Page 1 of 9 The Torsion Pendulum Introduction: This experiment helps to relate many of the concepts that we see in everyday life. Damped oscillations and pendulums are an everyday occurrence. You will have a chance to explore the pendulum behaviour that you have explored in class using the Science Workshop. The torsional pendulum consists of torsion wire attached to a rotary motion sensor with an object that will be mounted on it for the later exercises. Theory: The torque: τ = -қθ (1) where қ is the torsional spring constant and depends on the properties of the wire. It is essentially a measure of the amount of Torque required to rotate the wire 1 radian. Relating T (the period of oscillation and I (the rotational inertia): I = T 2 қ (2) 2π The rotational Inertia of a disk: I = 1MR 2 (3) 2 The rotational inertia of a cylinder is: I = 1M(R12 + R22 ) (4) 2 where R1 is the inner radius of the cylinder and R2 is the outer radius. The rotational inertia of a point mass rotating in a circle of radius R is given by: I = MR 2 (5) The rotational inertial of a thin rod of length L and mass M: I = 1 ML 2 (6) 12 The relationship between the torsional spring constant and the diameter of the wire қ = πgd 4 (7) 32 l where d is the diameter of the wire and l is the length of the wire. G is the shear modulus. For more on this relationship, see Reference (3).
Page 2 of 9 The general equation of a torsion pendulum is described by the following equation: 0 where no additional damping has been added to the system. I, c and қ are all constants that will be calculated throughout the course of this experiment. қ is calculated in exercise 1 and I for the disk is calculated in exercise 2. The solution to the above differential equation is: where α = c/2i OR α = β -1 This allows us to calculate c (damping coefficient) and β (amount of time to decay by 1/e). In this case, β is being used for the time constant instead of the more commonly used τ. This is because τ is already being used for torque in this experiment and using another letter such as β avoids unnecessary confusion. Another important formula is α = w0/2q, where Q -1 is the internal friction. The internal friction works against applied force to counter deformation. Materials: Wires of 3 different diameters (0.032, 0.047 and 0.063 ) Rotary Motion Sensor (RMS) Connected to Science Workshop Interface Force Sensor Lower Clamp Upper Clamp Disk Cylinder Rod with 2 point masses String (about 70 cm at least) Mass Scale Pliers Micrometer Ruler Preparatory Questions: 1. The torsional spring constant is defined as the amount of torque needed to rotate the wire by 1 radian. What do you think will happen to the torsional spring constant as you use wires with a larger diameter? 2. If a wire has a higher value for its torsional spring constant, is it harder to twist and bend? The Experiment: Obtain approximately 5 runs of data for each exercise.
Page 3 of 9 Exercise 1: Calculating the қ, the Torsional Spring Constant (all 3 wires) 1. This experiment contains wires with 3 different diameters and hence a different thickness. Although it is possible to guess the diameters of each wire based on the thickness, it is recommended that you use the micrometer to calculate the diameters of each wire. http://www.upscale.utoronto.ca/pvb/harrison/micrometer/micrometer.html 2. Use the pliers to bend each end of the wire into an L shape (about 90deg to the wire). Do this to all 3 wires. Try to keep the lengths of all 3 wires approximately the same. For example: if the first wire was 14 inches long after bending the ends then the other two wires should be 14 inches as well. 3. If the support rod is not already clamped to the table then clamp it to the table and make sure that it is tight and stable. 4. Now clamp the lower clamp near the bottom of the rod but above the table. Then clamp the Rotary Motion Sensor near the top of the rod. Make sure the pulley on the Rotary Motion Sensor is at the top and the shaft at the bottom. (Hint: The distance between the 2 should not be more than 18 inches long as the wires are 18 inches long and their ends have been bent and they will be used to attach the rotary motion sensor to the lower clamp. Of course, we will adjust the distance between them later as needed). Make sure that both the Rotary Motion Sensor and Lower Clamp are parallel. 5. Now slide the upper clamp onto the shaft of the Rotary Motion Sensor. Make sure that the guide mark on the upper clamp aligns with the shaft. 6. Take the wire with the 0.032 diameter and screw it to both the upper and lower clamps. Adjust the distance between the rotary motion sensor and the lower as needed to make sure that the wire is tight
