Rotational dynamics with constant CM velocity

[trulyfalse]

Hey PF!

Homework Statement

I've attached the problem to this post along with my free body diagram.

Homework Equations

Moment of inertia of a cylinder: 1/2MR^2

The Attempt at a Solution

Since the cylinder is moving at a constant velocity and is not slipping, ƩF = 0. For the torques around the instantaneous axis of rotation, we can see that Ʃt = -(1.2 m)(Ft1) - (0.4m)(Ft2). However, I'm not sure how to proceed from here, and also have a few conceptual questions related to rolling without slipping. In which direction does the force due to static friction act in these problems? Does it act in such a way that it opposes the translational motion of the object in question or does it act to oppose rotational motion?

EDIT: Just realized that I forgot Fn and Fg on my free body diagram, but they're not terribly pertinent to this problem anyway.

 

[TSny]

Hey trulyfalse!

My suggestion is to forget the forces and torques. This is just a kinematics problem.

 

[ehild]

and also have a few conceptual questions related to rolling without slipping.

That question is crucial: What does rolling without slipping mean? How is the linear velocity of the centre related to the angular velocity of rolling ?

TSny is right, this is a pure kinematic problem.


ehild

 

[haruspex]

have a few conceptual questions related to rolling without slipping. In which direction does the force due to static friction act in these problems? Does it act in such a way that it opposes the translational motion of the object in question or does it act to oppose rotational motion?

As others have posted, you don't need to worry about forces here, just the geometry. (Hint: the instantaneous centre of rotation of a rolling wheel is the point of contact with the surface.)
But to answer your question about friction, it acts to oppose relative motion of the surfaces in contact. For a car accelerating forwards, the driving wheels, if there were no friction, would spin on the road, so the friction there acts forwards from the road onto the wheels (pushing the car along). The idling wheels, were there no friction, would slide along without turning, so the friction there acts backwards, causing those wheels to rotate faster.

 

[HalfLostAndSearching]

Hello,

Great job on your attempt at the solution! You are correct in your approach of considering the torques around the instantaneous axis of rotation. To proceed, you can use the equation for rotational motion, τ = Iα, where τ is the net torque, I is the moment of inertia, and α is the angular acceleration. Since the cylinder is moving at a constant velocity, the angular acceleration is zero and therefore the net torque must also be zero. This will allow you to solve for the unknown forces, Ft1 and Ft2.

To answer your conceptual questions, the force of static friction acts in such a way that it opposes the tendency of the object to slip. In this case, since the cylinder is rolling without slipping, the static friction force will act in the direction opposite to the direction of motion. This is because the force of static friction is necessary to provide the necessary angular acceleration to prevent slipping.

Keep up the good work!