Lab 6: Centripetal Force

Transcription

1 Description Lab 6: Centripetal Force In this lab, you will learn what forces are involved in moving a massive object along a curved path. You will learn about the relationships between force, centripetal acceleration, and instantaneous velocity. Equipment Centripetal Force Apparatus Pulley String Masses Mass Hanger Bubble Level Introduction When a body moves along a curved path, the direction of its velocity changes. For the purposes of this experiment, we will be considering motion in a circle, at constant velocity, called uniform circular motion. Note that when the term constant velocity is used, it refers to a constant magnitude velocity, NOT constant direction. Even though the magnitude remains the same, the direction cannot, because themotion is inacircle. (Whatwould the motionbe ifthe directionwasconstant?) Consider a car driving around a circular path, as depicted in Figure 1. In order for the car to maintain the circular motion, its velocity vector must always be tangential to the circle. In addition, its acceleration vector must always point radially inward toward the center of the circle; thus, the velocity and acceleration vectors are always perpendicular to one another (this is only true for uniform circular motion). Because the acceleration is pointed toward the center of the circle, it is called centripetal acceleration, and its magnitude is expressed mathematically as: a c = v2 (1) r where the subscript c is used to denote centripetal acceleration, and r is the radius of the circular path. We can also express the magnitude of the velocity in terms of the distance traveled in some amount of time. We can easily express the distance of the circumference of the circle as 2πr. The amount of time it takes to travel one complete path around the circle is called the period of revolution, T. Thus, we can say: v = x x = 2πr (2) T Rearranging these equations gives: a c = 4π2 r T 2 (3) 41

2 Figure 1: Uniform circular motion. The force, acceleration, and velocity vectors are labeled at four points around the circle. Because the car is experiencing an acceleration to keep it moving in a circle, Newton s Second Law tells us that some external force or forces must be acting on the car to maintain the circular motion such that the vector sum of the forces is a vector pointing radially inward toward the center of the circle. (Why? What if the force is not pointed this direction?) We know that force can be expressed as a mass times an acceleration, and in this case, our acceleration is the centripetal acceleration; therefore, we can define a centripetal force as follows: F c = ma c (4) We can substitute the equation for centripetal acceleration to give an expression for the centripetal force: F c = m4π2 r T 2 (5) This is the equation you will be verifying in this experiment by studying the effects of varying the rotating mass, the radius of rotation, and the centripetal force on an object moving in a circular path. Procedure The apparatus you will be using in this experiment is depicted in Figures 2 and 2. It looks complicated and has many parts, so be sure that you are careful while working through this procedure. Before performing the experiment, it is important that the rotational platform is level or the mass and spring will oscillate as the platform rotates. To level the base, do the following: 1. Attach or move the 300-gram counter mass to either end of the aluminum track. Tighten the screw so the mass will not slide. Remove the brass mass if present. 42

3 Figure 2: The circular motion apparatus. 2. Rotate the track so that it is parallel to one side of the A-base and adjust the leveling screws until the track is level as measured using a bubble level sitting on the track. 3. Rotate the track again so that it is parallel to the other side of the A-base and adjust the leveling screws until it is level. Part 1: Varying the Radius 1. Measure the mass, M, and record it in your lab book. Replace it to its original position, hanging from the side post, and connect another string from M to the spring on the center post, making sure that the string runs under the pulley at the center post. 2. Attach the clamp on pulley to the end of the track nearer the mass M, as shown in Figure 2. Attach a string to M and run it over the pulley. 3. Measure the mass of the mass hanger and record it in your lab book. Choose a calibration mass, m. Hang it on the mass hanger and record the total mass in your lab book. Approximately 50 [g] will work fine. Calculate the weight of this mass, mg, which is equal to the centripetal force, and record it in your lab book. This will remain constant for this part of the experiment. By hanging a force on the mass when it is stationary, you know how much the spring will stretch under that force. Once you begin spinning the apparatus, when the pink ring lines up with the marker, you know that the force from centripetal acceleration is equal to the force you applied during calibration. 4. Create a data table with columns: Trial, Radius [m], Time of 10 Revolutions [s], Period (T) [s], T 2 [s 2 ]. Your table will need space for 5 trials. 43

