Rolling without slipping problem

In summary, the acceleration of the center of mass of a solid sphere rolling down a ramp at an angle of 32 degrees above horizontal can be calculated using the sum of torques and Newton's laws for translation and rotation. The acceleration is found to be 12.98 m/s^2, which may seem greater than the acceleration due to gravity, but this is due to the presence of friction and the condition for rolling without slipping.
  • #1
ph123
41
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A solid sphere rolls without slipping down a ramp that is at an angle of 32 above horizontal. The magnitude of the acceleration of the center of mass of the sphere as it rolls down the ramp is?


sum of torques = I(alpha)

rmgsin32 = (2/5)mr^2(a_tan/r)

the radii drop out as one would expect with no radius given. the masses also drop out since they weren't provided.

gsin32 = (2/5)a_tan

(9.8 m/s^2)sin32 = (2/5)a_tan

a = 12.98 m/s^2

This result clearly makes no sense because it is greater than the accelertion due to gravity. But that was the only approach I could think to use since I was only given the angle of the incline. Any ideas?
 
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  • #2
Hints: Does gravity exert a torque on the sphere? What other force acts on the sphere that exerts the torque?
 
  • #3
friction. but how can I calculate friction without mass or coefficient of friction?
 
  • #4
ph123 said:
friction. but how can I calculate friction without mass or coefficient of friction?
Just try it--maybe you won't need that information. :wink:

Apply Newton's law for translation and rotation. And the condition for rolling without slipping.
 
  • #5
omg duh. i forgot about the forces in the x-direction. i always do that. thanks.
 
  • #6
ph123 said:
sum of torques = I(alpha)

rmgsin32 = (2/5)mr^2(a_tan/r)

The second line is not correct. On the left hand side, you wrote the torque around the center which is the point where the sphere touches the slope, so you need to use inertial momentum I corresponding to the axis through above point.

On the right-hand side, it should be

I=2/5mr^2+mr^2=7/5 mr^2 (axis-parallel theorem, or Steiner's theorem)
 
  • #7
That's perfectly OK as well: Since the sphere rolls without slipping, you can view it as being in pure instantaneous rotation about the point of contact. With this approach, you need to use torques and rotational inertia about the point of contact, not about the center. Note that gravity does exert a torque about the point of contact. (I find the two step approach--analyzing translation and rotatation separately--to be more instructive. But it's all good! :smile: )
 
  • #8
Yes Doc Al, that's what I meant.
 

1. What is the difference between rolling with and without slipping?

Rolling without slipping refers to the motion of a rigid body where the point of contact with the ground is stationary, while the body rotates about that point. In contrast, rolling with slipping occurs when the point of contact is not stationary and there is relative motion between the body and the ground.

2. How do you calculate the velocity of a rolling object without slipping?

To calculate the velocity of a rolling object without slipping, you can use the equation v = ωr, where v is the linear velocity, ω is the angular velocity, and r is the radius of the rolling object.

3. What are the conditions for rolling without slipping to occur?

For a rolling object to move without slipping, two conditions must be met: first, the point of contact between the object and the ground must be stationary; and second, the point of contact must have a velocity of zero.

4. How does friction play a role in the rolling without slipping problem?

Friction is necessary for a rolling object to move without slipping. It creates a torque that opposes the rotational motion of the object, allowing it to roll instead of slide. Without friction, the object would simply slide instead of roll.

5. Can a rolling object ever have both linear and angular acceleration?

Yes, a rolling object can have both linear and angular acceleration. This occurs when an external force is applied to the object, causing it to accelerate both linearly and rotationally. However, the object will continue to roll without slipping as long as the conditions for rolling without slipping are met.

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