# Torque to speed

1. Dec 4, 2008

### saltine

1. The problem statement, all variables and given/known data
A wheel of mass m is given a torque to roll without slipping. What is the linear acceleration of the axle?

2. Relevant equations
Idisk = mr2/2

3. The attempt at a solution
From Torque to angular acceleration:

α = 2τ/mr2

The unit of α is rad/s2. Each radian correspond to r meters, so the axle is accelerating at a=2τ/mr (m/s2).

Since the whole wheel is moving, the force is ma= 2τ/r at the center of mass.

Since the wheel is rolling without slipping, there is a friction force at the ground. The force against the ground is F = τ/r. This force is the same at the top of the wheel, point tangentially clockwise. The acceleration at the top of the wheel from the turning is a = rα. If this acceleration is combined with the acceleration of the center, then the top is accelerating at 4τ/mr.

The center is accelerating at 2τ/mr.

The bottom is not accelerating.

Is there anything wrong or missing?

How would things change if there is a weight Fw acting on the axle?

- Thanks