Consider again the demonstration gyro from Example 11.13. Instead of the small mass having a fixed position, suppose it is connected to a spring of spring constant k and unstretched length ℓ0 that is attached to the post.
a. How many degrees of freedom does the system have?
b. Find the equations of motion for the entire system.
c. Suppose the gyro is spinning but otherwise balanced and an impulse is applied to the small mass. What is the total angular momentum before and after the impulse?
d. Let the mass of the gyro be 5 kg, the small mass be 2 kg, and set l = 0.5 m and l = 0.25 m. The gyro is spinning at a rate of 1 rad/s, and has the moment of inertia of a thin circular disk with a radius of 10 cm. The spring has a constant of 0.1 N/m and a rest length of zero. Numerically integrate the equations of motion after an impulse is applied (you can choose the size of the impulse). Qualitatively describe the resulting motion. Remember that the system is at rest before the impulse is applied when choosing your initial conditions.
e. Suppose a damper is added in addition to the spring. What is the steadystate configuration of the system?