Chapter 6 - Dynamics of Uniform Circular Motion w./ QuickCheck Questions

Chapter 6 - Dynamics of Uniform Circular Motion w./ QuickCheck Questions 2015 Pearson Education, Inc. Anastasia Ierides Department of Physics and Astronomy University of New Mexico September 24, 2015

Review of Last Time Uniform Circular Motion (UCM) definitions: period, frequency, speed, acceleration, force Centripetal acceleration/force; constant speed, but not constant velocity means acceleration; net force due to other forces, e.g., friction, tension Minimum speed to complete circle; maximum speed for walking Centrifuges

QuickCheck Question 3.19 A car is traveling around a curve at a steady 45 mph. Is the car accelerating? A. Yes B. No

QuickCheck Question 3.20 A car is traveling around a curve at a steady 45 mph. Is the car accelerating? A. Yes B. No

QuickCheck Question 3.19 A car is traveling around a curve at a steady 45 mph. Which vector shows the direction of the car s acceleration? A. B. C. D. E. The acceleration is zero.

QuickCheck Question 3.21 A toy car moves around a circular track at constant speed. It suddenly doubles its speed a change of a factor of 2. As a result, the centripetal acceleration changes by a factor of A. 1/4 B. 1/2 C. No change since the radius doesn t change. D. 2 E. 4

QuickCheck Question 3.21 A toy car moves around a circular track at constant speed. It suddenly doubles its speed a change of a factor of 2. As a result, the centripetal acceleration changes by a factor of A. 1/4 B. 1/2 C. No change since the radius doesn t change. D. 2 E. 4 ai = vi 2 /r af = vf 2 /r = (2vi) 2 /r = 4 (vi 2 /r) = 4ai

Velocity & Acceleration Speed is constant Velocity (magnitude and direction which changes) is not constant Acceleration (magnitude and direction which changes) is not constant

QuickCheck Question 6.2 A ball at the end of a string is being swung in a horizontal circle. The ball is accelerating because A. The speed is changing. B. The direction is changing. C. The speed and the direction are changing. D. The ball is not accelerating.

QuickCheck Question 6.3 A ball at the end of a string is being swung in a horizontal circle. What is the direction of the acceleration of the ball? A. Tangent to the circle, in the direction of the ball s motion B. Toward the center of the circle

Dynamics of UCM An object of mass m moving in a circle of radius r at a speed v exhibits a net force This is NOT a new kind of force

QuickCheck Question 6.4 A ball at the end of a string is being swung in a horizontal circle. What force is producing the centripetal acceleration of the ball? A. Gravity B. Air resistance C. Normal force D. Tension in the string

QuickCheck Question 6.5 A ball at the end of a string is being swung in a horizontal circle. What is the direction of the net force on the ball? A. Tangent to the circle B. Toward the center of the circle C. There is no net force.

QuickCheck Question 6.7 A coin is rotating on a turntable; it moves without sliding. At the instant shown in the figure, which arrow gives the direction of the coin s velocity?

QuickCheck Question 6.7 A coin is rotating on a turntable; it moves without sliding. At the instant shown in the figure, which arrow gives the direction of the coin s velocity? A

QuickCheck Question 6.8 A coin is rotating on a turntable; it moves without sliding. At the instant shown in the figure, which arrow gives the direction of the frictional force on the coin?

QuickCheck Question 6.8 A coin is rotating on a turntable; it moves without sliding. At the instant shown in the figure, which arrow gives the direction of the frictional force on the coin? D

QuickCheck Question 6.9 A coin is rotating on a turntable; it moves without sliding. At the instant shown, suppose the frictional force disappeared. In what direction would the coin move?

QuickCheck Question 6.9 A coin is rotating on a turntable; it moves without sliding. At the instant shown, suppose the frictional force disappeared. In what direction would the coin move? A

Maximum Walking Speed As you walk, your body is in circular motion, pivoting on the forward foot

Maximum Walking Speed Using Newton s 2nd law and setting the normal force, n = 0, we find that the weight is Leads to equation of maximum speed

QuickCheck Question 6.10 A physics textbook swings back and forth as a pendulum. Which is the correct freebody diagram when the book is at the bottom and moving to the right?

