Vectors. Physics 115: Spring Introduction

Transcription

1 Vectors Physics 115: Spring 2005 Introduction The purpose of this laboratory is to examine the hypothesis that forces are vectors and obey the law of vector addition and to observe that the vector sum of the forces acting upon an object at rest is zero. Theory In this exercise, forces are used to illustrate vectors because they are easy to control and measure (in particular, the gravitational forces on some calibrated weights). The object on which the forces act is, in this case, a light metal ring. Horizontal forces are applied to the ring by means of weights and pulleys. Using the fact that a vector can always be broken into its x and y components in a given coordinate system, (i.e. A = A x i+ A y j) we can add three vectors acting on the object at rest by finding the sum of individual x and y components separately (see Figure 1). It is also possible to add vectors graphically by placing them tail to head one after the other. The resultant vector is now just the vector from the tail of the first vector to the head of the last vector. Procedure In the three situations on the following pages, you will be adjusting the three different hanging weights and/or directions in which they point (refer to Figure 2) until the ring is stationary and centered around the nail holding it. In order to minimize the unknown frictional forces in the pulleys, take care that your ring is well centered in the table and that the strings pass straight over the pulley grooves. This will also assure that the measurement of the angles on the calibrated table is as accurate as possible. Determine the magnitude of the force and the angle at which it must act in order for the ring to be centered on the table for each of the following experiments. Record your data in the tables given. For each experiment, include a ``Top View'' diagram of the force configuration in your writeup as shown in Figure 1. Be sure to scale your drawings to indicate the magnitudes as well as the angles of your vectors. (Remember that the force F is mg, and not just equal to the mass.) Figure 1

2 Experiment 1: Three unequal forces, one along the x-axis and one along the y-axis After placing unequal forces along the x and y-axis, experimentally determine the force vector that will balance the two unequal forces. Fill out the table below: Experiment 1 Angle θ 1 Mass m 1 Angle θ 2 Mass m 2 Angle θ 3 Mass m 3 Force F 3 Note that the angle θ is measured in the conventional manner counterclockwise from the positive x-axis (i.e., θ =0 for those forces along the positive x-axis and θ= 90 for those forces along the positive y-axis.) Find the vector sum of the forces above by the following two methods: I. Direct graphical analysis (i.e., a scale drawing as shown in Figure 1). Be sure to label all vectors and explain which vector components add together to make the overall vector sum zero. Draw a second diagram which graphically shows, using the tail to head method, the addition of the vectors. If you put the tail of the first component at the origin, the head of the last component should terminate at the origin (or at least somewhere close to it-see the ensuing error discussion).

3 II. Calculating the x and y components of each force from the recorded magnitudes and angles using trigonometry (as described above). Do the x and y components add to zero? If not, be sure to state what your discrepancy is. III. Use propagation of error to obtain an estimate in the uncertainty of the x and y components of the third vector, the one not lying on the x or y axis.

4 Experiment 2: Three forces, one along the x-axis, and the other two of equal magnitude neither along the x-axis or the y-axis. Create the situation described above and fill out the table below. Experiment 2 Angle θ 1 Mass m 1 Angle θ 2 Mass m 2 Angle θ 3 Mass m 3 Force F 3 Find the vector sum of the forces above by the following two methods: II. Direct graphical analysis (i.e., a scale drawing as shown in Figure 1). Be sure to label all vectors and explain which vector components add together to make the overall vector sum zero. Draw a second diagram which graphically shows, using the tail to head method, that the vector s all add up to zero. III. Calculating the x and y components of each force from the recorded magnitudes and angles using trigonometry (as described above). Do the x and y components add to zero? If not, be sure to state what your discrepancy is.

5 Experiment 3: Three unequal forces in arbitrary directions, with none of the forces along either the x-axis or the y-axis. In this experiment, you will place two unequal forces in arbitrary directions, and then theoretically calculate the magnitude and direction of the third force which will cancel the first two forces. Experiment 2 Angle θ 1 Mass m 1 Angle θ 2 Mass m 2 Calculate the x and y components of these forces and show them in the table below. Show your work. Remember to be careful about your signs. F 1x F 1y F 2x F 2y Mathematically calculate the magnitude and direction of the third vector. Show your work. If you are stuck, drawing a diagram may help.

6 Once you have determined a theoretical value for the third vector, try it out! Does it work? Try adjusting only the force. Record the force change needed. If changing the force alone does not work, try adjusting both the angle and the force. What is the final experimental value for the third vector and how does it compare to the theoretical value? Scalar Dot Product Given the definition of the scalar (dot) product, A B= A B cosθ (1) where θ is the angle between vectors A and B, it is possible to find the dot product of any two of your vectors. Another method of finding the dot product will be used as well and involves using individual x and y components. This method uses the following formula: A B = (A x i + A y j) (B x i + B y j) = A x B x + A y B y (2) Now, for the three arbitrary forces in Experiment 3, calculate the scalar (dot) product of any two of the force vectors using: I. Magnitudes and angles. Show your work below. (See Equation 1 above). II. x and y components. Show your work below. (See Equation 2 above).

7 Do both methods result in the same answer? Prove it by showing mathematically that the two equations are identical. (A note of caution, θ in equation 1 always refers to the smallest angle between the two vectors.) Conclusion: Is the hypothesis stated in the introduction confirmed? Why or why not? One last question: In determining the theoretical vector in experiment 3, was this the only vector that would work? That is, while keeping F 1 and F 2 the same (same weight and same angles as they had in experiment 3), would another combination of a different weight and different angle for F 3 work? Why or why not?