# Equations of motion for angular acceleration

• jono90one
In summary, the conversation discussed the motion of a cylindrical shell going down a slope with a 36-degree angle to the horizontal. The equation of motion for angular acceleration was not explicitly stated, but it was mentioned that it must follow a similar structure to linear acceleration equations. The linear acceleration was also calculated and it was asked if it would change if the shell was changed to a solid cylinder. An energy approach was suggested to solve this problem, where the potential energy at the top equals the kinetic energy at the bottom. Differentiating this equation with respect to time can give a formula for the acceleration, which will involve the moment of inertia and can show the effect of changing to a solid ring.
jono90one

## Homework Statement

A slope angled 36* to the horizontal has a hoop cylindrical shell going down it, which has radius 3cm and mass 100g.
1) Write down an equation of motion for the angular acceleration.
2) Will the linear acceleration change if the hoop is changed to a cylinder (i.e. solid).

## Homework Equations

I=mr^2
F=ma
T=Iα (alpha)(T=Torque)

## The Attempt at a Solution

Logically angular acceleration equations of motion must follow a similar structure to linear acceleration equation of motion.
i.e. Iα = ∑T
But I am not sure what torque's are acting if so?

2. The linear acceleration is easy, that is just ma = ∑F, i get
sin(x)(gcos(x)+g) = 10.4 ms^-2 (Masses cancle)

But it wants to find out if it changes,
This one I'm also not sure on, I know α(angular accel) changes, as the moment of inertia changes. But I am not sure if this changes the linear acceleration as the center of mass remains the same...

Any advice is greatly appreciated :)

I hesitate to post because I don't see how to do this in the way you started. I think it can be done from an energy approach. If you write that
potential energy at the top = kinetic energy at the bottom
for some distance s along the hill, and have linear KE as well as rotational KE on the right side, you need only differentiate with respect to time to get a nice formula for the acceleration. It will have the moment of inertia in it (unless you put in I = mR² for the ring) so you can see the effect of changing to a solid ring. I used w = v/r in the rotational energy expression to avoid having two variables for the velocity.

## 1. What is angular acceleration?

Angular acceleration is a measure of how quickly the angular velocity of an object changes over time. It is a vector quantity and is typically measured in radians per second squared.

## 2. How is angular acceleration related to linear acceleration?

Angular acceleration and linear acceleration are related through the equation a = rα, where a is linear acceleration, r is the radius of the circular motion, and α is angular acceleration. This means that for the same angular acceleration, a smaller radius will result in a greater linear acceleration.

## 3. What is the difference between angular acceleration and angular velocity?

Angular acceleration is a measure of how quickly the angular velocity of an object changes, while angular velocity is a measure of how quickly an object is rotating. Angular velocity is a vector quantity that is measured in radians per second, while angular acceleration is also a vector quantity but is measured in radians per second squared.

## 4. How do you calculate angular acceleration?

Angular acceleration can be calculated using the equation α = (ωf - ωi)/t, where α is angular acceleration, ωf is the final angular velocity, ωi is the initial angular velocity, and t is the time interval over which the change in angular velocity occurs.

## 5. What are some real-life examples of angular acceleration?

Some real-life examples of angular acceleration include the spinning of a top, the rotation of a wheel on a moving vehicle, and the swinging of a pendulum. Other examples include the rotation of a planet around its axis and the spinning of a figure skater during a jump.

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