Page 4 of 9 and fits perfectly into the grooves of the screws. 7. Now take the string and tie it around the small pulley on the RMS and wrap around the medium pulley about 3 times. (You probably won t need more than 50 cm. Remember to record how much you used). Also remember to record the radius of the medium pulley. After wrapping around the medium pulley, leave about 20 cm of string before cutting of the rest. Make a loop with the string near the end where you just cut off the excess string. This loop will allow you to hook on the Force Sensor. 8. Connect the RMS cables to Channel 1 & 2 on the Science Workshop Interface and the Force Sensor to Channel A. Use a sample rate of 50 Hz and Med(10x) sampling option for Force Sensor. 9. Open Datastudio and make a calculation for Torque. We are using the following formula: τ = r*f where r is the radius of the pulley we are using (in this case the medium pulley) and F is the amount of force exerted, which will be recorded by the Force Sensor. Therefore, r will be recorded as an Experimental Constant and F will be a Data Measurement on Datastudio. 10. Plot a Graph of Torque vs. Angle. Since we know that қ is the amount of torque required to rotate the wire by 1 radian, the slope of the Torque vs. Angle graph is therefore the value for қ. 11. Hook the Force Sensor onto the string loop and while the string is slack. Tare the Force Sensor. Hold the Force sensor parallel to the rotary motion sensor. Start the Datastudio Timer and pull on the force sensor. After about 1 revolution, stop the Data Studio timer. 12. The graph of Torque vs. Angle should be approximately linear. Perform a linear fit on the data and extract the value of the slope. This is the torsional spring constant for the wire with the 0.032 diameter.
Page 5 of 9 13. Perform the above steps using the other 2 wires (0.047 diameter and 0.063 diameter). Obtain their Torsional Spring Constants. Exercise 2: Comparing the Theoretical and Measured Values of the period of oscillation for the disk using the 0.032 wire 1. Remove the Force Sensor and untie the string from the setup from Exercise 1. Re-clamp the wire with the 0.032 wire. 2. Measure and Record the mass and radius of the disk included. Place the disk to top of the pulley and screw it in. 3. Open a New Experiment in Datastudio and plot a graph of Angle vs. Time. Use a sample rate of 200Hz. We are trying to obtain the period of oscillation of the disk. 4. Rotate the disk about ¼ of the way from its equilibrium position. Start the Datastudio timer at the same time as you let go of the disk. Let the disk oscillate for about 3-5seconds and then stop the Datastudio timer. 5. Obtain the period of oscillation through either a sine fit or by using the smart tool. Exercise 3: Comparing the Theoretical and Measured Values of the period of oscillation for the disk + cylinder using the 0.032 wire 1. Measure and Record the mass and radius of the cylinder included. Place the cylinder on to top of the disk. Make sure the screws in the cylinder fit into the grooves on the disk. 2. Open a New Experiment in Datastudio and plot a graph of Angle vs. Time. Use a sample rate of 200Hz. We are trying to obtain the period of oscillation of the disk + cylinder.
Page 6 of 9 3. Rotate the disk + cylinder about ¼ of the way from its equilibrium position. Start the Datastudio timer at the same time as you let go of the disk. Let the disk oscillate for about 3-5seconds and then stop the Datastudio timer. 4. Obtain the period of oscillation through either a sine fit or by using the smart tool. Exercise 4: Comparing the Theoretical and Measured Values of the period of oscillation for the rod with 2 point masses using the 0.032 wire 1. Remove the disk and the cylinder from the RMS. 2. Measure and Record the mass and length of the rod and the mass of each point mass included. Clamp the 2 point masses on the 2 ends of the rod. Invert the 3- step pulley on the RMS. Place the center of the rod near the center of the round pulleys and screw it in. 3. Open a New Experiment in Datastudio and plot a graph of Angle vs. Time. Use a sample rate of 200Hz. We are trying to obtain the period of oscillation for the rod with the 2 point masses. 4. Rotate the rod with the 2 point masses about ¼ of the way from its equilibrium position. Start the Datastudio timer at the same time as you let go of the rod. Let the rod oscillate for about 3-5seconds and then stop the Datastudio timer. 5. Obtain the period of oscillation through either a sine fit or by using the smart tool. Exercise 5: Repeat Exercise 2, 3 and 4 for the wires with 0.047 diameter and 0.063 diameter, (only if time permitting). It is recommended that you do Exercise 6 first. Exercise 6: Observing damping of the torsion wire with a disk mounted on the rotary motion sensor that is allowed to rotate freely. 1. Use the disk and the wire with the thickest diameter (0.063 ) and let them oscillate for about 10-15 s (make sure that it is not completely damped).
