Sphere Rolling vs. Sliding Down a Ramp

In summary, both spheres, one sliding and one rolling, are released from the same displacement up the ramp at the same time. The first sphere slides with no friction while the second sphere rolls without slipping. Using the equations for moment of inertia and motion with constant acceleration, it can be proven that the rolling sphere will reach the bottom of the ramp first due to its conversion of potential energy into kinetic energy.
  • #1
oranim2
4
0

Homework Statement



Let there be two spheres, of equal radius but unknown masses (mass isn't important).
Both move down a ramp of angle [tex]\theta[/tex], where [tex]\theta[/tex] can be any angle but 90 deg. or 0 deg.
The first sphere slides down the ramp with no (negligible) friction. This sphere does not roll at all.
The second sphere rolls down the ramp. This sphere does not slip, and thus, its only movement is caused by the rolling, not a sliding motion.

If both spheres are released from the same displacement up the ramp from the ramp's end, at the same time, which will reach the bottom of the ramp first?
Prove that this is so


Homework Equations


Moment of inertia equations. F=MA. Motion with constant acceleration equations.


The Attempt at a Solution


I tried manipulating some of the formulas, but was not able to get very far. I didn't find a way to relate time to the rolling (not sliding) ball.
 
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  • #2
Compare the forces acting on each. Then apply Newton's 2nd law.
 
  • #3
You can also use energy, might be easier.
 
  • #4
Any other help?... I still am a bit stuck.
 
  • #5
Sure, let's examine the energy transformations of the sliding ball. We start out with purely potential energy; however, as the ball reaches the end of the incline, all the potential energy has subsequently been converted into kinetic energy. Hence, PE = KE.

Now apply similar logic to the rolling ball and interpret the results.
 

What is the difference between sphere rolling and sliding down a ramp?

Sphere rolling and sliding down a ramp are two different ways that a spherical object can move down an incline. In rolling, the sphere rotates as it moves down the ramp, while in sliding, it does not rotate.

Which one is faster, sphere rolling or sliding down a ramp?

In most cases, sphere rolling is faster than sliding down a ramp. This is because rolling involves both translational and rotational motion, while sliding only involves translational motion. However, the speed at which a sphere rolls or slides down a ramp will depend on factors such as the angle of the ramp and the surface properties of the ramp and the sphere.

What causes a sphere to roll instead of slide down a ramp?

A sphere will roll instead of slide down a ramp when the force of friction between the sphere and the ramp is greater than the force of gravity pulling the sphere down the ramp. This typically occurs when the ramp is not too steep and the surface of the ramp and the sphere are not too smooth.

How does the shape of the ramp affect the motion of a rolling sphere?

The shape of the ramp can greatly affect the motion of a rolling sphere. A ramp with a steeper incline will cause the sphere to roll faster, while a ramp with a gentler incline will cause the sphere to roll slower. Additionally, a curved ramp will cause the sphere to roll in a curved path, while a straight ramp will result in a straight path of motion.

Can a sphere roll and slide down a ramp at the same time?

No, a sphere cannot roll and slide down a ramp at the same time. These are two different types of motion and the sphere will either roll or slide, depending on the conditions of the ramp and the sphere. However, it is possible for a sphere to transition from rolling to sliding or vice versa as it moves down the ramp, depending on changes in the conditions of the ramp or the sphere's motion.

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