# Cylinder on inclined plane

• NAkid
In summary, the conversation discusses the maximum angle at which a solid cylinder made of a certain material can roll without slipping on an inclined plane. The solution involves using equations for motion and torque, and substituting in values for acceleration and moment of inertia.
NAkid

## Homework Statement

A block of a certain material begins to slide on an inclined plane when the plane is inclined to an angle of 11.86o. If a solid cyclinder is fashioned from the same material, what will be the maximum angle at which it will roll without slipping on the plane (in degrees)?

## The Attempt at a Solution

drew a free body diagram, and have the following equations

mgsin(theta) + friction = ma
torque = friction*R = I(angular acceleration) --> friction=(I*angular acceleration)/R
mgsin(theta) + (I*angular acceleration)/R = ma
a=R*(angular acceleration) and I = .5MR^2
... substitute those in, but then I get

theta = (inverse sin)[(.5*a)/g] -- how do i solve for a?

NAkid said:

## The Attempt at a Solution

drew a free body diagram, and have the following equations

mgsin(theta) + friction = ma
torque = friction*R = I(angular acceleration) --> friction=(I*angular acceleration)/R
mgsin(theta) + (I*angular acceleration)/R = ma
a=R*(angular acceleration) and I = .5MR^2
... substitute those in, but then I get

theta = (inverse sin)[(.5*a)/g] -- how do i solve for a?

Have you used the relation between $v$ and $\omega$?

I would approach this problem by first understanding the concept of rolling without slipping. In this case, the maximum angle at which the cylinder will roll without slipping is when the friction force is equal to the component of the weight of the cylinder down the inclined plane. This means that the cylinder will just start to slip and roll at the same time.

To solve for the maximum angle, we can use the equation for torque: torque = friction * radius. We know that the torque due to friction is equal to the weight of the cylinder times the radius of the cylinder. We can also use the equation for torque for a cylinder: torque = moment of inertia * angular acceleration.

Setting these two equations equal to each other, we get:

mgRsin(theta) = (1/2)MR^2 * alpha

Where alpha is the angular acceleration and theta is the angle of the inclined plane. We can rearrange this equation to solve for alpha:

alpha = (2mgRsin(theta))/MR^2

Now we can use the equation for acceleration for a rolling cylinder: a = R * alpha. Substituting in our value for alpha, we get:

a = (2mgRsin(theta))/MR

Finally, we can use this equation to solve for the maximum angle at which the cylinder will roll without slipping:

theta = sin^-1((aMR)/(2mg))

We can plug in the known values for mass, radius, and gravitational acceleration to solve for theta. The resulting value will be the maximum angle at which the cylinder will roll without slipping on the inclined plane.

## 1. What is a cylinder on an inclined plane?

A cylinder on an inclined plane refers to a cylindrical object placed on a sloped surface, with one of its circular ends touching the surface.

## 2. How does the angle of the inclined plane affect the cylinder?

The angle of the inclined plane determines the acceleration of the cylinder as it rolls down the slope. The steeper the angle, the faster the cylinder will accelerate.

## 3. What forces are acting on the cylinder on an inclined plane?

The main forces acting on a cylinder on an inclined plane are gravity and normal force. Gravity pulls the cylinder down the slope, while the normal force from the surface pushes back against the cylinder to prevent it from sinking into the surface.

## 4. What is the relationship between the weight of the cylinder and the angle of the inclined plane?

The weight of the cylinder affects the normal force acting on it. As the angle of the inclined plane increases, the normal force decreases, and therefore the weight of the cylinder will have a greater impact on its acceleration down the slope.

## 5. How can the acceleration of the cylinder on an inclined plane be calculated?

The acceleration of the cylinder can be calculated using the formula a = g sinθ, where g is the acceleration due to gravity (9.8 m/s²) and θ is the angle of the inclined plane. This assumes that there is no friction between the cylinder and the surface.

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