Work and Energy

Work and Energy Work and Energy

WORK

Work done on an object is defined as the magnitude of the force multiplied by the distance moved by the object in the direction of the applied force.

We define work to be equal to the product of the force and displacement.
Work done = force × displacement
W = F× s

The unit of work is joule:
1 joule = 1 newton × 1 metre.

Work has only magnitude and no direction.

  • when force and the displacement are in the same direction,work done by the force is taken as positive, then work done by the force is =F× s
  • when force, F, is applied in the opposite direction the work done by the force, F is taken as negative and denoted by the  minus sign. The work done by the force is
    F × (–s) or (–F × s)
  • when either Force or displacement is zero then, Work done is zero. also if Force and displacement act on angle 90 then also work done is zero.

ENERGY

Energy is defined as the ability to do work.

The energy possessed by an object is thus measured in terms of its capacity of doing work.

  • The unit of energy is the same as that of work, that is, joule (J). Larger unit is kilojoule (KJ).
  • 1 J is the energy required to do 1 joule of work.

FORMS OF ENERGY

The world we live in provides energy in many different forms. The various forms include

  • mechanical energy (potential energy + kinetic energy),
  • heat energy,
  • chemical energy
  •  electrical energy and
  • light energy.

KINETIC ENERGY

Kinetic energy is the energy possessed by an object due to its motion. Anything moving is said to have kinetic Energy.

The kinetic energy of an object increases with its speed.

We say that the kinetic energy of a body moving with a certain velocity is equal to the work done on it to make it acquire that velocity.

W=F×s,  where F= ma, and s=v2-u2/2a

u=0 initially as object starts from a stationary position

thus we have W=(1/2)mv2 .  and work done is equal to the change in the kinetic energy

so K.E=(1/2)mv2 .

POTENTIAL ENERGY

The energy possessed by a body due to its change in position or shape is called the potential energy.

There are two common forms of potential energy, gravitational and elastic.

Gravitational Potential Energy :When an object is allowed to fall from one level to a lower level it gains speed due. to gravitational pull, i.e. it gains kinetic energy. Therefore, in possessing height, a body has the ability to convert its height into kinetic energy, i.e. it” possesses potential energy.

If a mass m is at a height h above a lower level ,work done on the object against gravity be  W. That is,
work done,( W )= force × displacement
= mg × h
= mgh

an energy equal to mgh units is gained by the object. This is the potential energy (P.E) of the object

P.E=mgh

Elastic Potential energy : Same work has to be done to change the shape of a body. This work gets stored in the deformed body in the form of elastic potential energy. Elastic potential energy is never negative whether due to extension or to compression.

LAW OF CONSERVATION OF ENERGY

Whenever energy gets transformed, the total energy remains unchanged.

According to this law, energy can only be converted from one form to another: it can neither be created or destroyed.

The total energy before and after the transformation remains the same.

  • The sum of kinetic energy and potential energy of an object is its total mechanical energy(M.E).
  • M.E=K.E+P.E

RATE OF DOING WORK

The time rate of doing work is defined as power (P). More quickly work is done; power will be more.

Power= work/time,

P=W/t

  • The unit of power is watt [in honour of James Watt (1736 – 1819)] having the symbol W
  • 1 watt = 1 joule/second

The power of an agent may vary with time. This means that the agent may be doing work at different rates at different intervals of time. Therefore the concept of average power is useful. We obtain average power by dividing
the total energy consumed by the total time taken.

Q1)The potential energy of a body is 39600J. How high is the body if its mass is 20kg?

The potential energy of a body = mgh h = PE/mg = 39600j/20kg x9.8m/s2=198m

Q2)A force of 20 N displaces a body through a distance of 1 m at an angle of 60° from its own direction. Calculate the amount of work done.
Ans. Here, force F = 20 N, displacement, s = 1 m. Angle between force and displacement 60°.
Work done,W =Fscosθ =20 X 1 X cos60°=20X 1 X 1/2 = 10J.
A man of 50 kg jumps up to a height of 1.2 m. What is his potential energy at the highest point?
The potential energy of man = mgh = = 50 X 10 X 1.2 J = 600 J

Q3)How much work is done by a force of 10 N in moving an object through a distance of 4 m in the direction of the force.
Ans. Work done Force x Displacement =F.s = (10 N) x (4 m) = 40 joule or 40J.

Q4)A rocket is moving up with a velocity v. If the velocity of this rocket is suddenly tripled, what will be the ratio of two kinetic energies?
Ans. Initial KE/Final KE = ( ½ mu2) /( ½ mv2) = ( ½ mu2) /{ ½ m(3v2)} =1:9

Q5)Calculate the work done in lifting 200 kg of water through a vertical height of 6 m.
Ans. Work done in lifting a body = Weight of body X vertical distance
The work done in lifting  = W = mgh = 200 kg x 10m/s2 x6 m = 1200J

Q6) How much force is applied on the body when 150 joule of work is done in displacing the body through a distance of 10m in the direction of force?(15 N)

Solution: W = F x S ⇒ F = W/S = 150/10 = 15 N

Q7)A force of 5N acting on body at angle of 30 deg. with the horizontal direction displace it horizontally through of distance of 6 m . Calculate the work done. (15√3 J)

Solution: w = F S cos θ = 5 x 6 x √3/2 = 15√3 J

Q8)A moving body of 30kg has 60 J of KE. Calculate the speed.

Solution: KE = 0.5 mv² = 0.5 x 30 x v²
15 x v² =60

v =2m/sec

Q9) A hammer of mass 1kg falls freely from a height of 2 m .Calculate

(I) The velocity and

(II) The KE. of the hammer just before it touches the ground.

Does the velocity of hammer depend on the mass of hammer? (6.29m-2 , 19.6 J )

Solution: PE = mgh = 1 x 9.8 x 2 = 19.6 j

PE = KE = 0.5 mv² = 19.6

mv²= 39.2

v² =39.2

v =6.26 m/s

No, velocity of hammer does not depend on the mass of the hammer as v = u + at

Q10) Can any object have mechanical energy even if its momentum is zero? Explain.
Ans. Yes, mechanical energy comprises of both potential energy and kinetic energy. Zero momentum means that velocity is zero. Hence, there is no kinetic energy but the object may possess potential energy.