Exp03 Gyro

Experiment 3

Gyroscope Apparatus: Gyroscope, light barrier with counter, barrel base, power supply 5 V DC/2.4 A, Digital stopwatch, support rod with clamp, thread, Slotted weight and hook, meter scale. Purpose of experiment: (1) To determine the moment of inertia of the gyroscope disc by measuring the angular acceleration of the disc. (2) To determine the moment of inertia of the gyroscope disc by measuring the spin and precession frequencies of the gyroscope. Basic methodology: (1) Torque is applied to the gyroscope disc by a falling weight and the resultant spin frequency is measured. (2) Torque is applied to the gyroscope disc by hanging a weight at the long end and the presession frequency is measured as a function of the spin frequency.

counter−weight Gyro disc

notch for hanging weight axle rod support stand

thread drum Light barrier with counter

Figure 1: The gyroscope

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Theory

The gyroscope consists of a uniform heavy disk which is free to spin about its axes and which is pivotted in a frictionless manner such that the axle can assume any orientation. It is a striking illustration of the principles of rigid body rotation and of angular momentum conservation. The gyroscope effect finds applications in many areas such as motion stabilization and navigation. We first attempt to measure the moment of intertia of the gyroscope disk. The gyro. axle is 1

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PHYSICS LABORATORY MANUAL

fixed in the horizontal plane and the disc is made to spin by a falling weight. The weight is tied to a string which is wound around the thread drum, so that as it falls, the gyro. rotates. The weight does not fall freely but with an acceleration a < g. As a result, tension T = m(g − a) is applied off center to the gyro disc. This causes a torque which rotates the disc. The relationship between the applied torque τ , angular acceleration α and the moment of inertia 2r about the spin axes I is, τ = Iα.

(1)

T = m(g − a)

m where the torque is given by τ = T r = m(g − a)r, r being the radius of the thread drum. h The acceleration a of the falling mass m is related to the angular acceleration of the disc by α = a/r. The acceleration a can be found from Figure 2: Setup for measuring moment of the height h from which the weight is released and inertia the time tF of the fall, by the relation a = t2h 2 . SubF stituting for τ , α and a in the first equation gives, t2F =

2h(I + mr2 ) mgr2

(2)

Thus the plot of t2F vs. h is a straight line. From the slope of this straight line one can find the moment of inertia I. Precession of the Gyroscope: If the axle of the spinning gyroscope is not held fixed, and a torque acts on the axle, then the axis rotates in a direction perpendicular to the force causing the torque. This is known as precession. If the gyroscope is initially spinning about a fixed axes in space with angular velocity ωR , the angular momentum is L = IωR . Now a mass m1 is added at the far end of the gyroscope at distance d from the pivot point, resulting in an applied torque, τ = m1 gd. ~ it causes As the applied torque is normal to the direction of the spin angular momentum L, ~ to rotate. If L ~ rotates by angle dφ in time dt then the corresponding change in angular L momentum is dL = Ldφ. Since the applied torque causes the change in angular momentum according to dL dφ =L = Lωp = τ = m1 gd, (3) dt dt where ωp is the precession frequency. For spin time period tR and precession time period tp , we have ωp = 2π/tp and ωR = 2π/tR . Substituting for ωp , ωR and L in the equation for dL/dt we get, m1 gdtp 1 = . (4) tR 4π 2 I Thus 1/tR and tp are linearly related. From the slope of this line the moment of inertia I can be calculated.

Experiment 3. Gyroscope

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Set-up and Procedure

Part A: 1. Fix the axle of the gyro. in the horizontal direction with the help of the support rod and clamp as shown in the figure 2. 2. Position the gyro. so that the thread drum projects over the edge of the table. 3. Pass the loop of the thread over the pin on the thread drum and wind the thread around the drum. At the other end of the thread hang the given weight. Measure the height h of the weight above the floor. 4. Release the weight and measure its time of fall a stopwatch. Start the stopwatch as soon as you release the disc. 5. Repeat for various initial heights. Part B:

Figure 3: The gyroscope 1. Start with a gyro. that is free to rotate about all its axes (i.e, remove the support rod). Stick a small paper strip on the rotating disk in such a way that about 2–4 cm of the strip projects outside the edge. 2. On the forked light barrier, set the counter to the third switch, which measures the time between two consecutive obstructions in the light path. 3. Balance the gyro. horizontally and spin the disk by pulling on a thread wound around the thread drum. You will need to hold on the axle rod to keep the axis horizontal while you pull the thread.

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PHYSICS LABORATORY MANUAL 4. Measure the initial rotation time period tR1 by holding the light barrier in your hand and allowing the paper strip to pass through it twice. 5. Immediately after tR1 measurement, gently hang a weight m1 in the slot at the other end of the axle rod. The gyro. will start precessing. (It may wobble a bit initially. This is called nutation. To reduce this, try to give the disc a large spin and use a small weight m1 .) 6. Measure the time to complete half a precession and calculate the precession time period tp by doubling it. 7. Immediately after tp measurement, gently remove m1 and measure the final time period of rotation tR2 , using the forked barrier as before. Note down the average of tR1 and tR2 , measured before and after precession. 8. Repeat with different initial rotation frequencies.

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Exercises and Viva Questions

~ and τ at various stages of the experiment? 1. What are the directions of L 2. If a clamp is not used in the first part what will happen? 3. What is the advantage of measuring tp /2 instead of tp in part 2? 4. Why does the gyroscope not topple when an additional weight causes a force to act on its axis? 5. What happens to the precession frequency if the weight hung on the axle is doubled? What if the weight is halved? 6. If the direction of spin of the gyroscope is reversed, what happens to the precession? 7. Suppose the Gyroscope is spinning with its axle at an angle θ to the horizontal and is caused to precess. What will be its precession frequency compared to that when it is spinning with the same angular frequency but with horizontal axis? Try it out experimentally! 8. Theoretically derive the formula for precession frequency if a weight w is hung from a gyroscope spinning with frequency ωs with its axis making an angle θ with the horizontal.

References: 1. Mechanics, by Kleppner and Kolenkow , Tata McGraw-Hill (Indian ed.2007). 2. PHYWE Physics experiments manual.