Work-Energy Principle

The S.I. unit of work done is N.m or Joule
W.D. is a scalar quantity has only magnitude but no direction.

Work-Energy Principle for a Particle

$U_{1→2} = KE_2 – KE_1$

where, $U_{1→2}$ = Algebric sum of work done by all forces from position 1 → 2

$KE_1$ = KE of system at position 1
$KE_2$ = KE of system at position 2

Work Done Calculations for Different Forces:

W.D. by External Force: (constant)

W.D. = F * S
(Positive if work is done in the direction of force.

W.D. by Variable Force: (When F v/s S curve is given)

W.D. by variable force = Area under F v/s S diagram. (Force v/s displacement diagram)

W.D. by Frictional Force:

W.D. by frictional force = $F_r$ * S
but frictional force $F_r$ = $μ_k$ R

W.D. by frictional force = – $μ_k$*R * S

Important tips:

(a) W.D. by frictional force is always negative.
(b) Always use co-efficient of kinetic force $μ_k$.
(c) If a block is moving on horizontal plane R = mg
(d) If a block is moving on inclined plane R = mg cosθ

W.D. by Gravitational Force:

W.D. by gravitational force = mgh

If block of mass m is moving up along an incline.

W.D. by gravity = -mgh = -mg s sinθ
(where s is incline distance between the two positions of block)

Note: Gravity force does no work if a body moves horizontally.

Work Done by Spring Force

W.D. by a spring force = $1/2 k({x_1^2} – {x_2^2})$

where,
k = spring constant (N/m) = stiffness of the spring
$x_1$ = Deflection in spring in position 1 of the particle.

$x_2$ = Deflection in spring in position 2 of the particle.
The above relation is directly used when spring is not connected with the particle

$1/2 k [(l_1 – l_o)^2 + (l_2 – l_o)^2]$
where,
$l_o$ = undeformed or unstretched length of the string

$l_1$ = length of spring in position 1 of the particle.
$l_2$ = length of spring in position 2 of the particle
The above relation is useful when spring is connected with the particle during motion.

Power

Power = F * v

Unit: N.m/sec or Joule/sec or Watt.

In MKS system power is expressed in H.P.

Remember 1 H.P. = 746 watt = 0.76 K.W

Efficiency

$η = {\text"Output Power"}/{\text"Input Power"}$