# Definition of Work

**Work is said to be done when a force acting on a body displaces it.**

**Consider two porters each carrying a load of 10 kg on their head. Let the first porter (A) climb up a staircase with 20 steps. Let the second porter (B) climb only 10 steps. It is clear that the first porter (A) has done more work than the second porter (B). Now, let us consider another instance, where the second porter (B) carries half the load as the first porter (A). Let both climb up 20 steps. In this case, it is clear that the first porter (A) has done more work than the second porter (B) (Figure 6.1).**

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**Figure 6.1 Porters Carrying Load on Their Head**

Therefore, in general, we can say that work done by a force increases with

- the magnitude of the force and
- the displacement produced by it.

# Dot Product of Vectors

Both force and displacement are vector quantities. But work is a scalar quantity since it is done at the expense of energy which is a scalar quantity.

The product of two vectors can either be a scalar or a vector. If the product is a scalar, it is called scalar product or dot product of two vectors; if the product is a vector, it is called vector product or cross-product of two vectors.

Let be two vectors represented by , respectively, with

The product of two vectors can either be a scalar or a vector. If the product is a scalar, it is called scalar product or dot product of two vectors; if the product is a vector, it is called vector product or cross-product of two vectors.

Let be two vectors represented by , respectively, with

*Î¸**as the angle between them (see Figure 6.2). Then, the dot product of is given by*â€‹

**Figure 6.2 Dot Product of Vectors**

When a constant force represented by acting on a body displaces it through a distance represented by , the work done

*W*on the body is given by **Figure 6.3 Work as a Dot Product**

The displacement caused by the force need not be in the direction of . Work done is the product of the magnitude of and the magnitude of the component of along .

From the above mathematical expression for work, we arrive at the following important points:

From the above mathematical expression for work, we arrive at the following important points:

- The work done is directly proportional to the magnitude of as well as to the magnitude of . Hence, if the magnitude of either or both the quantities increase(s), work done increases.
- The work done also depends on the direction of the displacement with respect to the direction of the force.

**Following are the three special cases depending on the angle between**

**and**

**.**

**Case I: Displacement in the direction of the force.**

Let us consider a body falling freely under the action of gravity. In this case, the displacement is in the direction of the force. Similarly, if we push a book along a table, the displacement of the book is along the direction of the force.

When the displacement is in the direction of the force, the angle between the two vectors, i.e., force and displacement is zero. Therefore,

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*W* = *Fs*. This is the maximum value of work done. Thus,

*W* = *Fs* > 0

**Case II: Displacement in a direction perpendicular to the force.**

Consider a porter carrying load on his shoulders and walking on a horizontal platform. He exerts a vertically upward force on the load. But the displacement of the load is in the horizontal direction. In this case, the displacement is in a direction perpendicular to the force

**Case III: Displacement in a direction opposite to the force.**

When we throw a ball in the vertically upward direction, the work done by us on the ball is positive as the displacement of the ball is in the direction of the force exerted by us. But, the force of gravity due to the earth is in the downward direction. Therefore, the work done by the force of gravity is negative as the displacement is in the opposite direction to the force of gravity. Therefore,

Work done by gravity = (â€“Force) Ã— Displacement

In the following cases work done is zero.

- A man pushing a wall. Here, the displacement of the wall is zero even though a force is exerted by the man.
- A body executing uniform motion along a circular path.

When the body moves along a circular path, its displacement along the radius which is the direction of the force on the body is equal to zero and the motion of the body is along the tangential direction. Therefore, the angle between the radius and the tangent is 90Â°. Therefore, the work done is equal to zero.

# Units of Work

Work done is the dot product of force and displacement.

- SI units of force and displacement are
**newton (N)**and**metre (m)**, respectively. Therefore, the unit of work is**newton metre (Nm)**. This unit is called joule (J) in honour of the British scientist James Prescott Joule.*Work done is one joule if a force of 1 newton acting on a body displaces it by 1 metre in the direction of the force.* - The CGS unit of work is
**erg**.*Work done is one erg if a force of 1 dyne acting on a body displaces it by 1 centimetre in the direction of force.**The joule and the erg are the absolute units of work.*

- 1 joule (J) = 1 Nm.
- 1 kilo joule (kJ) = 10
^{3}J - 1 mega joule (MJ) = 10
^{6}J - 1 erg = 1 dyn cm
- 1 joule = 10
^{7}erg