# Pulley Problem: Find the acceleration of M2 & M3

In summary: Keep in mind that the accelerations will be in the same direction as the net force acting on each mass, so for m1, the acceleration will be downward since the net force is downward. And for m2 and m3, the net force is upward, so the acceleration will be downward relative to the pulley. In summary, the acceleration of m1 is -1.13 m/s2 and the acceleration of m2 and m3 is -2.87 m/s2.

1. Homework Statement

Pulley is mass less accelerating downward at 2.0 m/s2.
M1 = 4.0 kg
M2 = 2.0 kg
M3 = 3.0 kg

Find the acceleration of M2 & M3.

## Homework Equations

I made up the slope positive pointing up.
Since M2 and M3 have the same acceleration, a2=a3 ⇒ a
Since it's accelerating downwards, Apy = -2.0 m/s2

Mass 1:
ΣF1y = m1a1y
T-m1g = m1a1y
T = m1a1y + m1g (1)

Mass 2:
ΣF2y = m1ay
T-T1 = m2ay (2)

Mass 3:
ΣF3y = m3ay
T1-m3g = m3ay (3)

Pulley:
apy = (a1y+ay) / 2 (4)

## The Attempt at a Solution

I added eq. (2) & (3) to get rid of T1
T-m3g = m2ay+m3ay (5)

I plugged eq. (1) into (5)
m1a1y + m1g - m3g = ay(m2+m3)

Solve for a1y:
a1y = (ay(m2+m3) - m1g + m3g) / m1 (6)

Plug (6) into (4)

apy = [ (ay(m2+m3) - m1g + m3g) / m1) + ay ] / 2

Solve for ay:
ay = (m1g-m3g+2m1apy) / m2+m3+m1) = 2.87 m/s2

By the answer key, the acceleration for m2 and m3 is -2.87 m/22. How do I know that is negative and not positive? Is it because m2 + m3 have a higher mass than m1, thus when the pulley is accelerating downwards, m1's acceleration points up and m2 & m3's acceleration point down?

Last edited:
I get 2.87 downward. Let "a" be the downward acceleration of m2 and m3 relative to the pulley. So the total downward acceleration of m2 and m3 is (a+2). Using this approach, what is the upward acceleration of m1?

Chestermiller said:
I get 2.87 downward. Let "a" be the downward acceleration of m2 and m3 relative to the pulley. So the total downward acceleration of m2 and m3 is (a+2). Using this approach, what is the upward acceleration of m1?
The acceleration of m1 would be -1.13 m/s2. I think I finally caught my mistake. Since the pulley is accelerating downwards, the accelerations for m1, m2 & m3 are downwards as well. Then, solving the following equations using a negative sign on all accelerations, I get 2.87 downward:

Mass 1:
T = -m1a1y + m1g (1)

Mass 2:
T-T1 = -m2ay (2)

Mass 3:
T1-m3g = -m3ay (3)

Is this the right approach for this problem?

The acceleration of m1 would be -1.13 m/s2. I think I finally caught my mistake. Since the pulley is accelerating downwards, the accelerations for m1, m2 & m3 are downwards as well. Then, solving the following equations using a negative sign on all accelerations, I get 2.87 downward:

Mass 1:
T = -m1a1y + m1g (1)

Mass 2:
T-T1 = -m2ay (2)

Mass 3:
T1-m3g = -m3ay (3)

Is this the right approach for this problem?
I can't follow your notation, but if you get 2.87, you must have done it right.

## 1. What is a pulley problem?

A pulley problem is a type of physics problem that involves calculating the acceleration or motion of objects connected by a pulley system.

## 2. How do you solve a pulley problem?

To solve a pulley problem, you need to use Newton's laws of motion, specifically the second law (F = ma), and apply it to the objects involved in the pulley system. You also need to consider the tension and weight of the objects, as well as the properties of the pulley itself.

## 3. What is the purpose of finding the acceleration of M2 and M3 in a pulley problem?

Finding the acceleration of M2 and M3 allows you to understand how the objects in the pulley system will move and how their velocities and positions will change over time. This information is important in predicting the overall behavior of the system.

## 4. What is the role of friction in a pulley problem?

Friction plays a crucial role in a pulley problem because it affects the tension in the rope or string that is connecting the objects. Friction can either increase or decrease the tension, which in turn affects the acceleration of the objects.

## 5. How does the number of pulleys affect the acceleration of the objects?

The more pulleys there are in a system, the less the objects will accelerate. This is because the tension in the rope or string is divided among the multiple pulleys, resulting in less force being exerted on the objects.

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