QuickCheck Question 6.10 A physics textbook swings back and forth as a pendulum. Which is the correct freebody diagram when the book is at the bottom and moving to the right? Centripetal acceleration requires an upward force. C.

QuickCheck Question 6.11 A car that s out of gas coasts over the top of a hill at a steady 20 m/s. Assume air resistance is negligible. Which free-body diagram describes the car at this instant?

QuickCheck Question 6.11 A car that s out of gas coasts over the top of a hill at a steady 20 m/s. Assume air resistance is negligible. Which free-body diagram describes the car at this instant? Now the centripetal acceleration points down. A.

QuickCheck Question 6.12 A roller coaster car does a loop-theloop. Which of the free-body diagrams shows the forces on the car at the top of the loop? Rolling friction can be neglected.

QuickCheck Question 6.12 A roller coaster car does a loop-theloop. Which of the free-body diagrams shows the forces on the car at the top of the loop? Rolling friction can be neglected. Remember the roller coaster The track is above the car, so the normal force of the track pushes down. E.

Centrifuges Centrifuges are used to separate liquids Heavier ones flow to the outside Lighter ones remain at the top of the tube

QuickCheck Question 6.13 A coin sits on a turntable as the table steadily rotates counterclockwise. What force or forces act in the plane of the turntable?

QuickCheck Question 6.13 A coin sits on a turntable as the table steadily rotates counterclockwise. What force or forces act in the plane of the turntable? A.

QuickCheck Question 6.14 A coin sits on a turntable as the table steadily rotates counterclockwise. The freebody diagrams below show the coin from behind, moving away from you. Which is the correct diagram?

QuickCheck Question 6.14 A coin sits on a turntable as the table steadily rotates counterclockwise. The freebody diagrams below show the coin from behind, moving away from you. Which is the correct diagram? Net force points to the center which is to the left of the coin

QuickCheck Question 6.14 A coin sits on a turntable as the table steadily rotates counterclockwise. The freebody diagrams below show the coin from behind, moving away from you. Which is the correct diagram? Net force points to the center (the left) which is to the left of the coin C.

QuickCheck Question 6.15 A car turns a corner on a banked road. Which of the diagrams could be the car s free-body diagram?

QuickCheck Question 6.15 A car turns a corner on a banked road. Which of the diagrams could be the car s free-body diagram? Net force points to the center of circle Road creates static friction down ramp n n n n w w w E.

Orbital Motion - Satellites

Orbital Motion - Satellites A projectile with high enough velocity that the curve of its trajectory is parallel to the curve of the earth has a closed trajectory

Orbital Motion - Satellites A projectile with high enough velocity that the curve of its trajectory is parallel to the curve of the earth has a closed trajectory An orbit is a closed trajectory

Orbital Motion - Satellites A projectile with high enough velocity that the curve of its trajectory is parallel to the curve of the earth has a closed trajectory An orbit is a closed trajectory An orbiting projectile is in constant free fall

Orbital Motion Force of gravity, or weight, causes orbital motion, i.e., centripetal acceleration

Orbital Motion Force of gravity, or weight, causes orbital motion, i.e., centripetal acceleration The speed of the object in orbit, vorbit, can then be determined from the centripetal acceleration equation as

Orbital Motion Force of gravity, or weight, causes orbital motion, i.e., centripetal acceleration The speed of the object in orbit, vorbit, can then be determined from the centripetal acceleration equation as The object travels parallel to the earth s surface

Orbital Motion Orbital speed, vorbit, of a projectile skimming the surface of the earth of radius, Re, ignoring air resistance,

Orbital Motion Orbital speed, vorbit, of a projectile skimming the surface of the earth of radius, Re, ignoring air resistance, The period, or time it takes for an object in orbit with speed, vorbit, to complete one revolution at radius, r, is

Weightlessness in Orbit Astronauts, space shuttles, and satellites are in constant free fall around the earth/other planetary objects

Weightlessness in Orbit Astronauts, space shuttles, and satellites are in constant free fall around the earth/other planetary objects No contact forces means wapp = 0 N

QuickCheck Question 6.19 Astronauts on the International Space Station are weightless because A. There s no gravity in outer space. B. The net force on them is zero. C. The centrifugal force balances the gravitational force. D. g is very small, although not zero. E. They are in free fall.

