Work Energy Equation of Particles

5.11.1. Work and Energy. Work is defined as the product of force and displacement of the body on which force is acting. The force and displacement should be in the same direction. Energy is defined as the capacity to do work. The energy exists in many forms-like, mechanical, electrical, heat, chemical and light etc. But in engineering mechanics, we only consider mechanical energy. This article deals with work and energy.

Work. As defined above, work is the product of force and distance. The distance should be in the direction of the force. If a force P is acting on a body and the body moves a distance of S in the direction of the force, as shown in Fig. 5.49 (a), then the work done on the body is given by:

Problem 5.83.A chain of length 50.5 m is placed on a pulley whose radius is 16 cm, One end of the chain hangs down 40 In below the point it leaves the pulley. A man holds the other end or the chain and pulls it down until the length of the chain on both sides of the pulley is equal. Find the work done by the man. Given the weight of the chain as 10 N per metre length and the pulley is frictionless.

Sol. Given:

Total length of chain, L = 50.5 m

Radius of pulley, R = 16 cm = 0,16 m

Length of chain between A and C = 40 m

Weight of chain = 10 N per m length

= 10 × Total length of chain 0

= 10 × 50.5 = 505 N

Length BD = Total length of chain – Length AB – Length AC

5.11.5. Energy. The capacity of doing work is known as energy. It is the product of
power and time. The energy is expressed in Nm. It exists in many forms i.e., mechanical, electrical, heat, chemical, light etc. In engineering mechanics, we are only concerned with mechanical energy and the same will be dealt with.

Mechanical energy consists of the following two types:

1. Potential energy (or position energy or datum energy)
2. Kinetic energy.

Depending upon the state of rest or that of motion, a body may possess anyone or both of the above energies.

1.   Potential energy. Potential energy is also known as position energy or datum energy. It is the energy by virtue of position of a body with respect to any given reference or datum. A weight W lying on the top of a tower of height h possesses a potential energy of W × h with respect to the ground, as the weight W is capable of doing W × h work if it falls on the ground.
A compressed spring has potential energy, because it can do work in, recovering its original shape. Similarly, compressed air also possesses potential energy because it is capable of doing work whim ail owed to expand.

2.   Kinetic energy. The energy possessed by a body by virtue of its velocity (or its motion) is known as kinetic energy. It is represented by KE.

The expression for the kinetic energy is derived as follows: Consider a body of mass m starting from rest. Let it be subjected to an accelerating force F and after covering a distance S, its velocity becomes v.

Initial velocity, u = 0

Now work done on the body = Force × Distance

= F × S                                                                                … (i)

But Force = Mass × Acceleration

F=m × a

Substituting the value of F in equation (i),

Work done = m × a × S

= m × (a × S)                                                                                   … (ii)

But from equation of motion, we have

v2 – u2 = 2a × S            or                 v2 – 02= 2a × S                                (u = 0)