# Three hanging masses and two pulleys, why does m3 accelerate?

• goraemon
In summary, the figure shows three hanging masses connected by massless strings over two massless, frictionless pulleys. In part (a), given the masses of the three objects, the acceleration was calculated using a derived equation. In part (b), it was explained that the 4.0 kg mass accelerates due to the system being in an unbalanced state, as the tension in the rope pulling on it is less than the combined weight of the other two masses. The concept of Newton's laws in non-inertial frames was also mentioned.
goraemon

## Homework Statement

Figure CP7.57 shows three hanging masses connected by massless strings over two massless, frictionless pulleys.

a) Suppose: m1 = 2.5 kg, m2 = 1.5 kg, and m3 = 4.0 kg. Find the acceleration of each.

b) The 4.0 kg mass would appear to be in equilibrium. Explain why it accelerates.

## Homework Equations

Tension of Rope A = 4*m1*m2*m3*g / (4*m1*m2 + m2*m3 + m1*m3)

## The Attempt at a Solution

I managed to solve a) using the above equation which I derived thru some lengthy algebra (5 equations involving 5 unknowns -- a1, a2, a3, Tension of Rope A, and Tension of Rope B), and got the following answers: a1 = -2.21m/s^2, a2 = 2.85m/s^2, and a3 = -0.316m/s^2. These answers match up with the answer key.

But I'm kind of stuck at part (b) -- even though I worked out mathematically in part (a) that box 3 does indeed accelerate, I'm unsure how to explain why it accelerates in a way that makes intuitive sense. There was no answer key for part (b), so I'd appreciate any helpful guidance. I've attached a picture illustration of the problem as well.

#### Attachments

• physics.jpg
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Were m1 and m2 equal, the system should stay still, since m1 is heavier it pulls on the cord, adding momentum to the smaller pulley system and breaking the equilibrium state between m1, m2 and m3. It has been a long day, though, I haven't run any numbers, just an intuitive guess.

Oh...thanks for posting lendav_rott. I think I might have gotten this. Just now, I imagined an extreme case in which m1 was massless, and m2 = m3. Well m2 would accelerate right down without any resistance, and due to this m3 would also accelerate right down. So, going back to this problem, m3 accelerates because m1 and m2 are not in equilibrium, so even though m1 + m2 = m3, the force pulling upward on m3 is less than m1 + m2. Does that sound right?

Newtons laws don't work very well in non inertial frames. As a thought experiment, consider a block m1 resting on another block m2. There is no friction between blocks m1 and m2 or between the floor and m2. Now apply a horizontal pull force T to the lower block, and an equal but opposite force T to the top block. While there is no acceleration of the center of mass of the system, the blocks each accelerate separately. They do not remain at rest.

Let me drop a coin in the hat as well: Hold B fixed for a moment.
The center of gravity of m1 and m2 is accelerating downwards. You can calculate that a, and it must be the resultant from g(m1+m2) - tension in AB . So tension in AB < gm3. Now let go of B: acceleration again. m1+m2 accelerated upwards so a new balance equation is needed - the a has to be adapted slightly. How much follows from the tension, which is now (g - a')m3. Sort of, if I didn't miss something.

Funny they ask a first, then b...

## 1. What is the concept behind three hanging masses and two pulleys?

The concept behind three hanging masses and two pulleys is that it demonstrates the principles of mechanical advantage and how forces are transferred through a system of pulleys. The masses are connected by a series of ropes and pulleys, and the movement of one mass will cause the other masses to move as well.

## 2. Why does m3 accelerate in this setup?

M3, or the third hanging mass, accelerates in this setup because of the forces acting on it. The first and second hanging masses exert a downward force on m3, and the pulleys redirect this force to accelerate m3 in the upward direction. This is an example of Newton's second law of motion, which states that an object will accelerate in the direction of the net force acting on it.

## 3. How does the number of pulleys affect the acceleration of m3?

The more pulleys there are in the system, the greater the mechanical advantage and the slower the acceleration of m3. This is because the pulleys reduce the amount of force needed to lift m3, but also increase the distance over which the force is applied. Therefore, the acceleration of m3 will be inversely proportional to the number of pulleys in the system.

## 4. What factors can affect the acceleration of m3 in this setup?

The acceleration of m3 can be affected by various factors, such as the mass of m3, the mass and position of the other hanging masses, the angle of the ropes, and the friction in the pulleys. These factors can alter the forces acting on m3 and therefore impact its acceleration.

## 5. How is this setup relevant to real-world applications?

This setup is relevant to real-world applications, particularly in areas such as engineering and physics. The principles demonstrated by this system can be applied to various machines that use pulleys, such as elevators, cranes, and even exercise equipment. Understanding the mechanics of this setup can also help in designing more efficient and effective systems in various industries.

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