Vectors & Newton's Laws I

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1 Physics 6 Vectors & Newton's Laws I Introduction In this laboratory you will eplore a few aspects of Newton s Laws ug a force table in Part I and in Part II, force sensors and DataStudio. By establishing equilibriu between three forces on the force table in Part I, you will gain a better understanding of vectors and vector addition. In Part II, you will observe Newton's Third Law by connecting two force sensors to an elastic band and easuring the force on each at varying distances. eference Young and Freedan, University Physics, th Edition: Chapter, section.7-.9, Chapter 4 sections 4., 4., 4.5 Theory Part I: Forces are vector quantities and in order to be added, they ust be broken into their separate coponents. In this eperient, you will work in two diensions and resolve the and y coponents of each force relative to an assigned ais. There are two ajor ethods for adding vectors to find the su, or resultant vector. A graphical approach involves drawing the vectors by hand on a graph, where length is proportional to the agnitude of the vector and the angle of the vector describes its orientation or direction in space. In ug the "tail to tip" or parallelogra ethod to add vectors it is iportant that the orientation/angle of each vector be aintained and also that the agnitude is carefully easured ug a ruler to ensure that the vectors are the appropriate relative lengths. The length of the resultant vector can be converted back to agnitude and the angle easured as is to deterine the orientation of the vector. Graphical ethods have liited accuracy and therefore an analytical ethod is usually preferable. Analytical vector addition is outlined in your tet. For this eperient, the agnitude of each vector will correspond to the weights on the ends of the strings. The su of the forces eerted by each ass should be 0 according to Newton's Second Law: F a because the net acceleration is equal to 0. The force on eerted by each ass is g but as a short cut in this eperient, we can consider the ass () on each string, equal to the force because the value of g will be the sae for each and can be factored out of the equation. However, it is iportant to realize that it is the WEIGHT, or the FOCE that deterines the vector, not the ass.

2 Figure Forces are said to be in equilibriu when the acceleration of the object is zero. Therefore, when three asses balance with the knot where they are joined over the center point of the force table, the net force on the knot for both and y coponents is 0 (N). The asses give the agnitude, and the angle of each string gives the orientation of each vector. The agnitudes of the coponents of each vector can be found by ultiplying the weight by either the e or coe of its angle and the su can be found according the following equations: F y g g g 0 F g g g 0 For the purposes of the eperient, two asses will be assigned to specific angles, a third ass will then be adjusted and placed at such an angle as to eactly counter the other two asses, establishing equilibriu. This third ass will NOT be the resultant vector. It will have the sae agnitude as the resultant vector but it is oriented 80 in the opposite direction. Analytically, the resultant vectors of the and y coponents can be found with the following equations: y 4 In each case, the angles are defined with a coordinate syste where the +-ais is at 0 on the force table. These equations nicely illustrate the relationship between the third ass and the resultant vector. In this eperient, because "g" appears in all of the ters, it is canceled out and the agnitudes of each coponent ay be found ug just the ass and the e or coe of the

3 angle. This is how the table you create will be calculated. The resultant vector,, can be found by ug the and y coponent vector sus and the Pythagorean Theore. y 5 Part II: Newton's Third Law states that any force eerted on an object generates a reaction force equal and opposite in agnitude to the applied force. This can be seen very clearly in the case of two force sensors connected by a rubber band. The rubber band is present to enable sooth easureents of the two forces. When the sensors are pulled apart, the force reading on both should be equal and in opposite directions. Procedure Part I: You will vary the asses and the angles on two strings of the syste, each tie finding the vector that counters the resultant vector of the first two strings.. To begin, set one of the strings to 0 and the other to -5 by sliding the bases of the pulleys to where the center notch at the base lines up with the desired angle.. On each of the first two strings, place a 50 gra ass fro your set on the plastic hooks. The ass of each hook is about 5 gras--be sure to add this to the hanging ass to get the total ass of each vector.. With the third string, adjust the angle of the pulley and the ass on the hook until the knot which joins all three strings is above the center of the hole in the force table. This ay take soe practice, but if you are observant of how changing the angle and the ass affect the location of the knot, you will becoe ore efficient with each trial. Pay close attention to all of the strings to be sure that they are still on the tracks of the pulleys--they tend to slip out and get caught in the ale of the pulley, which distorts the forces on the knot and therefore the accuracy of your results. 4. When you are sure that all of the strings are correctly lined up with their pulleys and that the knot is at the center of the hole--this is very iportant to ensure the accuracy of your results-record the ass on the third string and the angle into a table in Ecel like this one: y y (g) (deg) (g) (deg) (g) (deg) (g) (g) (g) (g)

