#1. [Gyroscope Wheel] A rubber wheel on a steel rim spins freely on a horizontal axle that is suspended by a fixed Up A pivot at point P. When the wheel spins at a rate of 4.00 rev/s, it precesses smoothly about point P in a horizontal plane with a period of 3.50s. The wheel's outer Down radius is 15.0 cm, and it's total mass is measured to be 1.12 kg, 60% of this being the spinning wheel and 40% its P axle assembly. The wheel and its axle are both symmetric B about the middle of the wheel, and the pivot point P is 4.70 cm from the wheel's middle. (a) Draw an extended free-body diagram for the wheel + its axle considered as a body. (b) What is the direction of the net torque on the wheel about the pivot point P? (c) If the wheel is spinning counterclockwise (O) as seen from point B on the wheel's axle, what is the direction of its spin angular momentum referenced to the diagram? (d) As seen from point A above the pivot, which way is the wheel precessing (O or O)? Please give your reasoning. (e) Using this information, determine the moment of inertia I of the wheel about its axle. Please be sure your work is clear. [HINT: Various unit conversions need to be done before you can do numerical calculations.](f) If the wheel's moment of inertia about its axle is assumed to be of the form I = CMR, where R is the wheel's radius and M is the amount of mass that is spinning, what is the constant C? How does it compare with the range of values you might expect for a wheel whose spinning mass is not concentrated all at its outer rim nor is distributed uniformly between its axle and rim? Please explain. (g) In order to stop the wheel's precession without causing any other precessional motion of the axis up or down, in which direction should a force be applied to point B at the axle's end? Why? How should the magnitude of this force compare with the wheel's weight? Please explain. (h) What is the magnitude of the vertical component of force acting on the axle at point P due to the pivot? (i) Why must there be a horizontal component of force acting on the axle at point P due to the pivot? What is the magnitude of this component