Work Power Energy:

• Work is said to be done by a force if the point of application of force undergoes displacement either in the direction of force or in the direction of component of force.
• Amount of work done is equal to the dot product of force and displacement. If is the force acting on a body and is displacement, work done by that force is

• Work is a scalar

Units of work : erg in CGS system
joule in S I
one joule = 10⁷erg

• Work done by a force can be positive or negative or zero

(a) If the displacement is in the direction of force or in the direction of component of force, work done is positive

(b) If the displacement is in a direction opposite to that of force or component of force, work done is negative

• If displacement and force are perpendicular to each other, work done is zero e.g :
  
  (i) In pulling a body on a horizontal surface, no work is done against gravitational force.
  
  (ii) If a man with a load on his head walks horizontally no work is done against gravitational force.
  
  (iii) When a body is in circular motion no work is done by centripetal force.
• Work done against friction on a level surface. W = F.S = mgs. (' is coefficient of friction).
• If the work done in moving a body between two points is independent of path along which it is moved, the force acting on the body is known as conservative force.

e.g: gravitational force, electrostatic force. The work done by a conservative force in moving a body along a closed path is zero.

• If the work done in moving a body by the application of force depends on the path along which it is moved, such force is known non conservative force.

e.g : frictional force, viscous force.

• If a body is moved along a closed path by applying a non conservative force, work done by such force is not equal to zero.
• When a body of mass m is lifted to a height h above the ground without any acceleration,

a) work done by the applied force = mgh

b) work done by the gravitational
force = - mgh

c) work done by the resultant force = O

• The work done in lifting a body of mass m having density inside a liquid of density through a height h is (a = 0)

• When a gas expands work done by it is

W = Pdv
or

(P is pressure and dv is change in volume)

• A point sized sphere of mass m is suspended vertically using a string of length l . If the bob is pulled to a side till the string makes an angle q with the vertical, work done against gravity is
• A uniform rod of mass m and length l is suspended vertically. If it is lifted to a side till it makes an angle q to the vertical, work done against gravity is

• A uniform chain of mass m and length l is suspended vertically. If the lower point of the chain is lifted to the point of suspension, work done against gravity is
• A uniform chain of mass m and length 'l' is kept on a table such that of its length is hanging down vertically from the edge of the table. The work done against gravity to pull the hanging part on to the table is

• Work done in stretching or compressing a spring is

K is spring constant
x is compression or elongation

• A bucket full of water of total mass M is pulled up using a uniform rope of mass m and length l, work done is

=
(Bucket is treated as point mass)

• When a body of mass m moves on rough horizontal surface through a distance 's', work done against friction is

( is coefficient of friction)

• If a body of mass m slides down distance S, work done by gravity is

( is inclination to the horizontal)

• A block of mass m is suspended vertically using a rope of negligible mass. If the rope is used to lift the block vertically up with uniform acceleration a, work done by the tension in the rope is

W = m (g + a) h
(h is distance through which it is lifted up)
In the above case if the block is lowered with acceleration a
W = - m (g - a) h

• The ability or capacity to do work is called energy
• Energy is cause for doing work and work is effect.
• Energy is a scalar. Units of energy are same as that of work.
• Mechanical energy is of two types

a) potential b) kinetic energy

• The energy possessed by a body by virtue of its state or position is known as potential energy.

e.g : Water on hill top and stretched or compressed spring possess potential energy.

• The energy possessed by a body by virtue of its motion is called kinetic energy. Any body in motion possesses kinetic energy.
• Gravitational potential energy of a body which is at a height 'h' above the ground is mgh (h is small so that g remains same almost)

Here surface of the earth is taken as reference. Kinetic energy of a body = (m is mass, v is velocity)

• A body can have both potential and kinetic energies e.g a flying bird, freely falling body
• A body at rest cannot have kinetic energy but it can have potential energy A body in motion can have potential energy.
• When a spring is compressed or stretched, elastic potential energy stored in it is
• Work done by an in balanced or resultant force on a body is equal to change in its kinetic energy. This is known as work-energy theorem.

• According to the law of conservation of energy, the total energy of a closed system remains constant. Energy can neither be created nor destroyed but it can be transformed from one form to another form without any loss or gain.
• In the case of a freely falling body or a body projected vertically up, the sum of potential and kinetic energies at any point in its path is constant.
• Kinetic energy of a freely falling body

a) is directly proportional to height of fall
b) is directly proportional to square of time of fall.

• If p is linear momentum of a body moving with kinetic energy E, we can write

(m is mass of the body)

for a given body

• If two bodies of different masses have same kinetic energy, heavier body will have greater momentum. If two bodies of different masses have same momentum, lighter body will have greater kinetic energy.

eg: When a bullet is fired from a gun, bullet and gun have same linear momentum but kinetic energy of bullet is more than that of gun.

