the equator of the Earth is given by v2. The velocity of the earth can be found by taking the ratio of
Homewok VI Ch. 7  Poblems 15, 19, 22, 25, 35, 43, 51. Poblem 15 (a) The centipetal acceleation of a point on the equato of the Eath is given by v2. The velocity of the eath can be found by taking the
More informationTALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS 14. NEWTON'S RINGS
4. NEWTON'S RINGS. Obective Detemining adius of cuvatue of a long focal length planoconvex lens (lage adius of cuvatue).. Equipment needed Measuing micoscope, planoconvex long focal length lens, monochomatic
More informationGauss Law. Physics 231 Lecture 21
Gauss Law Physics 31 Lectue 1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing
More information2.2. Trigonometric Ratios of Any Angle. Investigate Trigonometric Ratios for Angles Greater Than 90
. Tigonometic Ratios of An Angle Focus on... detemining the distance fom the oigin to a point (, ) on the teminal am of an angle detemining the value of sin, cos, o tan given an point (, ) on the teminal
More informationLesson 32: Measuring Circular Motion
Lesson 32: Measuing Cicula Motion Velocity hee should be a way to come up with a basic fomula that elates velocity in icle to some of the basic popeties of icle. Let s ty stating off with a fomula that
More informationChapter 17 The Kepler Problem: Planetary Mechanics and the Bohr Atom
Chapte 7 The Keple Poblem: Planetay Mechanics and the Boh Atom Keple s Laws: Each planet moves in an ellipse with the sun at one focus. The adius vecto fom the sun to a planet sweeps out equal aeas in
More informationESCAPE VELOCITY EXAMPLES
ESCAPE VELOCITY EXAMPLES 1. Escape velocity is the speed that an object needs to be taveling to beak fee of planet o moon's gavity and ente obit. Fo example, a spacecaft leaving the suface of Eath needs
More informationExperiment MF Magnetic Force
Expeiment MF Magnetic Foce Intoduction The magnetic foce on a cuentcaying conducto is basic to evey electic moto  tuning the hands of electic watches and clocks, tanspoting tape in Walkmans, stating
More informationCharges, Coulomb s Law, and Electric Fields
Q&E 1 Chages, Coulomb s Law, and Electic ields Some expeimental facts: Expeimental fact 1: Electic chage comes in two types, which we call (+) and ( ). An atom consists of a heavy (+) chaged nucleus suounded
More informationPhysics 202, Lecture 4. Gauss s Law: Review
Physics 202, Lectue 4 Today s Topics Review: Gauss s Law Electic Potential (Ch. 25Pat I) Electic Potential Enegy and Electic Potential Electic Potential and Electic Field Next Tuesday: Electic Potential
More informationIn this section we shall look at the motion of a projectile MOTION IN FIELDS 9.1 PROJECTILE MOTION PROJECTILE MOTION
MOTION IN FIELDS MOTION IN FIELDS 9 9. Pojectile motion 9. Gavitational field, potential and enegy 9.3 Electic field, potential and enegy 9. PROJECTILE MOTION 9.. State the independence of the vetical
More information7 Circular Motion. 71 Centripetal Acceleration and Force. Period, Frequency, and Speed. Vocabulary
7 Cicula Motion 71 Centipetal Acceleation and Foce Peiod, Fequency, and Speed Vocabulay Vocabulay Peiod: he time it takes fo one full otation o evolution of an object. Fequency: he numbe of otations o
More informationThe Electric Potential, Electric Potential Energy and Energy Conservation. V = U/q 0. V = U/q 0 = W/q 0 1V [Volt] =1 Nm/C
Geneal Physics  PH Winte 6 Bjoen Seipel The Electic Potential, Electic Potential Enegy and Enegy Consevation Electic Potential Enegy U is the enegy of a chaged object in an extenal electic field (Unit
More informationChapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6
Chapte 9 lectic Chages, Foces, an Fiels 6 9. One in a million (0 ) ogen molecules in a containe has lost an electon. We assume that the lost electons have been emove fom the gas altogethe. Fin the numbe
More informationIntroduction to Electric Potential
Univesiti Teknologi MARA Fakulti Sains Gunaan Intoduction to Electic Potential : A Physical Science Activity Name: HP: Lab # 3: The goal of today s activity is fo you to exploe and descibe the electic
More informationPhysics HSC Course Stage 6. Space. Part 1: Earth s gravitational field
Physics HSC Couse Stage 6 Space Pat 1: Eath s gavitational field Contents Intoduction... Weight... 4 The value of g... 7 Measuing g...8 Vaiations in g...11 Calculating g and W...13 You weight on othe
More information2. An asteroid revolves around the Sun with a mean orbital radius twice that of Earth s. Predict the period of the asteroid in Earth years.
