# Angular Acceleration and Linear Acceleration of a Pulley

• EchoTheCat
In summary, there are two blocks, m1 and m2, hanging from a rope over a pulley. The mass of the pulley is 0.15 kg with a radius of 0.055 m. The mass of m1 is 0.165 kg and m2 is 0.18 kg. The linear acceleration of m1 is 0.3504 m/s/s downward and the linear acceleration of m2 is 0.3504 m/s/s upward. The magnitude of the angular acceleration of the pulley can be calculated using the equation a = α * r, where α is the angular acceleration and r is the radius of the pulley.
EchoTheCat

## Homework Statement

There are two blocks, m1 and m2, that are hanging from a rope which passes over a pulley.
The mass of the pulley is 0.15 kg.
The radius of the pulley is 0.055 m.
The mass of m1 is 0.165 kg.
The mass of m2 is 0.18 kg.
The linear acceleration is m1 is 0.3504 m/s/s downward, and the linear acceleration of m2 is 0.3504 m/s/s upward.
What is the magnitude of the angular acceleration of the pulley?

## Homework Equations

The tension on mass 1 is m1g-m1a.
The tension on mass 2 is m2a+m2g.
T1-T2 = 1/2 mp * a * r.

## The Attempt at a Solution

m1g-m1a - (m2a+m2g) = 1/2 * mp * a * r
0.165 (9.81 - 0.3504) - 0.18(0.3505+9.81) = 1/2 * 0.15 * a * 0.055
a = 64.98 rad/s/s

Ok, so I was able to solve for
(m1g+m1a)R – (m2g-m2a)R = -(1/2mr2)a, so, a = 6.37 rad/s/s.

EchoTheCat said:
Ok, so I was able to solve for
(m1g+m1a)R – (m2g-m2a)R = -(1/2mr2)a, so, a = 6.37 rad/s/s.
I get the same numerical result, but I can't get there using your equation. I am not sure how to parse -(1/2mr2)a. Do you mean ##-(\frac 1 2 m_p r^2) α##? It looks like you might be using the moment of inertia of the pulley multiplied by the angular acceleration to get the torque. And that could work if you knew that the pulley was a perfect disc. However, when I plug in the numbers I don't get the 6.37 ##\frac {rad} {s^2}##.
Here is another approach: the angular acceleration α must be equal to the linear acceleration of the cable divided by the radius of the pulley.

tnich said:
I get the same numerical result, but I can't get there using your equation. I am not sure how to parse -(1/2mr2)a. Do you mean ##-(\frac 1 2 m_p r^2) α##? It looks like you might be using the moment of inertia of the pulley multiplied by the angular acceleration to get the torque. And that could work if you knew that the pulley was a perfect disc. However, when I plug in the numbers I don't get the 6.37 ##\frac {rad} {s^2}##.
Here is another approach: the angular acceleration α must be equal to the linear acceleration of the cable divided by the radius of the pulley.
I see I made a mistake in my calculations, so my numbers now agree with yours. I think you have the right answer. Now that you have solved the problem that way, try the approach I outlined above. I think you will see that you get the same answer and you don't have to assume anything about the distribution of mass in the pulley.

EchoTheCat said:
The mass of m1 is 0.165 kg.
The mass of m2 is 0.18 kg.
The linear acceleration is m1 is 0.3504 m/s/s downward, and the linear acceleration of m2 is 0.3504 m/s/s upward.
How is it that the lighter mass is accelerating downwards?

Edit: You should be aware that there is a simple relationship between angular acceleration and the linear acceleration of the pulley's rim that doesn't involve any involvement of the masses, torques, etc. Do you know what it is?

Last edited:

## 1. What is a massive pulley with 2 blocks?

A massive pulley with 2 blocks is a simple machine consisting of a wheel with a groove around its circumference and two blocks attached to either side of the wheel. The blocks are connected by a rope or belt that runs around the wheel and allows for the transfer of force and motion between the blocks.

## 2. How does a massive pulley with 2 blocks work?

The massive pulley with 2 blocks works by using the principle of mechanical advantage, where the force applied to one block is transferred to the other block through the pulley. This allows for a smaller force to be applied over a greater distance to lift a heavier load on the other block.

## 3. What are the advantages of using a massive pulley with 2 blocks?

One advantage of using a massive pulley with 2 blocks is that it can provide a mechanical advantage, making it easier to lift heavy objects. Additionally, it can also change the direction of the force applied, allowing for more versatility in its use.

## 4. How is the mechanical advantage calculated for a massive pulley with 2 blocks?

The mechanical advantage of a massive pulley with 2 blocks is calculated by dividing the output force (the weight of the lifted object) by the input force (the force applied to one block). This ratio can also be expressed as the number of ropes supporting the lifted object.

## 5. What are some real-world applications of a massive pulley with 2 blocks?

Massive pulleys with 2 blocks are commonly used in construction and engineering for lifting heavy materials and equipment. They are also used in cranes and elevators to lift and move objects vertically. In addition, they can be found in exercise equipment and rock climbing systems for providing mechanical advantage when lifting and pulling objects.

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