Pendulum Force and Centripetal Acceleration 1 Objectives 1. To calibrate and use a force probe and motion detector. 2. To understand centripetal acceleration. 3. To solve force problems involving centripetal acceleration. 2 Introduction A swinging pendulum follows a path that is circular in shape. In order to maintain this path, an unbalanced force must be acting towards the center of the path. The size of this set force is mv 2 /r where m is the mass of the pendulum bob (assuming a massless string), v is the speed and r is the radius. A pendulum bob has two primary forces acting on it. One is the constant downward force of gravity, mg. The second is the tension, which changes its size and direction as the pendulum swings. When it swings through the bottom of its arc, the pendulum has maximum speed and requires a maximum tension force to hold it in its circular path. 2.1 Forces at the bottom of the swing Let s first analyze the forces at the bottom of the swing. 1. Draw a force diagram below for the pendulum bob at the instant in time when it is at the bottom of its swing, clearly labeling each force. r v 1
2. What is the acceleration of the pendulum bob when it is at the bottom of the swing, in terms of v and r? Since acceleration is a vector, make sure to indicate the direction too. 3. Use Newton s second law to solve for the tension in the string, which is what the force probe will measure. 2.2 Forces at the peak of the swing Now let s analyze the forces at the peak of the swing, when it is turning around so its velocity is zero. 1. Again draw a force diagram below for the pendulum bob at the instant in time when its velocity is zero, clearly labeling each force. r x θ y 2. Resolve your vectors into components parallel and perpendicular to the direction of the string. 3. Using Newton s second law, solve for the tension in the string, which is what the force 2
probe will measure. 4. Also solve for the acceleration of the string. Since acceleration is a vector, make sure to indicate the direction too. Make sure to check your answers with your instructor before moving on. 3 Experimental Procedure 3.1 Measuring Tension at the bottom of the swing pulley Force Sensor r Motion Detector Pendulum Bob Set up the physical arrangement shown above. Position a motion detector as shown to measure the speed of the pendulum bob throughout its swing. The tension force of the support string is transmitted to the force sensor by the pulley and can be recorded by the interface and the cpu. 3
1. Open up Logger Pro 3.4.6. Select Experiment/Set up Sensors/Labpro1 from the menu. Then click on DIG/SONIC 1/Select sensor/motion Detector. You should now see three graphs displayed. Select for the wide beam on the motion detector. 2. You will need to calibrate the force probe. Select Experiment/Calibrate/CH1: Dual Range Force Probe from the menu. First remove the string from the force probe and record that as 0 N. Then hang a 500 g mass on the string going over the pulley and record that force (make sure it is not swinging). Also record that value below. 3. Next place the pendulum bob back on the force probe and pulley. 4. Record both the length of the string and the mass of the pendulum. You should be able to measure the mass of the pendulum with the force probe (make sure it is not swinging). Record below. 5. Swing the pendulum and record a run of 3-4 complete swings of the pendulum. Use a small angle (< 30 ) to start your swing. You must keep the plane of the swing in the line of sight of the motion detector. If it precesses too much you will need to redo it until you get a nice clean run. It takes a little patience to get the motion detector to pick up the full swing of the pendulum. Make sure the detector is always at least 15 cm away from the pendulum. 4
Draw the Force versus time plot below (only draw a few complete swings of the pendulum): Force (N) time (s) Draw the corresponding position versus time plot below: Position (m) time (s) Draw the corresponding velocity versus time plot below: Velocity (m/s) time (s) 5
At what position in the swing does the pendulum bob achieve its maximum velocity (max/min/zero)? What is the size of the force there (max/min/zero)? At what position in the swing does the pendulum bob achieve its minimum velocity (max/min/zero)? What is the size of the force there (max/min/zero)? At what position in the swing does the pendulum bob achieve zero velocity (max/min/zero)? What is the size of the force there (max/min/zero)? 5. Click and drag over the clean portion of your velocity versus time graph and then obtain statistics (click on the stat icon). From that data, record the maximum speed of the pendulum bob. 6. Click and drag over the corresponding portion of the force versus time graph and then obtain statistics. Record the maximum force acting on the pendulum bob during its swing. 7. From the maximum speed you can compute the tension in the string. Use T bot = mv2 max r + mg (1) 6
which should be the formula you derived in part 2.1. 8. Change the variables of the experiment and repeat at least two more times. Some variables might be (a) length of the string, (b) mass of the pendulum bob (don t use the wood or cork as they are too light). 9. Compare the value you calculated from (1) to the maximum force you measured by calculating the percent difference. How well does the theory fit your experimental data? Run 1 2 3 Pendulum Mass Pendulum Length Maximum Speed Maximum Force Calculated Tension Percent Difference 3.2 Measuring Tension at the peak of the swing 1. Place a heavy bob on the pulley. 2. This time pull the string back by 30 (measure with a protractor) and release it. Take data again for about three-four complete swings, making sure to keep the plane of the swing in the line of sight of the motion detector. 3. Perform statistics on the force data, this time finding the minimum force. Where is the pendulum (max/min/zero) when the tension is a minimum? 7
4. Switch the velocity graph to acceleration and use statistics to find the maximum acceleration in the horizontal direction. Where is the pendulum (max/min/zero) when the acceleration is a maximum? 5. Compute the expected value for the tension in the string when bob is at its peak swing: T pk = mg cos θ (2) which should be the formula you derived in part 2.2. How well does the theory fit your experimental data? Minimum Force Calculated Tension Percent Difference 8