Sphere rolling down an inclined plane

In summary, the conversation discusses a simple scenario involving a sphere rolling down an inclined plane and the question of its linear and angular velocities at any given time. The speaker suggests using the conservation of energy to find an equation for linear velocity.
  • #1
aorrdalca
4
0
I've seen a number of posts on the following question, but don't believe any contain a solution to the following very simple scenario:

A sphere of radius r and mass m rolls down a plane inclined at theta degrees. What are its linear and angular velocities at any time t thereafter, assuming it starts at rest?

Perhaps I've missed the answer elsewhere; if so, apologies.
 
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  • #2
Here is a hint: write down the conservation of energy (potential energy + kinetic energy of center of mass + rotational energy around COM = const)
Then express angular velocity by linear velocity and you have an equation for linear velocity only!
 

1. How does the angle of the inclined plane affect the speed of the rolling sphere?

The speed of the rolling sphere is directly affected by the angle of the inclined plane. The steeper the angle, the faster the sphere will accelerate and roll down the plane due to the force of gravity. However, if the angle is too steep, the sphere may roll too quickly and lose control.

2. What is the relationship between the mass of the sphere and its acceleration down the inclined plane?

The mass of the sphere has a direct relationship with its acceleration down the inclined plane. The greater the mass of the sphere, the more force is required to accelerate it down the plane. This means that a heavier sphere will accelerate slower than a lighter sphere on the same inclined plane.

3. How does friction affect the motion of the rolling sphere on an inclined plane?

Friction plays a significant role in the motion of a rolling sphere on an inclined plane. If the surface of the inclined plane is rough, it will create more friction, slowing down the sphere's motion. However, if the surface is smooth, there will be less friction, allowing the sphere to roll faster.

4. What is the relationship between the height of the inclined plane and the distance the sphere will travel?

The height of the inclined plane has a direct relationship with the distance the sphere will travel. The higher the height, the longer the distance the sphere will travel before reaching the bottom of the inclined plane. This is because the potential energy of the sphere is converted into kinetic energy as it rolls down the plane.

5. How does the shape of the sphere affect its rolling motion down an inclined plane?

The shape of the sphere can affect its rolling motion down an inclined plane. A sphere with a smaller diameter will have less surface area in contact with the inclined plane, reducing friction and allowing it to roll faster. In contrast, a larger diameter sphere will have more surface area in contact with the plane, increasing friction and slowing its motion.

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