• A body can have energy without momentum. But it can not have momentum without energy.

• A bullet of mass m moving with velocity v stops in wooden block after penetrating through a distance x. If F is resistance offered by the block to the bullet,

• A sphere of mass m is dropped from a height 'h' above the ground. On reaching the ground, it pierces through a distance S and then stops finally. Resistance offered by the ground is R. Then

mgh = (R - mg) S


Here time of penetration is given by

• A body is projected vertically up from the ground. When it is at a height 'h' above the ground its potential and kinetic energies are in the ratio x:y. If H is the maximum height reached by that body,

• A bullet of mass m is fired horizontally with a velocity V which strikes a wooden block of mass M suspended vertically. After striking the block, the bullet stops in the block by piercing. If the combined system rises to a height 'h' above the initial position,

• The rate of doing work by a force is called power

power is a scalar quantity.
Units of power : erg /s (CGS system)
Js^-1or watt (S I)
Horse power or HP (FPS)
1 H.P = 746 W

• If is the force acting on a body and is its velocity at an instant, instantaneous power

• If a machine gun fires n bullets per second such that mass of each bullet is m and coming out with velocity v, power of that machine gun
• A conveyor belt moves horizontally with a constant speed 'v'. Gravel is falling on it at a rate . Then extra power required to drive the belt is
• If a pump/ crane is used to lift water/coal of mass m to a height h in time 't' power of that pump or crane is

If efficiency of that crane or motor is , then

• A car of mass m is moving on horizontal road with constant acceleration 'a'. If R is resistance offered to its motion, power of the engine when its velocity is V will be P = (R + ma) V
• A body of mass m is initially at rest. By the application of a constant force, its velocity changes to in time . Then kinetic energy of that body at any time 't' is

Instantaneous power is

• A car moves on a rough horizontal road with constant speed v. Power of its engine is

( is frictional force)

• Kilowatt hour is practical unit of energy

Electron volt is smallest unit of energy




Circular Motion
Kinematics of Circular Motion:

• If an object moves in a circular path with constant speed, its motion is called uniform circular motion.
• The line perpendicular to plane of motion of the body about which the body rotates is called axis of rotation.
• The line joining the instantaneous position of the body and centre of the circle is called radius vector.
• The angle swept by the radius vector in a given interval of time is called angular displacement .

a) S.I unit of is radian.
b) Finite angular displacements are scalars.
c) Infinitesimally small angular displacements are vectors
d) This is a pseudo vector.
e) = 2 N rad, N-Number of rotations.

• Angular velocity ''

a) Rate of angular displacement is called angular velocity.
b) S.I unit is rad s^-1
c) = /t or w = d/dt.
d) = 2 n. Where n is the frequency of rotation.
e) = 2 /T Where T is the time period of rotation.
f) If is the linear velocity of the body
= ×

v = r sin

v = r when = 900

• The angular velocity is a pseudo vector. The direction is perpendicular to the plane of rotation and given by the right hand thumb rule. The direction of angular velocity is same as that of the angular displacement.
• Angular accelerationa :

a) Rate of change of angular velocity is called angular acceleration

b) S.I unit is rad.s^-2

c) where ₁and ₂are the initial and final angular velocities and t is the time interval.


d) = where n₁and n₂are initial and final frequencies of rotation.

e) It is produced only when an unbalanced torque acts on the body

f) When angular velocity of the body is constant = 0

g) If 'a' is the tangential acceleration a = r. Where r is the radius of the circular path.

• a) The acceleration possessed by a body moving in a circular path and directed towards the centre is called centripetal or radial acceleration.

b) a = v = v²/r = r2, (in m/s2 ) where v, and r are the linear velocity, angular velocity of the body and 'r' radius of the circle

c) It arises due to change in the direction of velocity of the body.

d) r =

• Tangential acceleration 'a_T'

a) If a body moves in a circular path with changing speed it possesses tangential acceleration along with radial acceleration.

b) a_T= dv/dt. (in ms^-2 )

c) It arises due to change in speed of the object

d) A car going in a circular track with increasing speed possesses both a_r and a_T.

e) Resultant linear acceleration of the body
a =
If a is the angle made by 'a' with ar, then tan a =



Centripetal Force :

• The force required to keep the body in circular path is centripetal force.

The centripetal force changes the direction of the velocity but not the magnitude of the velocity. It acts perpendicular to the direction of motion of the body.

• The centripetal force is observed in the inertial frame of reference.
• The gravitational force of attraction provides the necessary centripetal force for the moon to go round the earth.
• The gravitational force of attraction between the sun and the planet provides the necessary centripetal force to the planet to be in its orbit.
• The work done by a centripetal force is zero.