The Orbit of the Moon The moon is a natural satellite

The Orbit of the Moon The moon is a natural satellite It is in free fall or falling around the earth

The Orbit of the Moon The moon is a natural satellite It is in free fall or falling around the earth Using the distance of the moon from the earth of r = 3.84 10 8 m, the period is found to be

The Orbit of the Moon The moon is a natural satellite It is in free fall or falling around the earth Using the distance of the moon from the earth of r = 3.84 10 8 m, the period is found to be about 11 hours, not 1 month (actual)

Gravity - A Universal Law Gravity is a universal force that affects all objects in the universe

Gravity - A Universal Law Gravity is a universal force that affects all objects in the universe The value of gravitational acceleration decreases with increasing distance from the earth

Gravity - Inverse-Square Law Newton s Law of Gravity (2 Rules)

Gravity - Inverse-Square Law Newton s Law of Gravity (2 Rules) 1. The force is inversely proportional to the square of the distance between the objects.

Gravity - Inverse-Square Law Newton s Law of Gravity (2 Rules) 1. The force is inversely proportional to the square of the distance between the objects. 2. The force is directly proportional to the product of the masses of the two objects.

Gravity - Inverse-Square Law Newton s Law of Gravity

Gravity - Inverse-Square Law Newton s Law of Gravity It can be seen from the inverse square law, that an increase in the distance between two objects decreases the gravitational pull between them

Gravity - Inverse-Square Law Newton s Law of Gravity It can be seen from the inverse square law, that an increase in the distance between two objects decreases the gravitational pull between them; doubling the distance, decreases it by 4

Example 6.12: Gravitational force between two objects You are seated in your physics class next to another student 0.60 m away. Estimate the magnitude of the gravitational force between you. Assume that you each have a mass of 65 kg.

Example 6.12: Gravitational force between two objects You are seated in your physics class next to another student 0.60 m away. Estimate the magnitude of the gravitational force between you. Assume that you each have a mass of 65 kg. PREPARE We will model each of you as a sphere. This is not a particularly good model, but it will do for making an estimate. We will take the 0.60 m as the distance between your centers.

Example 6.12: Gravitational force between two objects SOLVE The gravitational force is given by Equation 6.15:

Example 6.12: Gravitational force between two objects SOLVE The gravitational force is given by Equation 6.15: ASSESS The force is quite small, roughly the weight of one hair on your head. This seems reasonable; you don t normally sense this attractive force!

QuickCheck Question 6.16 The force of Planet Y on Planet X is the magnitude of FX on Y. A. One quarter B. One half C. The same as D. Twice E. Four times 2M Planet X M Planet Y

QuickCheck Question 6.16 The force of Planet Y on Planet X is the magnitude of FX on Y. A. One quarter B. One half C. The same as D. Twice E. Four times 2M Planet X M Planet Y Remember:

QuickCheck Question 6.16 The force of Planet Y on Planet X is the magnitude of FX on Y. A. One quarter B. One half C. The same as D. Twice E. Four times 2M Planet X Newton s third law M Planet Y Remember:

QuickCheck Question 6.17 The gravitational force between two asteroids is 1,000,000 N. What will the force be if the distance between the asteroids is doubled? A. 250,000 N B. 500,000 N C. 1,000,000 N D. 2,000,000 N E. 4,000,000 N

QuickCheck Question 6.17 The gravitational force between two asteroids is 1,000,000 N. What will the force be if the distance between the asteroids is doubled? A. 250,000 N B. 500,000 N C. 1,000,000 N rnew = 2 rold D. 2,000,000 N E. 4,000,000 N Remember:

