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Revision Notes on Work, Power and Energy

  • Work:-  Work done W is defined as the dot product of force F and displacement s.

Here θ is the angle between  and  .

Work done by the force is positive if the angle between force and displacement is acute (0°<θ<90°) as cos θ is positive. This signifies, when the force and displacement are in same direction, work done is positive. This work is said to be done upon the body.

  • When the force acts in a direction at right angle to the direction of displacement (cos90° = 0), no work is done (zero work).Work Done by a Man

  • Work done by the force is negative if the angle between force and displacement is obtuse (90°<θ<180°)  as cosθ is negative. This signifies, when the force and displacement are in opposite direction, work done is negative. This work is said to be done by the body.

  • Work done by a variable force:-

If applied force F is not a constant force, then work done by this force in moving the body from position A to B will be,

 

Here ds is the small displacement.

  • Units:  The unit of work done in S.I is joule (J) and in C.G.S system is erg.

1J = 1 N.m , 1 erg = 1 dyn.cm

  • Relation between Joule and erg:- 1 J = 107 erg

  • Power:-The rate at which work is done is called power and is defined as,

P = W/t = F.s/v = F.v

Here s is the distance and v is the speed.

  • Instantaneous power in terms of mechanical energy:- P = dE/dt

  • Units: The unit of power in S.I system is J/s (watt) and in C.G.S system is erg/s.

  • Energy:-

1) Energy is the ability of the body to do some work. The unit of energy is same as that of work.

2) Kinetic Energy (K):- It is defined as,

K= ½ mv2

Here m is the mass of the body and v is the speed of the body.

  • Potential Energy (U):- Potential energy of a body is defined as, U = mgh

Here, m is the mass of the body, g is the free fall acceleration (acceleration due to gravity) and h is the height.

  • Gravitational Potential Energy:- An object’s gravitational potential energy U is its mass m times the acceleration due to gravity g times its height h above a zero level.

In symbol’s,

U = mgh

  • Relation between Kinetic Energy (K) and momentum  (p):-

K = p2/2m

  • If two bodies of different masses have same momentum, body with a greater mass shall have lesser kinetic energy.

  • If two bodies of different mass have same kinetic energy, body with a greater mass shall have greater momentum.

  • For two bodies having same mass, the body having greater momentum shall have greater kinetic energy.

  • Work energy Theorem:- It states that work done on the body or by the body is equal to the net change in its kinetic energy .

  • For constant force,

W = ½ mv2 – ½ mu2

= Final K.E – Initial K.E

  • For variable force,

?

  • Law of conservation of energy:- It states that, “Energy can neither be created nor destroyed. It can be converted from one form to another. The sum of total energy, in this universe, is always same”.

  • The sum of the kinetic and potential energies of an object is called mechanical energy. So, E = K+U

  • In accordance to law of conservation of energy, the total mechanical energy of the system always remains constant.

So, mgh + ½ mv2 = constant

In an isolated system, the total energy Etotal of the system is constant.

So, E = U+K = constant

Or, Ui+Ki = Uf+Kf

Or, ?U = -?K

Speed of particle v in a central force field:

v = √2/m [E-U(x)]

  • Conservation of linear momentum:-

?In an isolated system (no external force ( Fext = 0)), the total momentum of the system before collision would be equal to total momentum of the system after collision.

So, pf = pi

  • Coefficient of restitution (e):- It is defined as the ratio between magnitude of impulse during period of restitution to that during period of deformation.

e = relative velocity after collision / relative velocity before collision

= v2v1/u1u2

Case (i) For perfectly elastic collision, e = 1. Thus, v2v1 = u1u2. This signifies the relative velocities of two bodies before and after collision are same.

Case (ii) For inelastic collision, e<1. Thus, v2v1 < u1u2. This signifies, the value of e shall depend upon the extent of loss of kinetic energy during collision.

Case (iii) For perfectly inelastic collision, e = 0. Thus, v2v1 =0, or v2 = v1. This signifies the two bodies shall move together with same velocity. Therefore, there shall be no separation between them.