Page 7 of 9 2. Extract the period of oscillation by performing a sine fit on the data and record the value. 3. Plot an FFT using the data. An FFT is a graph that plots the relative amplitude against the frequency. You can now determine the resonant frequency f0 from this graph. It is the frequency at the peak. The following picture is an example of an FFT where the placement of the peak on the x-axis yields the resonant frequency. f 0 = 101.5Hz values using the following 2 formulas: Method 1: w0 = 2πf0 from step 3 Method 2: w0 = 2π/T0 from step 2 4. Calculate the w0 5. Calculate the % difference between the 2 angular frequencies obtained above. If the difference is lower than 5%, use the average; otherwise use the more accurate value, which is Method 2 if the peak of the FFT has few points around it. 6. Calculate β which is the amount of time that it takes for the initial value to drop by 1/e or 37%. Again β can be calculated two different ways: Method 1: Using the smart tool on your angular position vs. time, calculate amplitude at t =0. Divide this value by 1/e. Find the new value of amplitude on this angular position vs. time graph. Record the time that this new amplitude occurs at. This is your value of β. Method 2: On the damped oscillations, draw an envelope using the prediction tool (pencil). Envelope: exponential decay Perform a natural exponent fit onto the envelope. Extract the exponent from the fit. This exponent is the inverse of β. Choose 1 of the 2 methods in step 6. Picture is an example of oscillations damping and showing
Page 8 of 9 an envelope. 7. The inverse of β is α. That is β = α -1. Since we know α, this can help us to evaluate many helpful quantities when studying damping. Both the internal friction: Q -1 and the damping coefficient: c can be calculated. Important Formulas: α = c/2i and α = w0/2q. Error Calculations: Some important propagation formulas: F = A ± b F = AB σf = FbσA σf 2 = σa 2 + σb 2 A F A B Remember to record all reading errors. Also remember to record errors when using the mass balance. It s best to use the error calibration provided by the manufacturer. Discussion: 1. Using mass and radius measurement: a) Calculate the Rotational Inertia of the disk? b) Calculate the Rotational Inertia of the disk + cylinder? c) Calculate the Rotational Inertia of the rod + 2 point masses? 2. Using the rotational inertia calculated in Question 1, the torsional spring constant obtained in Exercise 1, and Equation (2); Find the Theoretical Period of Oscillation (T) for: a) The disk? b) The disk + cylinder? c) The rod + 2 point masses? d) Comment on the values obtained for Theoretical vs. Measured Period of Oscillation (from ex 2, 3, 4 and 5). Calculate the % error and highlight any major deviations. 3. What does a graph of I vs. T 2 tell you? Plot the graph and write about any important characteristics. 4. Why do the wires with a larger diameter dampen more quickly when you perform oscillations with the disk, the disk + cylinder and the rod + 2 point masses? 5. Calculate Internal Friction, Q -1, and the damping constant c and explain what role these values played in the damping of the disk that was freely rotating. 6. a) Plot a Graph of қ (torsional spring constant) vs. (d 4 / l) on Datastudio. Perform a linear fit and extract the slope. Use the slope to calculate an average value of G. b) The value of G for steel is 79.3 GPa. Calculate the % error between your measured value of G and the value of G for steel. Comment on the composition of the 3 wires.
Page 9 of 9 References: 1. R.A.Serway and J.W.Jewett, Jr. Physics for Scientists and Engineers 8th ed. (Thomson, 2010). 2. R.D.Knight Physics for Scientists and Engineer with Modern Physics 2nd ed. (Pearson, 2008). 3. Mohazzabi, P., and B. M. Shefchik. "A Universal Relationship between Spring Constant and Torsion Constant." Journal of Physics and Chemistry of Solids 62 (2001): 677-81. Elsevier. Web. 26 May 2011. <http://www.elsevier.nl/locate/jpcs>. 4. Vise, J. B. 2000. Mechanical Oscillations-Resonance and Ringing in a Tuning Fork. University of Toronto, ON. <http://faraday.physics.utoronto.ca/iyearlab/tunfk.pdf>. The handout is written by Ms. Onaizah Onaizah, Summer Undergraduate Research Fellowship student in 2011.