QuickCheck Question 6.17 The gravitational force between two asteroids is 1,000,000 N. What will the force be if the distance between the asteroids is doubled? A. 250,000 N B. 500,000 N C. 1,000,000 N D. 2,000,000 N rnew = 2 rold Fold = G m1m2/rold 2 E. 4,000,000 N Remember:

QuickCheck Question 6.17 The gravitational force between two asteroids is 1,000,000 N. What will the force be if the distance between the asteroids is doubled? A. 250,000 N B. 500,000 N C. 1,000,000 N D. 2,000,000 N E. 4,000,000 N rnew = 2 rold Fold = G m1m2/rold 2 Fnew = G m1m2/rnew 2 Remember:

QuickCheck Question 6.17 The gravitational force between two asteroids is 1,000,000 N. What will the force be if the distance between the asteroids is doubled? A. 250,000 N B. 500,000 N C. 1,000,000 N D. 2,000,000 N E. 4,000,000 N rnew = 2 rold Fold = G m1m2/rold 2 Fnew = G m1m2/rnew 2 = G m1m2/(2 rold) 2 Remember:

QuickCheck Question 6.17 The gravitational force between two asteroids is 1,000,000 N. What will the force be if the distance between the asteroids is doubled? A. 250,000 N B. 500,000 N C. 1,000,000 N D. 2,000,000 N E. 4,000,000 N Remember: rnew = 2 rold Fold = G m1m2/rold 2 Fnew = G m1m2/rnew 2 = G m1m2/(2 rold) 2 = G m1m2/(4 rold 2 ) = 1/4 Fold

Gravity on Other Worlds Your mass, m, is always the same,

Gravity on Other Worlds Your mass, m, is always the same, but your weight, w, would change according the mass, Mobject, and radius, robject, of the planetary object you are on

Gravity on Other Worlds Your mass, m, is always the same, but your weight, w, would change according the mass, Mobject, and radius, robject, of the planetary object you are on Using Newton s law of gravity the weight is

Gravity on Other Worlds Your mass, m, is always the same, but your weight, w, would change according the mass, Mobject, and radius, robject, of the planetary object you are on Using Newton s law of gravity the weight is w = m gobject = GMobject m r 2

Gravity on Other Worlds Your mass, m, is always the same, but your weight, w, would change according the mass, Mobject, and radius, robject, of the planetary object you are on Using Newton s law of gravity the weight is w = m gobject = GMobject m r 2 The gravitational acceleration on that object is then

Gravity on Other Worlds Your mass, m, is always the same, but your weight, w, would change according the mass, Mobject, and radius, robject, of the planetary object you are on Using Newton s law of gravity the weight is w = m gobject = GMobject m r 2 The gravitational acceleration on that object is then gobject = GMobject r 2

Gravity on Other Worlds Using the mass, Mmoon, and radius, rmoon, of the moon, we find gmoon = 1.62 m/s 2

Gravity on Other Worlds Using the mass, Mmoon, and radius, rmoon, of the moon, we find gmoon = 1.62 m/s 2 A 70-kg astronaut wearing an 80-kg spacesuit would weigh more than 330 lb on the earth but only 54 lb on the moon.

QuickCheck Question 6.18 Planet X has free-fall acceleration 8 m/s 2 at the surface. Planet Y has twice the mass and twice the radius of planet X. On Planet Y A. g = 2 m/s 2 B. g = 4 m/s 2 C. g = 8 m/s 2 D. g = 16 m/s 2 E. g = 32 m/s 2

QuickCheck Question 6.18 Planet X has free-fall acceleration 8 m/s 2 at the surface. Planet Y has twice the mass and twice the radius of planet X. On Planet Y A. g = 2 m/s 2 B. g = 4 m/s 2 C. g = 8 m/s 2 D. g = 16 m/s 2 E. g = 32 m/s 2 gx = 8 m/s 2