4 Figure 3: A detailed view of the circular motion apparatus. 5. Select a radius by aligning the side post with any desired position on the scale on the track. The radius is the distance between the center post and the side post. Record this distance in your data table. 6. The mass, M, on the side bracket must hang vertically. On the center post, adjust the spring bracket vertically until the string from which M hangs on the side post is aligned with the vertical line on the side post. 7. Align the indicator bracket on the center post with the orange indicator. 8. Remove the mass hanging over the pulley and the pulley itself. 9. Rotate the apparatus, increasing the speed until the orange indicator is centered in the indicator bracket on the center post. When this occurs, the string supporting the mass, M, is once again vertical, and M is at the desired radius. 10. You will need to manually maintain the rotational speed which keeps the orange indicator in the bracket. As you are doing so, use a stopwatch to time ten revolutions. Divide this time by ten to obtain the period of one revolution, and then square it. Record these values in your data table. 44

5 11. Move the side post to a new radius and repeat steps 5-10 four more times, for a total of five radii. 12. Make a graph with radius on the vertical axis and the square of the period on the horizontal axis. Rearranging equation (4) gives: F c r = ( 4π 2 M )T2 (6) which is in the form y = mx, the equation of a line, where x in this case is T 2. Find the line of best fit from your graph, and calculate its slope. Note that the slope of the line you calculate should equal the quantity in parenthesis. From this, calculate the centripetal force. Calculate a percent difference between this centripetal force and the centripetal force you set in the beginning with the weights of the mass and hanger. (What should it be?) Record these values. Part 1: Varying the Force 1. In this part, you will repeat the steps from Part 1, but instead of varying in the radius each time, you will be varying how much force the centripetal acceleration puts on the mass. Record the mass M, mass of the hanger by itself, and the radius in your lab book. These will all be constant. 2. Create a new data table with columns: Trial, Mass on hanger [kg], Total Mass [kg], Total Weight [N], Centripetal Force F c [N], Time of 10 Revolutions [s], Period T [s], T 2, and 1. T 2 Again, you will need space for 5 trials. 3. To vary the centripetal force, clamp the pulley to the track again and hang a different mass over the pulley. Record this new mass and its weight in your data table. Remember to keep the radius constant. Repeat the procedure from Part I four more times for a total of five different forces. 4. Make a graph with the centripetal force on the vertical axis and the inverse square of the period on the horizontal axis. From equation (4), we have: F c = (4πMr)( 1 T2) (7) which is in the form y = mx, the equation of a line. Again, find the line of best fit from your graph, and calculate its slope. Note that the slope you calculate should equal the first quantity in parenthesis in the above equation. From this, calculate M. Calculate a percent difference between this value of M and the value you measured it to be in Part I. (What should it be?) Record these values. Part 1: Varying the Mass 1. In this part, you will repeat the experiment again, but the force and radius will be held constant and the mass M will be varied. Record the total mass of the hanger and the masses on the hanger, the total calibration weight, and the radius. These will all be constant. 45

6 2. Create a table with columns: Trial, Mass M [kg], Time of 10 Revolutions [s], Period T [s], T 2, Centripetal Force F c [N], and Percent difference in forces. You will need space for 3 trials in this part. 3. Vary the mass, M, by removing the side masses. Repeat the procedure above for a total of three different masses. 4. Calculate the centripetal force for each trial and record it in your data table. 5. Calculate the percent difference between the centripetal force calculated in step 3 and the Total Weight (Fc), mg, and record it in your data table. (What should it be?) Analysis 1. Why don t we draw a graph for Part III? Could we? 2. What direction does the centripetal acceleration vector point in uniform circular motion? 3. If an object is spun around in a circle on the end of a string and suddenly released, in what direction will the object move compared to its original path (negating the effects of gravity)? 4. If an object travels in a circle with a radius of 50 [m] at a velocity of 15 [m/s], what is the centripetal acceleration acting on the object? How does this compare with gravity on the surface of the earth? If the object has a mass of 42 [kg], what is the centripetal force acting upon the object? 46