Work, Energy, and Simple Machines

Work is done by a force on an object, when the force displaces the object in the direction of the force.

Work Done by a Constant Force

The work done by a constant force acting on an object in bringing about a certain displacement can be defined as:

work done on an object by a constant force = force applied × displacement in the direction of the force

Consider an object upon which a constant force F is acting, and it undergoes a displacement s in the direction of force. Then, the work done W by the force on the object is

Work done = force × displacement

$$ W = F s $$

The SI unit of work done is joule which is represented by J. The SI unit of force is the newton (N) and the SI unit of displacement is the metre (m). Thus, 1 joule can be defined as the work done on an object when a constant force of 1 newton is applied to it and it is displaced by 1 metre in the direction of the force.

1 J = 1 kg m s^-2 × 1 m = 1 kg m² s^-2

In the graph, the force on an object is plotted on the y-axis against the displacement in the direction of force on the x-axis. In this case, the work done on the object by the force is equal to the area of the shaded rectangle in the graph.

When is work done equal to zero?

If the force acting on an object is zero, i.e., F = 0, then no work is done on the object. The work done on an object is also zero if there is no displacement of the object, i.e., s = 0, regardless of the force being applied on it.

Positive and negative work done

The work done by a force on an object can either be positive or negative depending upon the relative directions of the force and the displacement. When the displacement is in the same direction as the applied force, the work done by the force on the object is said to be positive.

When the displacement is in the direction opposite to that of the force, then the work done by the force on the object is negative.

The Work-Energy Theorem

An object having the capacity to do work is said to possess energy.

When positive work is done on an object, it gains energy. Subsequently, the object can use that energy to apply a force on another object and cause it to move, thereby transferring energy. Thus, work done on an object and its energy are closely related to each other.

Work done on an object appears as a change in its energy. The relation between the work done on an object and the change in its energy is called the work-energy theorem, which can be stated as

work done on an object = change in its energy

The SI unit of energy is the same as the SI unit of work, the joule (J).

Forms of Energy

Energy can exist in many forms. It is possible to change energy from one form to another. For example, electrical energy is converted into light energy in a bulb, and into thermal energy of the water in an electric water heater. The chemical energy in the food we eat powers our muscles and gets converted into mechanical energy. Similarly, a ringing bell converts mechanical energy into sound energy.

Kinetic Energy

The energy possessed by an object due to its motion is called kinetic energy. All moving objects possess kinetic energy, such as a moving bicycle or a rolling ball.

Assume that the force F acting on the object is constant. In that case, the acceleration a is also constant, and the kinematic equations can be used.

$$ v^2 = u^2 + 2as $$

$$ s = \frac{v^2 - u^2}{2a} $$

$$ F = ma $$

$$ W = F \times s $$

$$ W = ma \times s $$

$$ W = (ma) \times \left( \frac{v^2 - u^2}{2a} \right) $$

$$ W = \frac{1}{2}m (v^2 - u^2) $$

If the initial velocity of the object is u = 0, the change in the energy of the object is equal to its final kinetic energy.

$$ W = \frac{1}{2}m v^2 $$

$$ K = \frac{1}{2}m v^2 $$

The SI unit of kinetic energy is the joule (J). The kinetic energy has no direction. When positive work is done on the object and its velocity increases, its kinetic energy also increases. If negative work is done on the object and its velocity decreases, its kinetic energy will also decrease. If no work is done by the force on the object (W = 0) and its velocity does not change, then its kinetic energy remains constant.

Potential Energy

The energy stored by an object as a result of its deformation or in a system of objects due to their relative positions is called the potential energy.

Consider an object of mass m lying on the ground. We can define the potential energy to be zero in this configuration. To raise the object gradually from the surface of the Earth to a height h, we need to apply a force equal to the force mg between the Earth and the object. The work done W by this applied force on the object is

W = force × displacement

$$ W = mg \times h = mgh $$

Thus, the potential energy U of the object at a height h is given by

$$ U = mgh $$

The unit of potential energy is the joule (J), which is the same as the unit of work done or that of kinetic energy.

Conservation of Mechanical Energy

The sum of the kinetic energy and the potential energy of the object is called its mechanical energy.

As an object moves due to the gravitational force, its mechanical energy remains the same, i.e., the mechanical energy of the object is conserved if no other external forces act on it. This is called the conservation of mechanical energy.

Power

Power is defined as the rate at which work is done. Mathematically, the average power P is the work done W, divided by the time taken t.

$$ \text{Power} = \frac{\text{Work}}{\text{Time}} $$

$$ P = \frac{W}{t} $$

To do more work in the same time interval, requires more power. To do the same amount of work but in a shorter time interval also requires more power. The SI unit of power is the watt (W), where 1 watt is equal to 1 joule of work done per second, i.e., 1 W = 1 J s^-1.

Simple Machines

In everyday life, we often need to do work against gravity or other forces, such as lifting or moving heavy objects. Although the total work required for a task cannot be reduced, it can be made easier by changing the magnitude or direction of the force that needs to be applied. The devices that help us do this are called simple machines.

The force we apply to a machine is called the effort, and force that needs to be overcome is called the load. To describe how a machine changes the magnitude of the applied force, we define mechanical advantage as the ratio of the load to the effort.

$$ \text{Mechanical Advantage} = \frac{\text{Load}}{\text{Effort}} $$

Pulley

A pulley is a wheel with a groove that guides a rope. A fixed pulley does not reduce the magnitude of the force required, it only changes its direction. It is easier for us to pull downward than to lift a load by applying an upward force directly. Thus, the pulley provides convenience by changing the direction of the effort. Since the effort and the load are equal in magnitude, the mechanical advantage of a fixed pulley is 1.

Inclined Plane

An inclined plane is a simple machine that helps move a heavy load to a higher (or lower) level.

Lever

A lever is a rigid bar that can rotate about a fixed point. Levers are often used to lift heavy objects. A lever has three main parts:

Fulcrum: a fixed point about which the lever rotates
Load: the force to be overcome
Effort: the force applied

The distance of the load from the fulcrum is called the load arm, and the distance of the effort from the fulcrum is called the effort arm.