QuickCheck Question 6.18 Planet X has free-fall acceleration 8 m/s 2 at the surface. Planet Y has twice the mass and twice the radius of planet X. On Planet Y A. g = 2 m/s 2 B. g = 4 m/s 2 C. g = 8 m/s 2 D. g = 16 m/s 2 E. g = 32 m/s 2 gx = 8 m/s 2 = GM X rx 2 gy = GM Y ry 2

QuickCheck Question 6.18 Planet X has free-fall acceleration 8 m/s 2 at the surface. Planet Y has twice the mass and twice the radius of planet X. On Planet Y A. g = 2 m/s 2 B. g = 4 m/s 2 C. g = 8 m/s 2 D. g = 16 m/s 2 E. g = 32 m/s 2 gx = 8 m/s 2 = GM X rx 2 GMY gy = = ry 2 G(2MX) (2rX) 2

QuickCheck Question 6.18 Planet X has free-fall acceleration 8 m/s 2 at the surface. Planet Y has twice the mass and twice the radius of planet X. On Planet Y A. g = 2 m/s 2 B. g = 4 m/s 2 C. g = 8 m/s 2 D. g = 16 m/s 2 E. g = 32 m/s 2 gx = 8 m/s 2 = GM X rx 2 GMY gy = = = ry 2 2GMX 4rX 2 G(2MX) (2rX) 2

QuickCheck Question 6.18 Planet X has free-fall acceleration 8 m/s 2 at the surface. Planet Y has twice the mass and twice the radius of planet X. On Planet Y A. g = 2 m/s 2 B. g = 4 m/s 2 C. g = 8 m/s 2 D. g = 16 m/s 2 E. g = 32 m/s 2 gx = 8 m/s 2 = GM X rx 2 GMY gy = = ry 2 2GMX = = 4rX 2 G(2MX) (2rX) 2 1 GMX 2 rx 2 = 1/2 gx

QuickCheck Question 6.18 Planet X has free-fall acceleration 8 m/s 2 at the surface. Planet Y has twice the mass and twice the radius of planet X. On Planet Y A. g = 2 m/s 2 B. g = 4 m/s 2 C. g = 8 m/s 2 D. g = 16 m/s 2 E. g = 32 m/s 2 gx = 8 m/s 2 = GM X rx 2 GMY gy = = ry 2 2GMX = = 4rX 2 G(2MX) (2rX) 2 1 GMX 2 rx 2 = 1/2 gx = 4 m/s 2

QuickCheck Question 6.22 A 60-kg person stands on each of the following planets. On which planet is his or her weight the greatest?

QuickCheck Question 6.22 A 60-kg person stands on each of the following planets. On which planet is his or her weight the greatest? Remember: w = m gobject = GMplanet m rplanet 2 Mplanet rplanet 2

QuickCheck Question 6.22 A 60-kg person stands on each of the following planets. On which planet is his or her weight the greatest? M R 2 2M (2R) 2 3M (3R) 2 Remember: w = m gobject = GMplanet m rplanet 2 Mplanet rplanet 2

QuickCheck Question 6.22 A 60-kg person stands on each of the following planets. On which planet is his or her weight the greatest? M R 2 M 4R 2 M 3R 2 Remember: w = m gobject = GMplanet m rplanet 2 Mplanet rplanet 2

QuickCheck Question 6.22 A 60-kg person stands on each of the following planets. On which planet is his or her weight the greatest? A M R 2 M 4R 2 M 3R 2 Remember: w = m gobject = GMplanet m rplanet 2 Mplanet rplanet 2

Example 6.14: Find the speed to orbit Deimos Mars has two moons, each much smaller than the earth s moon. The smaller of these two bodies, Deimos, isn t quite spherical, but we can model it as a sphere of radius 6.3 km. Its mass is 1.8 10 15 kg. At what speed would a projectile move in a very low orbit around Deimos?

Example 6.14: Find the speed to orbit Deimos Mars has two moons, each much smaller than the earth s moon. The smaller of these two bodies, Deimos, isn t quite spherical, but we can model it as a sphere of radius 6.3 km. Its mass is 1.8 10 15 kg. At what speed would a projectile move in a very low orbit around Deimos? SOLVE The free-fall acceleration at the surface of Deimos is small:

Example 6.14: Find the speed to orbit Deimos Given this, we can use Equation 6.13 to calculate the orbital speed:

Example 6.14: Find the speed to orbit Deimos Given this, we can use Equation 6.13 to calculate the orbital speed: ASSESS This is quite slow. With a good jump, you could easily launch yourself into an orbit around Deimos!

Gravity and Orbits An orbit is a closed trajectory of a projectile

Gravity and Orbits An orbit is a closed trajectory of a projectile An object in circular orbit exhibits a net centripetal force

Gravity and Orbits The speed of the object in circular orbit is then

QuickCheck Question 6.20 Two satellites have circular orbits with the same radius. Which has a higher speed? A. The one with more mass. B. The one with less mass. C. They have the same speed.

QuickCheck Question 6.21 Two identical satellites have different circular orbits. Which has a higher speed? A. The one in the larger orbit B. The one in the smaller orbit C. They have the same speed.

QuickCheck Question 6.21 Two identical satellites have different circular orbits. Which has a higher speed? A. The one in the larger orbit B. The one in the smaller orbit C. They have the same speed. Remember:

Gravity and Orbits For a satellite in orbit at radius, r, with speed, v, the period, T, is the time it takes to complete one full orbit:

Gravity and Orbits For a satellite in orbit at radius, r, with speed, v, the period, T, is the time it takes to complete one full orbit: T = 2πr/v

Gravity and Orbits For a satellite in orbit at radius, r, with speed, v, the period, T, is the time it takes to complete one full orbit: T = 2πr/v Plugging in the equation for the speed of an object in circular obit, yields the period of a satellite

QuickCheck Question 6.23 A satellite orbits the earth. A Space Shuttle crew is sent to boost the satellite into a higher orbit. Which of these quantities increases? A. Speed B. Angular speed C. Period D. Centripetal acceleration E. Gravitational force of the earth

QuickCheck Question 6.23 A satellite orbits the earth. A Space Shuttle crew is sent to boost the satellite into a higher orbit. Which of these quantities increases? Remember: A. Speed B. Angular speed C. Period D. Centripetal acceleration E. Gravitational force of the earth

Example 6.15: Locating a Geostationary satellite Communication satellites appear to hover over one point on the earth s equator. A satellite that appears to remain stationary as the earth rotates is said to be in a geostationary orbit. What is the radius of the orbit of such a satellite?

Example 6.15: Locating a Geostationary satellite Communication satellites appear to hover over one point on the earth s equator. A satellite that appears to remain stationary as the earth rotates is said to be in a geostationary orbit. What is the radius of the orbit of such a satellite? PREPARE For the satellite to remain stationary with respect to the earth, the satellite s orbital period must be 24 hours; in seconds this is T = 8.64 10 4 s.

Example 6.15: Locating a Geostationary satellite SOLVE We solve for the radius of the orbit by rearranging the equation for the period, T 2

Example 6.15: Locating a Geostationary satellite SOLVE We solve for the radius of the orbit by rearranging the equation for the period, T 2 The mass at the center of the orbit is the earth:

Example 6.15: Locating a Geostationary satellite ASSESS This is a high orbit, and the radius is about 7 times the radius of the earth. Recall that the radius of the International Space Station s orbit is only about 5% larger than that of the earth.

Gravity on a Grand Scale No matter the distance, stars and other objects in space are attracted to one another via gravity

Gravity on a Grand Scale No matter the distance, stars and other objects in space are attracted to one another via gravity Galaxies are held together by gravity

Gravity on a Grand Scale No matter the distance, stars and other objects in space are attracted to one another via gravity Galaxies are held together by gravity Stars vary in distance from the galactic center and orbit with different periods

Summary

Things that are due Reading Quiz #7 Due September 29, 2015 by 4:59 pm Homework #5 Due September 30, 2015 by 11:59 pm

QUESTIONS?