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Pulleys

- Calculus
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It is not easy to lift heavy materials. This problem can be solved with the help of a simple machine like a pulley. A pulley comprises a wheel and a fixed axle, with a groove along the edges to guide a rope or a cable.

Pulleys are what we call simple machines. Without using an engine, they can help multiply the amount of force applied to lift an object. When you put two or more wheels together and run a rope around them, you have created a great lifting machine. The load is the amount of mass of the object to be moved, and effort is the force required to move the object.

As you increase the number of pulleys in a system, you would require more rope to pull the load across a distance. Hence, you will have to apply the force for a longer period of time when you have more pulleys.

As you add more pulleys, you increase your mechanical advantage and it becomes very easy to lift the same load. However, before we continue, let us define what mechanical advantage is, and be clear on the differences between weight and mass.

Weight is the amount of force with which gravity pulls an object towards the ground. It is measured in Newtons (N).

Mass is the measure of the amount of substance contained in an object which is measured in kilograms (kg).

Mechanical advantage is the ratio of the force that performs the useful work to the force applied. It is the measurement of how simple machines multiply forces.

If you had a sack with a mass of about 40kg, you can calculate the weight as follows:.

.

Earth's gravity in Newtons is approximately ten times the mass in kilograms.

A simple pulley helps you lift loads by reversing the direction of the load. If you pull the rope down, your load goes up, as shown in the diagram below.

By adding one more wheel to the system, you can now reduce the effort you would use to lift the same load using a one-wheel pulley system. If I had my sack with a mass of 40 kg and a weight of 400 N, this time the weight of the load will be split amongst the two pulleys. Therefore, you will need half the effort required to pull the load. Pulleys with more wheels give more mechanical advantage.

With the two-wheel system, when you pull 5 meters of the rope, you move the load only half the distance. That is 2.5 meters.

If four wheels were held together with a rope to form a pulley, it will be noticed that four sections of the weight will be hung over the wheels of the pulley. That will mean that our 400 N (40 kg) sack's weight will now be spread across each wheel, and we will need only a quarter of the force to move the load as much as we would have in the simple one-wheel pulley.

The mechanical advantage of this pulley system will be twice as good as it was in the two-wheel. However, to lift the load across a 5-meter distance, the rope will have to be pulled four times that distance. That is 20 meters.

Among simple machines there are three types of pulley systems:

Fixed pulleys: they are also called single spot. This type of pulley stays fixed while the load stays on one side of the wheel and the effort is applied on the other side. And because of this, you will need the same amount of force to lift the object.

Moveable pulleys: these are different from fixed pulleys because the pulley system here actually moves with the load. In contrast to the fixed pulley, this does not change the direction, rather, it multiples the force that is being exerted to the load. It is useful for heavier loads, and you won't need as much effort to lift the load.

Compound pulleys: These systems are a combination of both the fixed and the moveable pulley systems. It does not only multiply the force being applied to the load, it allows it to change direction too. They are the most convenient for lifting heavy loads.

Here are a few worked examples to find the acceleration of a load. We use SUVAT equations for some of these.

**Finding acceleration**

If these two particles in the diagram below are released from rest, what will the acceleration be?

Answer:

The particle with the highest mass will drop, and the particle with the lowest mass will rise. Let's take the particle of 5kg mass as particle a, and the 12kg mass as particle b.

To clarify the weight of each particle we have to multiply their mass with gravity. We therefore use g = 9.8 m / s².

Weight of a = 5g

Weight of b = 12g

Now you can model an equation for each particle's acceleration and tension.

T - 5kg **×** g = 5kg **×** a [Particle a] [Equation 1]

12kg **×** g - T = 12kg **×** a [Particle b] [Equation 2]

You now solve this simultaneously. Add both equations to eliminate the T variable.

7kg **×** g = 17kg **×** a

If you take g = 9.8 m / s².

**Investigating two possibilities**

Two particles of mass 8 kg and m kg are connected by a tight string passing over a smooth peg. Both particles hang vertically with one particle held at rest. The particle is released. Given that the acceleration is 5 m/s², find the mass m.

Answer:

Let us draw a diagram to suit the question.

Let us take the particle with the mass of 8kg as particle a, and the particle with unknown mass as b.

For all of this to work, the mass of particle b is either greater or less than particle a. This will determine which particle will accelerate. So we may have to investigate both possibilities.

So let's have a look at the situation where m > 8.

Resolving particle a vertically:

Resolving particle b vertically:

That would be the mass of particle b in a case where m > 8.

Let us now take the situation where m <8

8kg **×** g - D = 8kg ** ×** 5m/s²

T = 38.4N

Resolving particle b vertically:

m = 2.6 kg

We now have a mass for both scenarios. If the mass of particle a> b, b will accelerate upwards while a accelerates downwards. On the other hand, a <b will mean a will accelerate upwards while b accelerates downwards.

**SUVAT**

The diagram shows two particles that are connected by a light inextensible string. The 5kg particle a is on a rough horizontal surface and the 3 kg particle b hangs on the other end of the string. The string passes over a smooth light pulley. The initial speed of both particles isand a constant frictional force of 4N works against a. The particles slow down and stop before a reaches the pulley or b hits the fall. Find:

The acceleration of the particles.

The distance the particle travels before they come to rest.

Figure 7. Pulley example on SUVAT

Answer:

T -4g = 5a [Equation 1] [Particle a resolved horizontally]

3g - T = 3a [Equation 2] [Particle b resolved vertically]

Adding equations together

-g = 8a

Take g to be

s = x

u =

v = 0

a =

t =?

We will use the equation that doesn't have t since information on that isn't available.

- A pulley is a collection of wheels over which you loop a rope to make it easier to lift things.
- There are three types of pulley systems. Fixed, moveable, and compound pulleys.
- The more wheels you have in your system, the more mechanical advantage you have in lifting your load.
- By including a wheel in your one-wheel pulley system, you reduce the amount of force needed to lift the load, but you will have to move your rope twice the distance for the load to cover the distance it would have if it were a one-wheel system.
- SUVAT equations can be used to solve certain pulley problems.

Images

One wheel pulley https://commons.wikimedia.org/w/index.php?curid=643667

A pulley is a collection of wheels over which you loop a rope to make it easier to lift things.

Fixed pulleys

Moveable pulleys

Compound pulleys

A flag pole.

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