# Cylinder rolling thanks to external torque.

• bobon123
In summary, a cylinder of mass m and radius R is set on a plane and a constant torque is applied along the axis passing through the center of mass of the cylinder. The central equation for this problem is I dw / dt = M. The angular acceleration can be found by setting up the torque equation relative to the center of mass, taking into account the frictional torque and the net force, and solving for the angular acceleration. The linear acceleration can then be found by substituting the angular acceleration into the equation aG = dw / dt R, which yields aG = 2 M / (3 m R). This solution is valid for both the center of mass and the contact point between the cylinder and the plane, as they are

## Homework Statement

A cylinder of mass m and radius R is set on a plane, with large enough friction coefficient to ensure at any moment rolling without slipping. A constant torque is applied along the axis passing through the center of mass (G) of the cylinder and perpendicular to the basis of the cylinder.
What is the acceleration of the center of mass G?

## Homework Equations

The central equation in this problem is
I dw / dt = M
This equation should be valid if w (angular velocity) and I are referred to either an axis passing through the center of mass of the system (I_G) or through a point that is fixed along the motion.

I will also need to compute I for the cylinder,
I_G=1/2 mR^2,
while remembering that, if I need I for a different axis,
I_O=I_G+ m (GO)^2

## The Attempt at a Solution

To find a solution is easy, I have two! I can set myself either in the center of mass, or in the contact point between the cylinder and the plane, since it is fixed.

In the second case, I have I_O = 3/2 mR^2 and therefore the angular acceleration is 2 M/(3 mR^2), and the linear acceleration should be 2M/(3 mR)
In the first case, I have I_G=1/2 mR^2, and therefore the angular acceleration is 2 M/(mR^2) and the linear acceleration would be 2 M/(mR).

I am pretty sure the proper solution is the first, but I am not able to understand why I can't solve it easily in the center of mass... I think I am missing in this case the additional torque caused by the attrition force, but I don't understand how that would bring the total to 2M/(3 mR)

Welcome to PF!
bobon123 said:
... I am not able to understand why I can't solve it easily in the center of mass... I think I am missing in this case the additional torque caused by the attrition force
Yes.
but I don't understand how that would bring the total to 2M/(3 mR)
Try setting up the torque equation relative to the center of mass with the torque due to friction included. Don't forget that you also have Fnet = M aG.

bobon123
Thanks a lot! I think I am there:

In the center of mass, I have
1) M_T = M - F_a R
2) F_a = m a_G
3) M_T= I_G dw/dt
where M_T is the total torque and the friction is the only force parallel to the plane. I also have
4) a_G = dw / dt R,
since the cylinder is rolling without slipping.

Therefore, from 1) and 2),
5) M_T=M-m a_G R

While from 3) and 4),
6) M_T=1/2 m R a_g

Therefore, from 5) and 6),
7) M-m R a_G = 1/2 m R a_g,
from which a_g = 2 M / (3 m R)

Perfect! Finally the two solutions are only one!

Looks great. Good work!

## 1. How does external torque cause a cylinder to roll?

External torque is a force applied to an object at a distance from its center of mass. This causes a rotational motion, and in the case of a cylinder, it causes it to roll.

## 2. What is the relationship between external torque and rotational motion?

The relationship between external torque and rotational motion is described by Newton's Second Law of Motion, which states that the net torque on an object is equal to the product of its moment of inertia and its angular acceleration.

## 3. Can you give an example of how external torque can be applied to a cylinder to make it roll?

One example is pushing a cylinder on a flat surface with a stick. The force of the push is applied at a distance from the cylinder's center of mass, creating an external torque that causes the cylinder to roll.

## 4. How does the shape of a cylinder affect its ability to roll thanks to external torque?

The shape of a cylinder does not affect its ability to roll thanks to external torque. As long as there is a force applied at a distance from its center of mass, a cylinder will roll regardless of its shape.

## 5. Is there a limit to how much external torque can be applied to a cylinder before it stops rolling?

Yes, there is a limit to how much external torque can be applied to a cylinder before it stops rolling. This is determined by the coefficient of friction between the cylinder and the surface it is rolling on. Once the external torque exceeds this limit, the cylinder will stop rolling and begin to slide.

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