4 5. The table should include values for each ass and its corresponding angle. In the table, Calculate the and y coponents of the resultant vector and copare with the and y coponents of vector. In Ecel, angles ust be in radians, however the function 'radians()' will convert degrees into radians. 6. To fill rows and in the above table, repeat steps -4 with a 50 gra ass and a 00 gra ass as the starting asses--pay attention to the angle of each so that you do not confuse the later. 7. To fill rows, and 4 in the above table, repeat steps -4 ug angles of -0 and 00 with a trial of two 50 gra asses followed by a trial with a 50 gra and a 00 gra ass. You will have four trials total in your spreadsheet. 8. To get a feel for the graphical ethod of adding vectors, take a protractor teplate sheet (see last page) and with a ruler, draw the two vectors for the first trial. You ay also want to draw and y aes as an aid. To siulate the agnitude of each, draw for every gra of ass on the pulley. Pay attention to the angle of each with respect to the origin (efer to Figure ). Figure 9. To deterine the agnitude and angle of the resultant vector, use the "tail to tip" ethod, placing the tail of the second vector on the tip of the first. Use the ruler to help you draw a straight line. Measure the length of the resultant vector and record it net to the vector. Make sure that the translated vector is parallel to the original vector that it represents. Draw the resultant vector fro the origin to the "tip" of the vector that was added. 4

5 0. To find the vector you deterined eperientally in Steps -4, use the ruler to etend the resultant vector in the opposite direction by eactly the length of the resultant vector. The length in illieters can be converted to the agnitude to give the ass of the resultant and equilibrant and the angles of both ay be read fro the teplate.. epeat Steps 8-0 for the second trial (second row in your table).. Create a second table to calculate the values of the resultant vector based on the asses and angles of the first two asses. First, use Equations -4 as a reference to deterine the agnitudes of the and y coponents of each resultant vector. List the and y coponents in separate coluns. y y (g) (º) (g) (º) (g) (g) (g) (g) (g) (g) (º) In the last colun, use the Pythagorean Theore (see Equation 5 if you need a reference) to calculate the overall agnitude of each of resultant vector,, fro the and y coponents. (eeber, the resultant,, should have the sae agnitude as the third ass you found eperientally.) 4. In the net colun, calculate the angles of the resultant vectors by taking the inverse tangent (tan ) of the y and coponents. Make sure you know which (y or ) goes in the nuerator and denoinator of the epression. See the appendi of your tet if you need a review of trigonoetry rules. 5. The theoretical angle of the vector which counters the resultant (which you found eperientally) can be found by adding 80 to the angles of the resultant vectors. Copare these to the eperiental values you obtained in Steps -4. Additionally, copare these values to the two values obtained in the graphical ethod (Steps 8-). 5

6 Part II: Newton's Third Law - a quick illustration. Figure. To set up the eperient plug in two force sensors into the Science Workshop interface bo. In DataStudio, drag the Force Sensor icons to the appropriate inputs and doubleclick on the Force Sensor icons and set the saple rate to 5 Hz. epeat this process for the other force sensor. It should be noted that the force sensor gives a negative reading when you pull on it, so epect negative values.. Create a graph that plots Force Sensor vs. tie. Add Force Sensor to the graph by dragging the icon fro the Data window to the -ais. Now the -ais should display Force, Ch B (N) and the y-ais should say Force, Ch A (N).. Connect the force sensors to each other by a rubber band. Press the start button and without stretching the rubber band press the Tare button on each force sensor when there is no force on the. This establishes the "zero" of the force. It should be noted that when you pull on a force sensor it gives a negative answer, so both forces will be negative. 4. Vary the tension in the rubber band for about 0 seconds while taking data and then press the Stop button. Place an appropriate trendline in the plot. What type of a trendline best fits the data? Why? 6. Eaine the graph. Are the forces easured by each sensor equal in agnitude at all ties? Is there any evidence of a tie delay between the two sensors? Do your results confir Newton s Third Law? For your Lab eport Write out a saple calculation of the steps you took in Ecel to calculate the first row of the first table. Assue there is a ± 5 g uncertainty for the asses (σ = 5g) and º uncertainty for the angles (σ θ = º). Ug the data in the first row of the second table, calculate the values of σ, σ y and σ. How any siga(s) is away fro? Based on this value was your eperient successful? Eplain. Copare the result (both agnitude and angle) of the graphical ethod (step 9) to the (step 4) of the first trial. Is the result close to each other? Discuss. 6

7 Protractor teplate sheet: (for Steps 8. Make etra sheets copies of these pages as needed). 7

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