Centrifugal Reaction :

• Centrifugal reaction is the radial force which acts outwards on the agency making the body to move in a circular path.
• Centripetal force and centrifugal reaction form action - reaction pair.
• The sun exerts centripetal force on earth and the earth in turn applies centrifugal reaction on the sun.
• Centrifugal reaction.

Centrifugal Force :

• The accelerated frame of reference is the frame of reference which is moving with an acceleration.
• When a body revolving round in a circular path is in the accelerated frame of reference, it is observed that the body is experiencing a centrifugal force.
• The magnitude of centrifugal force is equal to the centripetal force. The centrifugal force acts away from the centre.
• The centrifugal force is a pseudo force.
• Centripetal and centrifugal forces are not action and reaction pair.
  

Applications Of Centrifugal Force :

Centrifuge: "Different substances of different densities experience different centrifugal forces". Basing on this principle the centrifuge works.



Cyclist On A Curved Road :

• A cyclist negotiating a curve of radius 'r' with a speed 'v' has to lean through an angle '' to the vertical in order not to fall down.
• The angle of leaning q is given by

The cyclist obtains the necessary centripetal force from the frictional force, mg.

• The angle of leaning is equal to the angle of repose or angle of friction between the tyres and the road is equal totan^-1 .
• If the road is of less friction, the angle of leaning is less and the velocity must also be less.
• The safe velocity of a cyclist =

Banking Of Roads And Railway Tracks :

• The roads and rail roads are banked to provide the necessary centripetal force from the normal reaction.
• Centripetal force = horizontal component of normal reaction = N sin

Centripetal force = Resultant of the normal reaction and weight =

• The angle of banking q is given by where 'v' is the velocity of the vehicle on the road of radius 'r'.
• The maximum velocity for a vehicle on a banked road is .
• For a given banking of a road the upper limit of the velocity is constant for any vehicle i.e., it is independent of the mass of the vehicle.
• The height 'h' of the outer edge over the inner edge of a banked road of banking angle is given by h = l sin . Where 'l' is the width of the road.
• If 'l' is the distance between the two rails and is the angle of banking, the height of the outer rail over inner rail is given by l sin .
• For a vehicle moving on an unbanked road of radius 'r', the safe velocity v,

i) , half of the wheel base

ii) , h is the height of the centre of gravity from the ground.


iii) , r the radius of the path.

• If the wheel base is more the safe velocity is more. Broad gauge trains have greater safe velocity than the meter gauge trains.
• Sports cars which have greater safe velocity have less h. They are designed to have less height.

i) When a vehicle is moving over a bridge of convex shape, the maximum velocity with which the vehicle should move so that it does not leave the surface is v and where 'r' is the radius of the road.
ii) When the vehicle is on the maximum height the reaction on the vehicle


iii) When the vehicle is moving in a 'dip' the reaction on the vehicle


motion in a vertical circle :

• A particle of mass 'm' suspended by a thread is given a horizontal speed 'u', when it is at 'A', it moves in a vertical circle of radius 'r'
• When the angular displacement of the particle is '', ie when the particle is at p.

a) Speed of the particle
v = where u is the velocity at A, the lowest point

b) Centripetal force mv²/r = T - mg cos

c) The speed of the particle continuously changes. It increase while coming down and decreases while going up

d) This is an example for non-uniform circular motion.

e) Tangential acceleration = g sin

f) Tangential force = mg sin

g) Tension in the string T = mv2/r + mg Cos

h) Velocity, speed, K.E, linear momentum, angular momentum, angular velocity, all are variables. Only total energy remains constant

• Tensions at lowest point (i.e, = 0) is given by

T = (mv²/r) + mg and is maximum.
Where V is the velocity at the lowest point.

• Tension at the highest point (i.e, = 180º) is

where T = (mv²/r) - mg. and is minimum 'V' is the velocity at highest point

• Condition to complete the circle by the particle

a) Tension at the highest point should be zero
T = 0
(mv²/r) - mg = 0
v² = rg
v =
Where 'v' is the minimum velocity at the highest point.


b) Minimum velocity 'v' is independent of mass of the particle.

• The minimum velocity at the lowest point to just complete the circle is

u =

• If u < , the body oscillates about A.
• If< u < the body leaves the path without completing the circle.
• When the body is projected horizontally with a velocity u = from the lowest point A,

a) It completes the circle

b) Velocity at the top B is and called the critical speed

c) Tension in the string at the top T_{2 = 0}

d) Tension in the string at the lowest point T₁ = 6mg. It is the maximum tension in the string.)

e) T₁ - T₂= 6mg.

f) Velocity at the horizontal position ie, at C
V_C =

g) Tension in the string at C = 3 mg

h) Ratio of velocities at A,B and C =

i) Ratio of K.E at A,B and C =

j) Velocity at an angular displacement is given by V =

k) