# Velocity of the Center of Mass

• ryankunzzz
In summary, a spherical shell with a mass of 1kg and a radius of 2.0cm is released from rest at the top of an inclined plane with a height of 1.00m. The ball rolls down the incline without slipping. The problem involves finding the velocity of the center of mass at the bottom of the incline using equations for rotational and translational kinetic energy, potential energy, and tangential acceleration. The unknowns can be represented by variables such as "I", "omega", "m", "r", and "v" and can be manipulated to create a single formula for total energy with only a few unknowns.
ryankunzzz

## Homework Statement

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A shperical shell of mass = 1kg and radius 2.0cm is released from rest at the top of an inclined plane at a height of 1.00m. The ball rolls down the incline without slipping, what is the velocity of the center of mass at the bottom of the incline.

## Homework Equations

I'm assuming it will be using

KE rotational, which is 1/2 IW^2
Potential Energy which is mgh
KE translational which is 1/2 mv^2
tangential acceleration: at= αR

## The Attempt at a Solution

I have no idea where to begin. If we were given the angle of the incline, this would be much easier, but we are not.

If you do not know the value of a parameter, leave it as a variable. HINT: use r for the radius of the ball.

Now see if you can manipulate the formula for rotational kinetic energy to eliminate "I" and "omega" leaving only m, r and v. The goal is to have a single formula for total energy with only a few unknowns in it (ideally only one).

I got it, thank you!

## 1. What is the definition of velocity of the center of mass?

The velocity of the center of mass is the rate of change of the position of the center of mass of a system over time. It is a vector quantity that takes into account the mass and velocity of all the individual particles that make up the system.

## 2. How is the velocity of the center of mass calculated?

The velocity of the center of mass is calculated by taking the sum of the individual masses multiplied by their respective velocities and dividing by the total mass of the system. This can be expressed as: vcm = (m1v1 + m2v2 + ... + mnvn) / (m1 + m2 + ... + mn)

## 3. What is the importance of the velocity of the center of mass?

The velocity of the center of mass is important because it gives us information about the overall motion of a system. It allows us to understand how the system as a whole is moving, even if the individual particles within the system are moving in different directions.

## 4. How does the velocity of the center of mass relate to conservation of momentum?

The velocity of the center of mass is directly related to the conservation of momentum. According to the law of conservation of momentum, the total momentum of a system remains constant unless acted upon by an external force. The velocity of the center of mass is a measure of the total momentum of the system, and it remains constant as long as there are no external forces acting on the system.

## 5. Can the velocity of the center of mass be greater than the velocity of any individual particle in the system?

Yes, the velocity of the center of mass can be greater than the velocity of any individual particle in the system. This is because the velocity of the center of mass takes into account the mass of each particle, so even if one particle has a very high velocity, it may have a smaller mass compared to other particles in the system. This can result in a higher velocity of the center of mass.

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