Equilibrium (3.2.3)

Imagine you’re using a wrench to tighten a bolt. To maximise the tightening effect, where on the wrench should you apply force?

The answer is at the end of the wrench. The reason behind this observation lies in what is called the moment of a force (torque).

By definition, the moment of a force (torque) is the turning effect of a force about some axis or pivot. The SI unit of the moment of a force is expressed in newton metres

An axis of rotation is simply an imaginary line about which a body rotates. In our example, the axis of rotation passes vertically through the centre of the bolt.

The moment of a force can be expressed as:

Where:

• is the magnitude of the force in newtons , and
• is the perpendicular distance from the line of action of the force to the axis of rotation (pivot) in metres .

The SI unit of the moment of a force is expressed in newton metres

The line of action of a force, as shown in the figure below, is an imaginary line that extends infinitely in both directions along the direction of the force vector.

When dealing with moments of force, we have three cases:

1. Perpendicular line of action
   In this case (diagram A), the line of action of the force is perpendicular to the axis of rotation. The moment of a force is easily calculated by multiplying the magnitude of the force by the line connecting the axis of rotation and the point of application of the force.
2. Non-perpendicular line of action
   Here, the force’s line of action is non-perpendicular (diagram B). Given the angle that the force makes with the horizontal, the perpendicular distance between the line of action and the axis of rotation should be calculated using trigonometric relations.
3. Line of action through the axis of rotation
   In the third case, the line of action of the force passes directly through the axis of rotation (diagram C). Consequently, the moment of the force is equal to zero, as there is no distance to create a turning effect.

It is useful to note that drawing all the forces acting on the object in question can help you see which forces are perpendicular to the distance from the pivot. Not all the forces will provide a turning effect, and it is not unusual for a question to provide more forces than required to mislead you.

How can you make any free object spin without causing any translational motion? The trick is to apply a pair of opposite but equal forces to the object, as shown in the figure below. This pair of forces is called a couple. The couple should always be parallel or coplanar, have different lines of action, and be perpendicular to the distance between the pair of forces.

The moment of the couple is known as a torque. The magnitude of a torque can be calculated using the expression:

where:

• is the magnitude of one of the forces, and
• is the perpendicular distance between the two forces.

Some tips to keep in mind while dealing with a force couple:

• The resultant force of a couple is zero.
• Unlike moments of single forces, a couple does not need a pivot.

The principle of moments states that for a body in equilibrium, the sum of clockwise moments about any point must equal the sum of counterclockwise moments about that same point.

For a body to remain in equilibrium, the total moment generated by all the forces causing clockwise rotation must balance the total moment generated by all the forces causing counterclockwise rotation:

Consider a rod that can rotate around a pivot , as illustrated in the figure above.

• Force produces a CW rotation.
• Force produces a CCW rotation.
• Force produces a CW rotation.
• Force produces a CCW rotation.

The rod is in equilibrium as the total moment around the pivot is balanced.

It is hard to study the motion of an irregular rigid body when a force is applied to it. In physics, to simplify calculations, we use a concept called the centre of mass (COM) to analyse the translational motion of an object. The centre of mass of an object is the point at which the mass of the object can be considered to be concentrated.

The centre of mass of an object:

• does not depend on the orientation of the body
• can be located outside or inside the physical boundaries of a body
• does not depend on the gravitational field strength.

The centre of mass of simple geometric shapes is located at their geometric centres and can be found through symmetry. The centres of mass of a circle, a box, a triangle, and a doughnut are represented below by a red dot.

To determine the centre of mass of any regular shape:

1. Start by dividing it into two equal parts using a dashed line.
2. Next, divide the same shape into equal parts using a different dashed line.
3. The intersection of the two dashed lines indicates the centre of mass.

The centre of gravity (COG) is the point through which an object’s entire weight is considered to act, the resultant of the small weight forces distributed throughout the object. Because moments depend on the location of a force’s line of action, this is the point used when calculating the turning effect of weight in equilibrium and rotational problems.

On a free-body diagram, an object’s weight is drawn as a single vector acting downward from its centre of gravity, rather than as multiple forces spread across the object.

The centre of gravity of an object has the following properties:

• It can be located outside or inside the physical boundaries of a body.
• It can change in a non-uniform gravitational field.

In cases where the gravitational field strength can be considered uniform, such as a small object near the surface of Earth, a body’s centre of mass coincides with its centre of gravity.

However, in cases involving a non-uniform gravitational field, such as a large celestial object in orbit of a massive body, the centre of mass (COM) is not in the same location as the centre of gravity (COG):

• Image A: the triangle’s COM is identical to its COG because the gravitational field acting on it is uniform.

• Image B: the Moon’s COM is at its geometric centre. However, the COG is closer to the Earth because the Moon’s gravitational field is significantly stronger on the side facing the Earth.

The location of the centre of mass of a body affects its stability. An object is:

• Stable if its centre of mass lies vertically above its base (image A).
• Unstable and will topple if its centre of mass lies vertically outside its base (image B).

Widening an object’s base or lowering its centre of mass increases its stability.

To determine the centre of gravity of an irregular-shaped object, complete the following steps:

1. Hang the body from a pivot near its edge and allow it to settle.
2. Use a market to draw a vertical line passing through the pivot.
3. Hang the body from a different pivot and allow it to settle.
4. Draw another vertical line passing through the new pivot.
5. The intersection of the two lines is the centre of gravity.

Since you are applying the procedure to a small object on the surface of the Earth, its centre of gravity is the same as its centre of mass.

How to set up a triangle of force is shown in the illustration below, where a box rests in equilibrium on a slope.

1. Start by drawing the free-body diagram of the forces acting on the body.
2. Select any vector and draw it separately.
3. From the tip of that vector, start the tail of a second vector.
4. From the tip of the second vector, start the tail of the third vector.
5. Write down all the known angles formed between the vectors.
6. In most cases, the triangle of forces will be a right triangle.
7. The most common forces are weight, tension, normal reaction, and friction.