  • Elastic collision:- In an elastic collision, both the momentum and kinetic energy conserved.

  • One dimensional elastic collision:-

?Elastic collision in One Dimension

After collision, the velocity of two body will be,

v1 = (m1-m2/ m1+m2)u1 + (2m2/ m1+m2)u2

and

v2 = (m2-m1/ m1+m2)u2 + (2m1/ m1+m2)u1

Case:I

When both the colliding bodies are of the same mass, i.e., m1 = m2, then,

v1 = u2 and v2 = u1

Case:II

When the body B of mass m2 is initially at rest, i.e., u2 = 0, then,

v1 = (m1-m2/ m1+m2)u1 and v2 = (2m1/ m1+m2)u1

(a) When  m2<<m1, then, v1 = u1 and v2 = 2u1

(b) When  m2=m1, then, v1 =0  and v2 = u1

(c) When  m2>>m1, then, v1 = -u1 and v2 will be very small.

  • Inelastic collision:- In an inelastic collision, only the quantity momentum is conserved but not kinetic energy.

v = (m1u1+m2u2) /(m1+m2)

and

loss in kinetic energy, E = ½ m1u12+ ½ m2u22 - ½ (m1+ m2)v2

or,

E= ½ (m1u12 + m2u22) – ½ [(m1u1+ m2u2)/( m1+ m2)]2

= m1 m2 (u1-u2)2 / 2( m1 + m2)

  • Points to be Notice:-

(i) The maximum transfer energy occurs if m1= m2

(ii) If Ki is the initial kinetic energy and Kf is the final kinetic energy of mass m1, the fractional decrease in kinetic energy is given by,

KiKf / Ki = 1- v12/u21

Further, if m2 = nm1 and u2 = 0, then,

KiKf / Ki = 4n/(1+n)2

  • Conservation Equation:

(i)  Momentum – m1u1+m2u2 = m1v1+m2v2

(ii) Energy – ½ m1u12+ ½ m2u22 = ½ m1v12+ ½ m2v22

  • Conservative force (F):- Conservative force is equal to the negative gradient of potential V of the field of that force. This force is also called central force.

So,  F = - (dV/dr)

  • The line integral of a conservative force around a closed path is always zero.

So,

\oint \vec{F}.d\vec{r} = 0

  • Spring potential energy (Es):- It is defined as,

Spring Energy

Es = ½ kx2

Here k is the spring constant and x is the elongation.

  • Equilibrium Conditions:

(a) Condition for equilibrium, dU/dx = 0

(b) For stable equilibrium,

U(x) = minimum,

dU/dx = 0,

d2U/dx2 = +ve

(c) For unstable equilibrium,

U(x) = maximum

dU/dx = 0

d2U/dx2 = -ve

(d) For neutral equilibrium,

U(x) = constant

dU/dx = 0 

d2U/dx2 = 0

  • UNITS AND DIMENSIONS OF WORK, POWER AND ENERGY

Work and Energy are measured in the same units. Power, being the rate at which work is done, is measured in a different unit.

Quantity and  Units/Dimensions  

Work (Energy)

Power

Dimension

ML2T-2

ML2T-3

Absolute

MKS

Joule

Watt

FPS

ft-Poundal

ft-poundal/sec

CGS

erg

Erg/sec.

Gravitational

MKS

kg-m

Kg-m/sec

FPS

ft-lb

ft-lb/sec.

CGS

gm-cm

Gm-cm/sec

Practical
(Other)

 

kwh, eV, cal

HP, kW, MW

  • Conversions between Different Systems of Units

1 Joule = 1 Newton ´ 1 m = 105 dyne ´ 102 cm = 107 erg

1 watt = 1 Joule/ sec = 107 erg/sec.

1 kwh  = 103 watt ´ 1 hr  = 103 watt ´ 3600 sec 

= 3.6 ´ 106 Joule

1HP = 746 watt.

1 MW = 106 watt.

1 cal = 1 calorie = 4.2 Joule

1eV = "e" Joule  = 1.6 ´ 10-19 Joule

(e = magnitude of charge on the electron in coulombs)

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