Physics Department Electricity and Magnetism Laboratory CIRCULAR MOTION 1. Aim The aim of this experiment is to study uniform circular motion and uniformly accelerated circular motion. The motion equation for a rotating rigid body will be studied, and a moment of inertia will be calculated.. Overview Circular motion is a curved motion where the path (or trajectory) is a circumference, e.g. the motion of any point on a rotating disk or wheel. On a first approximation, the motion of the Moon around Earth and that of an electron around a proton in a hydrogen atom are circular motions. Due to the Earth's rotation, all bodies on its surface are on circular motion around the axis of rotation of the Earth..1 Uniform circular motion equations. In circular motion with constant speed, the velocity vector, v, is tangential to the circumference (as velocity is always tangential to the path). Distance covered, s, is always measured along the path, which in this case is an arc of circumference. Equations are similar to those of uniform rectilinear motion (constant velocity), but instead of distance covered s there is the angle swept out, and instead of linear velocity v, angular velocity,, (angle swept out per time unit). See Figure 1. v R d dt t [1] 0 0 [3] R is the distance from the body to the axis of rotation. [] In this motion, the velocity vector v has a constant modulus but variable direction and sense. This implies that there is acceleration: normal acceleration v R a N directed towards the axis of rotation. 1
Figure 1.. Uniformly accelerated circular motion equations. When the angular velocity of a particle in circular motion changes over time, angular acceleration is defined as The equations obtained are: d [4] dt 0 [5] t 1 t 0 0 t [6] Aside from [1] and the normal acceleration, now there is also a tangential acceleration, a T R (See Figure ). Figure.
.3 Relationship between moment of force and angular acceleration. Newton's second law of motion indicates the relationship between the forces applied onto a F ma body and the resulting acceleration, [7]. Now we are studying circular motion, and instead of force we speak about moment of force, also called torque. The moment of a force with respect to a certain axis is the product of the force and its distance to the axis, M R F (see Figure 3). This torque does not cause displacement but rotation, by causing angular acceleration. In a similar fashion as [7] but changing force for moment and acceleration for M I angular acceleration, we arrive to, [8], where I is the moment of inertia, which depends on the mass distribution and the geometry of the body, and has dimension ML (in the case of a coin it is 1 mr I ). Figure 3 3
3. Equipment Figure 4. 1. Photogate.. Rod support. 3. Rod. 4. Power source. 5. Motor. 6. Wires. 7. Pulley, fixed to the edge of the table. 8. Weight support. 9. Weights. 10. String. 11. Ruler. 1. Vernier caliper. 4
4. Experimental Procedure 4.1 Uniform circular motion. The experimental set-up is shown in Figure 5. Connect the motor to the power source. Turn on the power source and set a voltage so that the speed of rotation of the rod is as small as possible but stable. Never go beyond 1.5A for the current or the motor might be damaged! Now we are going to measure the linear speed of the rod (at the position of the photogate) and the angular velocity. The procedure will be: Learn how to use the photogate (sec. 4.3). To obtain the linear speed, use the photogate measure t (time that takes for the end of the rod to go through the photogate) three times. To obtain the angular velocity, use the photogate measure t 3 (time between two passes of the rod's ends, which corresponds to a 180º angle swept out) three times. This procedure has to be done for 5 different velocities of the rod. In order to set new velocities values, increase the voltage of the power supply. The velocity values chosen has to be: Different enough so the values t and t 3 change their values on the display. Low enough so the current is never higher than 1.5A. Figure 5. 5
4. Uniformly accelerated circular motion. Prepare the set-up shown in Figure 6. Note that the motor is not required for the set-up. Wind up the string around the rod support so that when dropping the weight you will see the rod start to spin increasingly fast. Ensure that the rod makes several turns before stopping when the string is unwound. r Is the radius of the cylinder where the string is wound up. Set a 10 g weight on the weight support. Now we are going to obtain experimentally the relation between the linear velocity v of the rod position where the photogate is placed and the angle swept. Later we will compare the result with the expected using the theory (see appendix 5.1). We are going to measure the linear velocity for 5 different angular positions /, 3 /, 5 /, 7 / y 9 /. In order to obtain the linear velocity for every one of the angular positions we will measure with the help of the photogate the time t. The linear velocity v is obtained dividing the width of the rod by t. We will follow the next procedure: Select the photogate measure mode t. To measure on / set the rod at a 90º angle from the photogate, as shown in Figure 6. Activate the photogate trigger (set button) and release the rod. Thus, the rod will start moving in uniformly accelerated circular motion. When the end of the rod goes through the photogate, the angle swept out is 90º. In that instant the sensor measures t, and from this, linear velocity v ( / ) can be calculated. To measure on 3 / angle is obtained when the opposite end of the rod goes through the photogate. The set-up of the previous measure is repeated (wind up the stricg around the rod and hold the rod with your hand) but we don t activate the photogate trigger. Release the rod and it will start spinning and, after the first rod end passes through the photogate, you have to activate the trigger. Thus, when the second rod end passes through the gate, a measure of t will be taken when the displacement is 3 /. Repeat the process for the other angles, taking into account that the gate has to be triggered just before the angle we want to measure. For example, if we are measuring on 9 /, the rod has to pass through the gate four times. After the fourth pass you should activate the gate and it will measure the fifth pass (that correspond to the desired angle). Of course, four passes of the rod correspond with two passes of one end of the rod and two passes of the other end of the rod! Obtain one single measurement of each angular position and fill in the table with the values, the measured t, and the computed v. Repeat the procedure using the 0g weight. 6
Figure 6. 4.3. Photogate. Figure 7. Check that the sensor is correctly plugged. To measure t set the switch (bottom-right on the figure) on the second position (next to Count). Measure the time interval in seconds that takes the rod to go through the sensor. 7
Reset the count with the Set button. To measure t 3 set the switch in the third position. The sensor starts counting when one end goes through, and stops when the other end goes through. So, t 3 is the time taken in sweeping a 180º angle. 5. Appendix Relationship between linear velocity and angle swept out in uniformly accelerated circular motion. v wr we can substitute w t. From [6] / Motion starts at rest. In which yields: v Rt t and substituting in we arrive to: Take the logarithm from both sides: v R [9] 1 1 ln v ln( ) ln ln R [10] 5. Relationship between moment of force and angular acceleration. Figure 8. 8
(See Figure 8). From equation [8] and bearing in mind that the only force present is the tension of the string, rt I [11] To obtain the tension of the string, we apply [7] on the weight holder. m g [1] T m a From a r we substitute in [11] and the moment of inertia results: mgr I m The moment of inertia is always positive, I 0 : r [13] r g [14] 9
Relationship between linear velocity and angle swept out in uniformly accelerated circular motion. v wr we can substitute w t. From [6] / Motion starts at rest. In which yields: v Rt t and substituting in we arrive to: Take the logarithm from both sides: v R [9] 1 1 lnv ln( ) ln lnr [10] 5. Relationship between moment of force and angular acceleration. Figure 8. (See Figure 8). From equation [8] and bearing in mind that the only force present is the 10
tension of the string, rt I [11] To obtain the tension of the string, we apply [7] on the weight holder. m g [1] T m a From a r we substitute in [11] and the moment of inertia results: m gr I m The moment of inertia is always positive, I 0 : r [13] r g [14] 11
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