CHAPTR 7 Gavitation Pactice Poblems 7.1 Planetay Motion and Gavitation pages 171 178 page 174 1. If Ganymede, one of Jupite s moons, has a peiod of days, how many units ae thee in its obital adius? Use
More informationAP Physics Electromagnetic Wrap Up
AP Physics Electomagnetic Wap Up Hee ae the gloious equations fo this wondeful section. F qsin This is the equation fo the magnetic foce acting on a moing chaged paticle in a magnetic field. The angle
More information1.1 KINEMATIC RELATIONSHIPS
1.1 KINEMATIC RELATIONSHIPS Thoughout the Advanced Highe Physics couse calculus techniques will be used. These techniques ae vey poweful and knowledge of integation and diffeentiation will allow a deepe
More informationPhysics: Electromagnetism Spring PROBLEM SET 6 Solutions
Physics: Electomagnetism Sping 7 Physics: Electomagnetism Sping 7 PROBEM SET 6 Solutions Electostatic Enegy Basics: Wolfson and Pasachoff h 6 Poblem 7 p 679 Thee ae si diffeent pais of equal chages and
More informationExperiment 6: Centripetal Force
Name Section Date Intoduction Expeiment 6: Centipetal oce This expeiment is concened with the foce necessay to keep an object moving in a constant cicula path. Accoding to Newton s fist law of motion thee
More informationMon., 3/9 Tues., 3/10 Wed., 3/11 Thurs., 3/12 Fri., 3/ 13. RE19 HW19:RQ.42, 49, 52; P.61, 66, 69 RE20, Exp new RE ,34 Magnetic Force
Mon., 3/9 Tues., 3/10 Wed., 3/11 Thus., 3/12 Fi., 3/ 13 Mon., 3/16 Tues., 3/17 Wed., 3/18 Thus., 3/19 Fi., 3/20 20.1,34 Magnetic Foce 20.2,5 Cuent and Motional Emf Quiz Ch 19, Lab 8 Cycloton & Electon
More informationrotation  Conservation of mechanical energy for rotation  Angular momentum  Conservation of angular momentum
Final Exam Duing class (13:55 pm) on 6/7, Mon Room: 41 FMH (classoom) Bing scientific calculatos No smat phone calculatos l ae allowed. Exam coves eveything leaned in this couse. Review session: Thusday
More informationFluids Lecture 15 Notes
Fluids Lectue 15 Notes 1. Unifom flow, Souces, Sinks, Doublets Reading: Andeson 3.9 3.12 Unifom Flow Definition A unifom flow consists of a velocit field whee V = uî + vĵ is a constant. In 2D, this velocit
More informationChapter 13. VectorValued Functions and Motion in Space 13.6. Velocity and Acceleration in Polar Coordinates
13.6 Velocity and Acceleation in Pola Coodinates 1 Chapte 13. VectoValued Functions and Motion in Space 13.6. Velocity and Acceleation in Pola Coodinates Definition. When a paticle P(, θ) moves along
More informationThe Grating Spectrometer and Atomic Spectra
PHY 19 Gating Spectomete 1 The Gating Spectomete and Atomic Specta Intoduction In the pevious expeiment diffaction and intefeence wee discussed and at the end a diffaction gating was intoduced. In this
More informationUnit Vectors. the unit vector rˆ. Thus, in the case at hand, 5.00 rˆ, means 5.00 m/s at 36.0.
Unit Vectos What is pobabl the most common mistake involving unit vectos is simpl leaving thei hats off. While leaving the hat off a unit vecto is a nast communication eo in its own ight, it also leads
More information4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first nonzero digit to
. Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate
More informationAlgebra and Trig. I. A point is a location or position that has no size or dimension.
