Work Energy and Power CBSE Class 11 Physics Revision Notes Chapter 6

 

• 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 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 E_total of the system is constant.

So, E = U+K = constant

Or, U_i+K_i = U_f+K_f

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 ( F_ext = 0)), the total momentum of the system before collision would be equal to total momentum of the system after collision.

So, p_f = p_i

• 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

= v₂ – v₁/u₁ – u₂

Case (i) For perfectly elastic collision, e = 1. Thus, v₂ – v₁ = u₁ – u₂. This signifies the relative velocities of two bodies before and after collision are same.

Case (ii) For inelastic collision, e<1. Thus, v₂ – v₁ < u₁ – u₂. 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, v₂ – v₁ =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:-

?

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 K_i is the initial kinetic energy and K_f is the final kinetic energy of mass m₁, the fractional decrease in kinetic energy is given by,

K_i – K_f / K_i = 1- v₁²/u²₁

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

K_i – K_f / K_i = 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,

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

E_s = 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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