# Mechanics question involving pulleys: where is the problem?

• Anonymouse
In summary, the system has two tensions in the strings- one with a mass of 4kg and one with a mass of 2kg- and two accelerations: the mass of the 4kg string is being accelerated upwards by 34.29N, and the mass of the 2kg string is being accelerated downwards by 15.24N.

#### Anonymouse

Question

A string is hung over a fixed pulley, and a 4kg mass is suspended from one end of the string. The other end of the string supports a massless pulley, over which a second string is hung. This string has a 2kg mass attached to one end, and a 1kg mass attached to the other end. See the diagram below:
View attachment pulleys2.bmp
If the system is released from rest, what are the tensions T1 and T2 in the strings? What are the accelerations of each mass, aA, aB and aC?

Solution

Firstly, let's define the upwards direction as the positive direction. Take the gravitational constant g to be 10, so the weight of mass m is given by 10m. Then if we apply Newton's 2nd law (F=ma where F is force, m is mass and a is acceleration) to each of the masses, we get three equations:

For mass A: T1 - 40 = 4aA

For mass B: T2 - 20 = 2aB

For mass C: T2 - 10 = 1aC

Now it can also be shown that 2aA + aB + aC = 0.

Now we have four equations in total, but five unknown quantities. How can we solve the problem?

I guess we could also consider masses B and C as a composite, and apply Newton's 2nd law again to arrive at the following equation:

T1 - 30 = -3aA

In this equation, there is a minus sign on the right-hand side because the acceleration of the composite of B and C must be of equal magnitude but opposite direction to that of A, in other words -aA, right?

So now we have five equation with five unknowns, which can be solved to give the following answers:

T1 = 240/7 = 34.29 N

T2 = 320/21 = 15.24 N

aA = -10/7 = -1.43 m/s²

aB = -50/21 = -2.38 m/s²

aC = 110/21 = 5.24 m/s²

Is this solution correct?

Problem

If we consider just the massless pulley, it has an upward force of T1 on it, and a downward force of 2T2.

In principle we could apply Newton's 2nd law, but since mass is zero, the net force must also be zero in order to avoid getting an infinite acceleration.

So if the net force is zero, then T1 = 2T2, but this is clearly not the case if the values calculated above are correct.

So, how can this discrepancy be accounted for?

Anonymouse said:
I guess we could also consider masses B and C as a composite, and apply Newton's 2nd law again to arrive at the following equation:

T1 - 30 = -3aA

The acceleration of the "composite mass" is not -aA. The acceleration of the massless pulley is -aA but the CG of the "composite mass" has an acceleration relative to the pulley.

Your other equation for the equilibrium of the massless pulley, T1 = 2T2, is correct.

The acceleration of the "composite mass" is not -aA. The acceleration of the massless pulley is -aA but the CG of the "composite mass" has an acceleration relative to the pulley.

Yes, that makes sense.

So the acceleration of the centre of mass of the BC composite aBC is given by 3aBC = 2aB + aC (this is just saying the rate of change of momentum of the composite is equal to sum of the rates of change of momentum of B and C separately).

So we can use Newton's 2nd law for the composite:

T1 - 30 = 3aBC = 2aB + aC

So now using the five equations, we can solve and get the following values:

T1 = 32 N

T2 = 16 N

aA = -2 m/s²

aB = -2 m/s²

aC = 6 m/s²

As can be seen, now T1 = 2T2 as the case should be.

## 1. What is the purpose of pulleys in a mechanical system?

Pulleys are used to change the direction of a force and to provide mechanical advantage in a system. They allow for the lifting of heavy objects with less effort by distributing the weight over multiple ropes or cables.

## 2. How do you calculate the mechanical advantage of a pulley system?

The mechanical advantage of a pulley system is calculated by dividing the output force (the weight being lifted) by the input force (the force applied to the rope or cable). This can also be expressed as the number of ropes supporting the weight in the system.

## 3. What are the different types of pulleys used in mechanics?

There are three main types of pulleys used in mechanics: fixed, movable, and compound. Fixed pulleys are attached to a structure and only change the direction of the force. Movable pulleys are attached to the object being lifted and provide mechanical advantage. Compound pulleys combine fixed and movable pulleys for even greater mechanical advantage.

## 4. How does friction affect the efficiency of a pulley system?

Friction can decrease the efficiency of a pulley system by creating resistance and reducing the amount of force that is transferred from the input to the output. To minimize friction, pulleys should be well lubricated and the ropes or cables should be in good condition.

## 5. What are some real-life applications of pulleys in mechanics?

Pulleys are used in a variety of real-life applications, including elevators, cranes, sailboats, exercise equipment, and even simple machines like window blinds. They are also commonly used in transportation systems, such as ski lifts and zip lines. Pulleys are essential in many industries for lifting and moving heavy objects efficiently and safely.