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Work, Energy And Power

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Work, Energy and Power PDF Notes, Important Questions and Formulas

WORK:

Work is said to be done by a force when the force produces a displacement in the body on which it acts in any direction except perpendicular to the direction of the force.

Work done by constant force

Consider an object undergoes a displacement S along a straight line while acted on a force F that makes an angle θ with S as shown.

The work done W by the agent is the product of the component of force in the direction of displacement and the magnitude of displacement.

i.e., W=FS Cosθ  …..(1)

Work done is a scalar quantity and its S.I. unit is N-m or joule (J). We can also write work done as a scalar product of force and displacement.

W=F.S       …..(2)

Where S is the displacement of the point application of the force.

From this definition, we conclude the following points

Work done by a force is zero If displacement is perpendicular to the force (θ 90o)

Important points about work:

  1. Work is said to be done by a force when its point of application moves by some distance. Force does no work if point of application of force does not move (S = 0)
  2. Work is defined for an interval or displacement. There is no term like instantaneous work similar to instantaneous velocity.
  3. For a particular displacement, work done by a force is independent of type of motion i.e. whether it moves with constant velocity, constant acceleration or retardation etc.
  4. If a body is in dynamic equilibrium under the action of certain forces, then total work done on the body is zero but work done by individual forces may not be zero.
  5. When several forces act, work done by a force for a particular displacement is independent of other forces.
  6. A force is independent of reference frame. Its displacement depends on frame so work done by a force is frame dependent therefore work done by a force can be different in different reference frame. 

UNITS OF WORK:

In case system, the unit of work is erg. One erg of work is said to be done when a force of one dyne displaces a body through one centimetre in its own direction.

1 erg = 1 dyne × 1 cm = 1 g cm s–2 × 1 cm = 1 g cm2

s–2

Note: Another name for joule is newton metre

Relation between joule and erg

1 joule = 1 newton ×1 metre

1 joule = 105 dyne × 102 cm = 107 dyne cm

1 joule = 107 erg

1 erg = 10–7 joule

Dimensions of Work:

[Work] = [Force] [Distance] = [MLT–2] [L] = [ML2T–2]

Work has one dimension in mass, two dimensions in length and ‘–2’ dimensions in time, On the basis of dimensional formula, the unit of work is kg m2 s–2.

Note that 1 kg m2s–2 = (1 kg m s–2) m = 1 N m = 1 J.

Work done by multiple Forces:

If several forces act on a particle, then we can replace F in equation W=F.S by the force

begin mathsize 12px style table attributes columnalign left end attributes row cell sum straight F equals straight F subscript 1 plus straight F subscript 2 plus straight F subscript 3 plus. ..... end cell row cell therefore text   W= end text left square bracket sum straight F right square bracket. straight S text          end text. .. text (i) end text end cell end table end style

The gives the work done by the net force during a displacement S of the particle.

We can rewrite equation (i) as:

begin mathsize 12px style table attributes columnalign left end attributes row cell straight W equals straight F subscript blank subscript 1 end subscript. straight S plus straight F subscript 2. straight S plus straight F subscript 3. straight S plus. ..... end cell row cell Or text   W=W end text subscript 1 plus straight W subscript 2 plus straight W subscript 3 plus. ... end cell end table end style

So, the work done on the particle is the sum of the individual work done by all the forces acting on the particle.

Summary Points

  • Work done by a constant force is

            W= Fs cos Φ =begin mathsize 12px style straight F with rightwards arrow on top text  . end text straight s with rightwards arrow on top end style

  • Work done can be positive, negative or zero.
  • Work done by a constant force is 

            begin mathsize 12px style straight W integral from straight s subscript 1 to straight s subscript 2 of straight F left parenthesis straight s right parenthesis ds end style

  • Work-energy theorem: The work w done by the net force on a particle equals are change in the particle’s kinetic energy. 

           begin mathsize 12px style table attributes columnalign left end attributes row cell FS equals 1 half mv squared minus 1 half mu squared end cell row cell straight W equals straight k subscript straight f minus straight k subscript straight i end cell end table end style

  • Gravitational potential energy does not depend on the choice of the reference surface for measuring height.
  • Gravitational potential energy:
  1. Energy possessed by a body changes with height with respect to the surface of the earth.
  2. GPE= -WGravltational Force
  3. Law of conservation of mechanical energy: Total mechanical energy of the system always remains constant in the absence of dissipative forces.
  4. Total mechanical energy of the system equals the sum of potential energy and kinetic energy. Work done on a system by conservative forces implies that the mechanical energy of the system remains constant.
  5. Work done by the conservative force is the same along any path.
  6. Total work done by the gravitational force on the body moving along a closed loop is always zero.
  7. WGravity in closed loop = Zero
  8. Work done on a system by non-conservative forces implies that the mechanical energy of the system is not conserved.
  • A conservative force is the negative gradient of potential energy function.

            F(x)= -  ΔU /Δx

  • Power is the rate at which work is done or energy is transformed.
  • The unit of power is watt.

            1 watt= 1 joule/second

  • Linear momentum of an isolated system is always conserved in a collision.
  • A collision in which the total kinetic energy of the system is conserved is called elastic. A collision in which the total kinetic energy of the system is not conserved is called inelastic.
  • When two bodies collide, stick together and have a common final velocity, the collision is completely inelastic

Diagrams

Energy Curve

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