A gyroscope is precessing about a vertical axis. Describe what happens to the precession angular speed if the following changes in the variables are made, with all other variables remaining the same: (a) the angular speed of the spinning flywheel is doubled; (b) the total weight is doubled; (c) the moment of inertia about the axis of the spinning flywheel is doubled; (d) the distance from the pivot to the center of gravity is doubled. (e) What happens if all four of the variables in parts (a) through (d) are doubled?

Solution 55E Introduction We have to discuss how the precession changes if the given parameter changes. Step 1 The angular velocity of precession is given by Here I is the moment of inertia of the gyroscope about its axis, and is the angular speed of spin, r is the distance or center of mass from the pivot and w is the weight. (a) From the above equation we can see that the precession speed inversely proportional to the angular speed of spin. Hence if the angular speed of spin is doubled the angular speed of precision will be halved. (b) From the equation (1) we can see that the precession is directly proportional to the weight of the gyroscope, hence if the weight of the gyroscope is doubled, the precession will be doubled. (c) The angular speed of precession is inversely proportional to the moment of inertia of the gyroscope about spinning axis. Hence if the moment of inertia about the spinning axis is doubled the rotational speed precession will be halved. (d) The angular speed of the precession is directly proportional to the distance from the pivot to the center of gravity (r). Hence if the distance from the pivot to the center of gravity is doubled the angular speed of the precession will be doubled. (e) Now if all the parameter is doubled then we have Hence if all the parameters are doubled then the angular speed of precession will remain same.