John D. Cutnell &Kenneth W. Johnson

6. WORK AND ENERGY

6.1 Work Done by a Constant Force

To move a body work must be done on the body.

The product of force and the distance moved (s) in the direction of force (F) is defined as the work done (W).

W = F.s

Work done is a scalar quantity.

SI unit of work done: joule (J)

Point

When the force and the direction of motion are inclined, the work done is calculated by the force component in the direction of motion and the distance moved.

6.2 The Work-Energy theorem and Kinetic Energy

The Work done on a body will be converted in to energy of the body if there is no energy loss.

Definition of Kinetic Energy

The kinetic energy (KE) of an object with mass (m) and speed (v) is given by

SI unit of kinetic energy: joule (J)

Work-Energy theorem

When a net external force does work W on an object, the kinetic energy of the object changes from initial value of KE0 to a final value of KEf, the difference between the two values being equal to the work:

Vf – final speed of the object

V0 – initial speed of the object

6.3 Gravitational Potential Energy

Work done by the force of gravity

The gravitational force is a well known force that can do positive or negative work.

Gravitational potential energy

Gravitational potential energy is the energy of an object due to its position in a gravitational field. If an object of mass (m) is at a height (h) from a reference level in a gravitational field and the gravitation acceleration (g) then the gravitation potential energy (PE) is given by

PE = mgh

SI unit of gravitational potential energy: joule (J)

Work done by the force of gravity on an object of mass m is given by the equation below, where h0 and hf are the initial and final heights of the object, respectively.

6.4 Conservative versus Non-conservative Forces

The gravitational force has an interesting property that when an object is moved from one place to another, the work done by the gravitational force does not depend on the choice of path.

Definition of a conservative forces

Version 1 – A force is conservative when the work it does on a moving object is independent of the path between object’s initial and final positions.

Version 2 – A force is conservative when it does no net work on an object moving around a closed path, starting and finishing at the same point.

Some conservative forces

Gravitational force

Elastic spring force

Electric force

Non-conservative force

A force is non conservative if the work it does on an object moving between two points depends on the path of the motion between points.

For a closed path, the total work done by a non conservative force is not zero.

Some non conservative forces

Static and kinetic frictional forces

Air resistance

Tension

Normal force

Propulsion force of a rocket

If conservative forces and non conservative forces act simultaneously on an object, we can write work W done by the net external forces as

W = Wc +Wnc

Where Wc is the work done by conservative forces and Wnc is the work done by the non conservative forces.

6.5 The Conservation of Mechanical Energy

The concept of work and work-energy theorem has led us to the conclusion that an object can possess two kinds of energy: kinetic energy (KE) and gravitational potential energy (PE). The sum of these energies is called the total mechanical energy (E).

E = KE + PE

The work-energy theorem can be expressed in an alternate form as shown