# Acceleration of a pulley. Torque, and moment of inertia.

• EV33
In summary, the conversation discusses finding the expression for acceleration in a system with two blocks, a pulley, and a rope. After considering the equations Fnet=m2gsin(theta)-m1g and F/M=a, the individual solved for acceleration by drawing free-body diagrams and using Newton's second law. The final expression for acceleration is ma=Ia/(R^2), which includes the moment of inertia of the pulley.
EV33

## Homework Statement

There are two blocks, one of mass m1 and the other of m2. They are connected by a rope that runs over a pully of radius R and interia I. m2 slides down a frictionless incline at an angle theta, and m1 hangs virtically while being pulled up by the tension in the rope. Find the expression for acceleration.

## Homework Equations

Fnet=m2gsin(theta)-m1g
F/M=a

## The Attempt at a Solution

With the previous two equations I came to this conclusion...

(m2gsin(theta)-m1g)/(m1+m2+I)

I figured it would be just I because the mass of an object is its inertia.

Howwoud I derive the I/R^2?

How did you get an equation with an "I" when your initial equations:

Fnet=m2gsin(theta)-m1g
F/M=a

didn't have an "I" anywhere? Anyhow, for mechanics problems, you should always draw free-body diagrams for every object (including the pulley!), write down Newton's second law for each object, and then solve the resulting equations. That way, you'll see very clearly where the I/R^2 came from.

I got the I from thinking that I of the pulley would be the same thing as mass for the blocks because isn't mass just the inertia of an object? and I put my net force over the total mass.

I think I may have figured it out but I also may be making a mistake here...

T1 and T2 are equal to the tensions on each side, and alpha=a/R

T2R-T1R=Ialpha

R(T2-T1)=Ialpha

Tnet=I(a/R)/R

ma=Ia/(R^2)

m=I/R^2

## 1. What is acceleration of a pulley?

The acceleration of a pulley refers to the rate at which the velocity of the pulley changes. It is typically measured in meters per second squared (m/s²) and can be calculated using the formula a = (v² - u²)/2s, where v is the final velocity, u is the initial velocity, and s is the distance traveled.

## 2. What is torque?

Torque is a measure of the twisting force that is applied to an object. It is typically measured in Newton-meters (N·m) and can be calculated using the formula T = F x r, where F is the force applied and r is the distance from the axis of rotation to the point of application of the force.

## 3. How is torque related to acceleration?

Torque and acceleration are related through the moment of inertia, which is a measure of an object's resistance to changes in its rotational motion. The greater the moment of inertia, the more torque is required to produce a given amount of acceleration.

## 4. What is moment of inertia?

Moment of inertia is a measure of an object's resistance to changes in its rotational motion. It is typically denoted by the symbol I and is calculated by summing up the products of the mass of each particle in the object and the square of its distance from the axis of rotation.

## 5. How does the moment of inertia affect the acceleration of a pulley?

The moment of inertia of a pulley affects its acceleration by determining the amount of torque required to produce a given amount of acceleration. A pulley with a higher moment of inertia will require more torque to produce the same amount of acceleration as a pulley with a lower moment of inertia.

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