###### TM

## 1.3 Moments

In the previous chapter we learned about forces. We learnt how to add forces together to get a resultant force. However, in the examples we worked out, the forces never caused the objects to rotate about a fixed point. The ability of a force to cause an object to rotate is known as its moment.

In this chapter we are going to look the different factors that can affect the moment of a force.

### Principle of Moments

If we look at a windmill we can see that even though the wind is applying a force on the blades, they don't move along a straight line. Instead they keep moving in circles while still being in the same spot. Why is this? Its because it has a fixed point of rotation. Which basically means that the point is unmovable. The position of this point greatly affects the moment of a force.

The windmill above has its point of rotation right in the middle. This means that each of the four blades are equidistant from the point of rotation. If we assume that the force of the wind acts on the tips of each blade then it makes sense that each blade will be moving at the same speed. Therefore, we can say that all the blades are balanced when rotating.

Now imagine that the point of rotation is moved away from the center by shortening one of the blades. If the wind applies the same force on the tips of each blade, will the rotation be balanced? No. It will definitely rotate but the shorter blade will require a greater force to rotate at the same pace as the other blades. If the system is allowed to function like this, at a certain speed the whole windmill could break apart because its not balanced. In other words the moment of the shorter blade is smaller than the other blades.

So we know that the force affects the moment. But the system above had the same force acting on all blades. Then what makes the shorter blade have a smaller moment? Its the shorter length of the blade relative to the center of rotation that makes the shorter blade have a lower moment. From this we can come up with a equation for moments:

Is the above equation right? Well we are going to have to tweak it a bit. The direction of the force is very important when we are taking about moments. For example if the force was acting along the blade then the blade will not rotate. Which means its moment will be zero. Therefore to calculate moment we must always take the perpendicular distance from the direction of action of the force to the axis of rotation.

When we take into account the perpendicular distance we get the following equation. The units of moments are newton meters [Nm] as shown below:

So now that we know what moments are let's look at the principle of moments. Usually there is more than one force acting on an object, which means that it could have multiple moments. These moments cold be acting in opposite directions. The principle of moments states that in such situations, if the clockwise moments are equal to the anti-clockwise moments then the system will be in rotational equilibrium. In other words the system will remain stationary or if it is already rotating then it will keep turning at a fixed speed in the same direction.

Lets look at an example to better understand this.

Example 1

Assume you have a large platform being supported on a pivot. There is a car weighing 2 tons, a train engine weighing 150 tons, a truck weighing 6 tons and a bus weighing 10 tons. The lorry is 20 m from the pivot. The car 45 m from the pivot. The train is 10 m from the pivot. How far away should the bus be placed in order to have a stationary platform?

We have a total of 4 forces acting on the platform. Each of this forces has a moment because the platform is on a fulcrum, which means any of those forces could cause the platform to rotate clockwise or anti-clockwise. However, we need the platform to be stationary. That means all the moments must balance out, in other words we will have to use the principle of moments. Alright so let's set up our equation:

The forces will be the weights of each of the vehicles:

Now we juts have to plugin in the values inot our roiginal equation to calculate x4.

Therefore the bus has to be 147 m from the pivot for the platform to be stationary.

### Center of Gravity

In the previous example above we were balancing objects on a plank. In this case we were assuming that the weights being exerted on the platform by the vehicles were being generated from a single point on the vehicle. But we all know that the mass of the object is distributed through out itself. Then how can weight only be acting on one point? Well let's take a broom as an example.

If we place a broom on a finger and try to balance it can we do it? Yes, if we place the right point of the broom on the finger we can. But if we don't then the will rotate and fall off. Why is this? This is becasue of th center of gravity. The center of gravity is the single point on any object that gravity acts on. That means that the weight of the object will act on this point. If we place this point of the broom on our finger then the broom is balanced because the moment is zero since the force is acting on the pivot.

However if we place the broom on our finger such that its center if mass is to the right of our finger, the broom will rotate clockwise because the force has a moment now.

All objects have a center of mass and therefore have a center of gravity. The center of gravity is almost always the same as the center of mass. They differ when the gravitational pull is not uniform. This means that gravity is pulling part of the object stronger than the other part. However, since we are assuming that the gravitational field is uniform we don't have to worry about it.

All objects have a center of mass and therefore have a center of gravity. The center of gravity is almost always the same as the center of mass. They differ when the gravitational pull is not uniform. This means that gravity is pulling part of the object stronger than the other part. However, since we are assuming that the gravitational field is uniform we don't have to worry about it.

For a symmetrically uniform object the center of mass lies on every line of symmetry. Since these objects are uniform, the center of gravity is usually in the center of the object. What about uniform 3-D objects?

Centre of gravity

The same conditions apply. For example, if we are dealing with a sphere then its center of gravity will be in the center as shown below:

So finding the center of gravity for symmetrical objects is easy. How about irregular objects? How do we deal with them?

As we discussed earlier the center of gravity is the point from which gravity pulls on the object. Gravity is pulling on the mass of the object. Therefore if the irregular object has an area where its mass is concentrated more than on the other areas then it is very likely the center of gravity lies somewhere around there. For example if we have a vase with a thin neck and large bottom then we can safely say that its center of gravity is most certainly towards the bottom. But can we find the exact position?

The answer depends on the object. If it is an object that is not too big or heavy we perform a simple experiment to determine the center of gravity. First we freely hang the object from a point on its edge. Then we hang a pendulum from that point and draw the line of the pendulum on the object. Then we take another point on the object and hang it and repeat the procedure. The point of intersection of the two drawn lines should be the center of gravity.

But how can we deal with very large, irregular non-uniform objects. This would be difficult to do because we will have to use calculus and various other methods. We'll discuss this later. But it is important to note that an irregular object can have more than one center of gravity.