Application of force not acting through center of mass.

[meriadoc]

Hello PF!

I've got a lab on rigid body motion tomorrow, and I need help completing one of the prep questions:

A rigid body is acted on by a force F through its center of mass, and also experiences a torque caused by a similar force F at radius R. At time t, what are the linear and angular velocities of the rigid body? Show that they follow the relationship:

v = $\frac{1}{2}$ ω R

I understand that there is a translational as well as a rotational component to the F applied at radius R, but I'm not sure how to combine them.

I'm also unclear on whether the radius R is the radius of the body, or just the radius at which the second force is applied. But there was a previous question where R was defined as the radius of a uniform disc, so I'm leaning towards the former.

Thanks!

 

[jbsteele]

The linear velocity of the rigid body can be calculated using Newton's second law. The equation is F = ma, where F is the force applied to the center of mass and m is the mass of the rigid body. The acceleration of the rigid body is therefore F/m and its linear velocity is the integral of this acceleration with respect to time.The angular velocity of the rigid body can be calculated by taking the torque caused by the force F applied at radius R and dividing it by the moment of inertia of the rigid body. The equation is τ = Iα, where τ is the torque and I is the moment of inertia. The angular acceleration of the rigid body is therefore τ/I and its angular velocity is the integral of this acceleration with respect to time.To show that the linear and angular velocities follow the relationship v = 1/2ωR, we can use the definition of angular velocity as the linear velocity divided by the radius. This gives us v = ωR, which can be rearranged to v = 1/2ωR.