Gyroscopes, moment of inertia, rotational energy and friction.

[j-e_c]

Homework Statement

A gyroscope consists of a solid disk of diameter 1m and mass 100kg mounted on an axis passing through the center of mass of the disk.

(a) Calculate the moment of inertia of the disk about its rotational axis.
(b) What is the kinetic energy of rotation of the disk if its angular velocity is 75 revolutions per second.
(c) A brake is pressed against the edge of the disk to arrest its motion. The frictional force which the brake exerts on the disk is 100N. How long will it take to come to rest?

Homework Equations

N/A.

The Attempt at a Solution

(a) I=mR² = 0.5x0.5x100 = 25kgm²
(b) K=0.5I$$\omega$$² = 0.5x25x75x75 = 70312.5J
(c) Which I'm not sure about:

E=F.d (force x distance)
d=s.t (speed x time)
So E=F.s.t

The initial speed = s₁ = $$\omega$$R = 75x0.5 = 37.5m/s
The final speed = s₂ = 0m/s
So the average speed = 18.75m/s

t= E/(F.s) = 70312.5/(100x18.75) = 37.5s

 

[Doc Al]

(a) I=mR² = 0.5x0.5x100 = 25kgm²

The rotational inertia of a disk is not mR².

(b) K=0.5I$$\omega$$² = 0.5x25x75x75 = 70312.5J

ω is measured in radians per second.

(c) Which I'm not sure about:

E=F.d (force x distance)
d=s.t (speed x time)
So E=F.s.t

The initial speed = s₁ = $$\omega$$R = 75x0.5 = 37.5m/s
The final speed = s₂ = 0m/s
So the average speed = 18.75m/s

t= E/(F.s) = 70312.5/(100x18.75) = 37.5s

Once you correct your answers to the first two parts, this method will work. You can also calculate the torque and then find the rotational acceleration using Newton's 2nd law for rotation.