PC1141 Physics I Circular Motion

Transcription

1 PC1141 Physics I Circular Motion 1 Purpose Demonstration the dependence of the period in circular motion on the centripetal force Demonstration the dependence of the period in circular motion on the radius of the circular motion 2 Equipment Centripetal force apparatus Two satellites Various masses and mass hanger Laboratory balance Stopwatch 3 Theory When an object moves in a circle at constant speed, the velocity of the object is always tangent to the circle. This implies that the direction of the velocity is continuously changing. Thus, the object is accelerated because acceleration is, by definition, a change in velocity per unit time. The change in velocity always points toward the center of the circle. This acceleration a is called centripetal acceleration and its magnitude is given by a = v2 where is the radius of the circle. The centripetal acceleration is also pointed toward the center of the circle. According to Newton s second law, a constant force is required to keep an object in circular motion. The force that maintains an object s circular motion is called centripetal force and its magnitude is given by F = Mv2 Page 1 of 5

2 Circular Motion Page 2 of 5 where M is the mass of the object. If the object moves at constant speed v in a circle of radius, the time for one complete revolution around the circle is the period T. The period T is related to the speed v by the expression v = 2π T Figure 1: Equipment Setup The apparatus used in this experiment is schematically shown in Figure 1. A satellite with mass M is tied by a strong thread, over the pulley wheel, to the hanger of mass m 0, so that the hanger is on the axis of rotation. The satellite can be fixed at a distance from the axis of rotation and a motor is used to rotate the disc. The rotating speed of the disc can be varied and a few various masses m can be added on the hanger. When the satellite rotates, the centripetal force is provided by the tension in the thread T and the frictional force f, i.e. F = T + f The tension T is equal to the weight of the hanging masses W if the thread is being tensed. We therefore have F = W + f = mv2 The weight W of the hanging masses is given by the sum of the hanger s weight of mass m 0 and masses weight of mass m, W = (m 0 + m)g where g is the gravitational acceleration. For a given mass M of the satellite and a particular weight of the hanging mass W, the satellite will rotate at a given radius only for one particular rotating period T.

3 Circular Motion Page 3 of 5 4 Experimental Procedure 4.1 Period of otation and Weight of the Hanging Masses 1. Using a laboratory balance, measure the masses of the satellite M and hanger m 0. ecord your data as M and m 0 in the Data Table Screw the collar so that the center of the satellite is 20 cm from the axis of rotation which is marked on the top crossbar of the frame. 3. Tie the satellite with a strong thread over the pulley wheel to the hanger. 4. Adjust the rubber band so that it is at the same level as the bottom of the hanger. 5. Engage the motor and increase the supply voltage to the motor until the disc is rotating at such a speed that the satellite just begins to move outwards along the radius of its orbit. Hint: Observing the gap between the hanger and the rubber band may help in determining when the satellite just begins to slide. 6. Time at least 20 revolutions of the disc FIVE times and record your data as t 1, t 2, t 3, t 4, t 5 in the Data Table epeat the steps 4 6 FIVE times with different masses m added to the hanger. ecord the masses added to the hanger as m in the Data Table 1. Note that the radius of the orbit is kept constant and the rubber band may have to be lowered slightly as the hanging mass is increased. 4.2 Period of otation and adius of Orbit 1. Using a laboratory balance, measure the mass of the smaller satellite M. ecord your data as M in the Data Table Screw the collar so that the center of the satellite is 20 cm from the axis of rotation. 3. Tie the satellite with a strong thread over the pulley wheel to the hanger. Attach THEE masses to the hanger and record the total mass of these masses as m in the Data Table Adjust the rubber band so that it is at the same level as the bottom of the hanger. 5. Engage the motor and increase the supply voltage to the motor until the disc is rotating at such a speed that the satellite just begins to move outwards along the radius of its orbit. Hint: Observing the gap between the hanger and the rubber band may help in determining when the satellite just begins to slide.

4 Circular Motion Page 4 of 5 6. Time at least 20 revolutions of the disc FIVE times and record your data as t 1, t 2, t 3, t 4, t 5 in the Data Table epeat the procedures for FIVE different values of the radius of orbit while keeping the hanging mass m constant. ecord the radius of orbit as in the Data Table 2. Hint: It may be necessary to shorten the length of the thread for small orbits to prevent the hanging mass from behaving like a conical pendulum. 5 Data Analysis 5.1 Period of otation and Weight of the Hanging Masses D1. Enter your data in the Data Table 1 into the Excel spreadsheet. Calculate the weight of the hanging masses W for each masses added to the hanger m in the spreadsheet. D2. Calculate the mean t for the repeated trials of the time taken of at least 20 revolutions for each m in the spreadsheet. D3. Calculate the period of rotation T for each m in the spreadsheet. D4. Using the values of the period T determined, calculate the speed of circular motion v for each m in the spreadsheet. D5. Perform a linear least squares fit to the data, with the square of the speed v 2 as the y-axis and weight of the hanging masses W as the x-axis. Determine the slope and intercept with the corresponding uncertainties of the least squares fit to the data. D6. Use percentage discrepancy to compare the experimental value to the theoretical value of the slope. What does this imply about the accuracy of your results? Hint: The percentage discrepancy is defined as Percentage discrepancy = Experimental value Known value Known value 100% D7. How is the frictional force f involving in the experiment determined from your data? State the best experimental value of the frictional force f. D8. Plot a graph of the square of the speed v 2 against the weight of the hanging masses W in the spreadsheet. Also show on the graph the straight line that was obtained by the linear least squares fit to the data.

5 Circular Motion Page 5 of Period of otation and adius of Orbit D1. Enter your data in the Data Table 2 into the Excel spreadsheet. Calculate the mean t for the repeated trials of the time taken of at least 20 revolutions for each in the spreadsheet. D2. Calculate the period of rotation T for each m in the spreadsheet. D3. Perform a linear least squares fit to the data, with the square of the period T 2 as the y-axis and radius of orbit as the x-axis. Determine the slope and intercept with the corresponding uncertainties of the least squares fit to the data. D4. How is the frictional force f involving in the experiment determined from your data? State the best experimental value of the frictional force f. D5. Plot a graph of the square of the speed T 2 against the radius of the orbit in the spreadsheet. Also show on the graph the straight line that was obtained by the linear least squares fit to the data.