How are we supposed to solve for omega if we aren't given a velocity? The problem tells us that C is moving to the right with speed Vc, but doesn't give us a value. We need omega to find the acceleration at C, otherwise I believe there are too many unknowns...

Note that the acceleration of point A depends explicitly on the angular velocity and angular acceleration of the wheel. Since you are given complete information on the acceleration of A (xy-components), you have enough information to solve for both alpha and omega for the wheel. To do so, consider first writing down a rigid body acceleration equation relating A to the no-slip contact point (call it B), Then, next, writing down a rigid body acceleration equation relating C to B.

if I'm correct, we don't even need to find omega right? Omega^2 is dot product with R, and R is vertical between A, C and B, so any variables with Omega will strictly be in the Y direction. Since there will be no y-component for acceleration at C, any Y component accelerations at C should be canceled out by Y component acceleration at B anyways.

Can you assume that since the wheel is rolling without slipping that the acceleration of the center (point C) is only in the horizontal direction? Is this generally true for all wheels that are rolling on a horizontal plane no matter how they are driven?

Think about the path of the center C of the drum. C remains at a distance of R from the flat surface along which it moves. As a result, C moves on a STRAIGHT path (an infinite radius of curvature). Therefore C cannot have a component of acceleration perpendicular to its path, and therefore the acceleration is in the horizontal direction only.

If the surface on which the drum moves was curved, then C could have a vertical component of acceleration.

I set up equations relating the acceleration of points A and B as well as the acceleration of points B and C like another comment said, but I still am ending up with too many unknowns. I am able to get one equation from setting the sum of components in the j direction equal to each other, but this gives me an equation containing both omega and alpha. If I sum the components in the i direction I just introduce another unknown, a_C, which won't be helpful. What am I missing?

How are we supposed to solve for omega if we aren't given a velocity? The problem tells us that C is moving to the right with speed Vc, but doesn't give us a value. We need omega to find the acceleration at C, otherwise I believe there are too many unknowns...

Note that the acceleration of point A depends explicitly on the angular velocity and angular acceleration of the wheel. Since you are given complete information on the acceleration of A (xy-components), you have enough information to solve for both alpha and omega for the wheel. To do so, consider first writing down a rigid body acceleration equation relating A to the no-slip contact point (call it B), Then, next, writing down a rigid body acceleration equation relating C to B.

if I'm correct, we don't even need to find omega right? Omega^2 is dot product with R, and R is vertical between A, C and B, so any variables with Omega will strictly be in the Y direction. Since there will be no y-component for acceleration at C, any Y component accelerations at C should be canceled out by Y component acceleration at B anyways.

Please correct me if I'm wrong.

I believe you are correct, at least that is how I solved it.

Can you assume that since the wheel is rolling without slipping that the acceleration of the center (point C) is only in the horizontal direction? Is this generally true for all wheels that are rolling on a horizontal plane no matter how they are driven?

Think about the path of the center C of the drum. C remains at a distance of R from the flat surface along which it moves. As a result, C moves on a STRAIGHT path (an infinite radius of curvature). Therefore C cannot have a component of acceleration perpendicular to its path, and therefore the acceleration is in the horizontal direction only.

If the surface on which the drum moves was curved, then C could have a vertical component of acceleration.

I set up equations relating the acceleration of points A and B as well as the acceleration of points B and C like another comment said, but I still am ending up with too many unknowns. I am able to get one equation from setting the sum of components in the j direction equal to each other, but this gives me an equation containing both omega and alpha. If I sum the components in the i direction I just introduce another unknown, a_C, which won't be helpful. What am I missing?

We can say a_c is equal to R*omega^2 in j_hat. Explanation is on page 91 of the lecture book.