Algeba and Tig. I 4.1 Angles and Radian Measues A Point A A B Line AB AB A point is a location o position that has no size o dimension. A line extends indefinitely in both diections and contains an infinite
More informationXIIth PHYSICS (C2, G2, C, G) Solution
XIIth PHYSICS (C, G, C, G) 6 Solution. A 5 W, 0 V bulb and a 00 W, 0 V bulb ae connected in paallel acoss a 0 V line nly 00 watt bulb will fuse nly 5 watt bulb will fuse Both bulbs will fuse None of
More information2. Orbital dynamics and tides
2. Obital dynamics and tides 2.1 The twobody poblem This efes to the mutual gavitational inteaction of two bodies. An exact mathematical solution is possible and staightfowad. In the case that one body
More informationSection 53 Angles and Their Measure
5 5 TRIGONOMETRIC FUNCTIONS Section 5 Angles and Thei Measue Angles Degees and Radian Measue Fom Degees to Radians and Vice Vesa In this section, we intoduce the idea of angle and two measues of angles,
More informationUNIT CIRCLE TRIGONOMETRY
UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + =   
More information10. Collisions. Before During After
10. Collisions Use conseation of momentum and enegy and the cente of mass to undestand collisions between two objects. Duing a collision, two o moe objects exet a foce on one anothe fo a shot time: F(t)
More informationThe Critical Angle and Percent Efficiency of Parabolic Solar Cookers
The Citical Angle and Pecent Eiciency o Paabolic Sola Cookes Aiel Chen Abstact: The paabola is commonly used as the cuve o sola cookes because o its ability to elect incoming light with an incoming angle
More informationTheory and measurement
Gavity: Theoy and measuement Reading: Today: p11  Theoy of gavity Use two of Newton s laws: 1) Univesal law of gavitation: ) Second law of motion: Gm1m F = F = mg We can combine them to obtain the gavitational
More informationReview of Vectors. Appendix A A.1 DESCRIBING THE 3D WORLD: VECTORS. 3D Coordinates. Basic Properties of Vectors: Magnitude and Direction.
Appendi A Review of Vectos This appendi is a summa of the mathematical aspects of vectos used in electicit and magnetism. Fo a moe detailed intoduction to vectos, see Chapte 1. A.1 DESCRIBING THE 3D WORLD:
More informationOrbital Motion & Gravity
Astonomy: Planetay Motion 1 Obital Motion D. Bill Pezzaglia A. Galileo & Fee Fall Obital Motion & Gavity B. Obits C. Newton s Laws Updated: 013Ma05 D. Einstein A. Galileo & Fee Fall 3 1. Pojectile Motion
More informationChapter 6. GraduallyVaried Flow in Open Channels
Chapte 6 GaduallyVaied Flow in Open Channels 6.. Intoduction A stea nonunifom flow in a pismatic channel with gadual changes in its watesuface elevation is named as gaduallyvaied flow (GVF). The backwate
More informationMultiple choice questions [70 points]
Multiple choice questions [70 points] Answe all of the following questions. Read each question caefull. Fill the coect bubble on ou scanton sheet. Each question has exactl one coect answe. All questions
More informationMechanics 1: Work, Power and Kinetic Energy
Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).
More informationK.S.E.E.B., Malleshwaram, Bangalore SSLC MathematicsModel Question Paper1 (2015) Regular Private Candidates (New Syllabus)
K.S.E.E.B., Malleshwaam, Bangaloe SSLC MathematicsModel Question Pape1 (015) Regula Pivate Candidates (New Syllabus) Max Maks: 100 No. of Questions: 50 Time: 3 Hous Code No. : Fou altenatives ae given
More informationReview Module: Dot Product
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics 801 Fall 2009 Review Module: Dot Poduct We shall intoduce a vecto opeation, called the dot poduct o scala poduct that takes any two vectos and
More information6.2 Orbits and Kepler s Laws
Eath satellite in unstable obit 6. Obits and Keple s Laws satellite in stable obit Figue 1 Compaing stable and unstable obits of an atificial satellite. If a satellite is fa enough fom Eath s suface that
More informationGeostrophic balance. John Marshall, Alan Plumb and Lodovica Illari. March 4, 2003
Geostophic balance John Mashall, Alan Plumb and Lodovica Illai Mach 4, 2003 Abstact We descibe the theoy of Geostophic Balance, deive key equations and discuss associated physical balances. 1 1 Geostophic
More information