Please see the question prompt below:
3. A 4 kg cylinder with a diameter o 30 cm is attached to a spring (k 300 N/m) at point A e 38 cm The cylinder is released from rest in the equilibrium position shown. The cylinder rolls without slipping. Determine the frequency fn of the resulting vibration. Assume small oscillations about the equilibrium position 10
Solution
moment of inertia I = mr^2/2
moment of inertia about the point in contact with the incline = 3mr^2/2
let us displace the cylinder by a small angle theta about the contact point.
extension in spring = e*theta
restoring force = k*e*theta
restoring torque about contact point O = k*e*theta*(r+e)
I*alpha = -k*e*(r+e)*theta
(3mr^2/2) d^2(theta)/dt^2 + ke(r+e)theta = 0
so omega = (2*ke(r+e)/3mr^2)^1/2
f n = (2*ke(r+e)/3mr^2)^1/2/ (2*pi)
f n = 1.0177 s
.
Please see the question prompt below- 3- A 4 kg cylinder with a diamet.docx
1. Please see the question prompt below:
3. A 4 kg cylinder with a diameter o 30 cm is attached to a spring (k 300 N/m) at point A e 38
cm The cylinder is released from rest in the equilibrium position shown. The cylinder rolls
without slipping. Determine the frequency fn of the resulting vibration. Assume small
oscillations about the equilibrium position 10
Solution
moment of inertia I = mr^2/2
moment of inertia about the point in contact with the incline = 3mr^2/2
let us displace the cylinder by a small angle theta about the contact point.
extension in spring = e*theta
restoring force = k*e*theta
restoring torque about contact point O = k*e*theta*(r+e)
I*alpha = -k*e*(r+e)*theta
(3mr^2/2) d^2(theta)/dt^2 + ke(r+e)theta = 0
so omega = (2*ke(r+e)/3mr^2)^1/2
f n = (2*ke(r+e)/3mr^2)^1/2/ (2*pi)
f n = 1.0177 s
