Question

Consider a bicycle wheel of mass M and radius R that sits on a flat, level surface, such that the surface is tangent to the wheel. One end of a spring (spring constant, k) is attached to bicycle wheel's hub, and the other end is fixed to a vertical wall. The spring is horizontal. There is sufficient friction to prevent the wheel from sliding at the point of contact with the surface. When the center of the wheel is directly over the point of contact with the ground the spring is not stretched. If the spring is initially stretched a small amount and then released the wheel will rock back and forth about the contact point. Use torque ideas to obtain an expression for the angular frequency of this oscillation. Your expression should be in terms of g, M, R and k. Hints: 1. Don't forget to draw a picture and a free body diagram. 2. Remember the rotational inertia of the wheel needs to be calculated about the pivot point, not the center of mass

Answer 1

Fig Step 2 (k.E)_T + (k.E)_R + P.E. = constant = 1/2 Mx^2 + 1/2 I theta^2 + 1/2 kx^2 = constant 1/2 M (R theta)^2 + 1/2 (MR^2) theta^2 + 1/2 k(R theta)^2 = constant step 3 Differentiating above eqn we get MR^2 theta theta + MR^2 theta theta + kR^2 theta theta = 0 = 2 MR^2 theta + kR^2 theta = 0 theta + k/2 M theta = 0 ------(1) This is eqn of SHM w = squareroot k/2 M rad/s

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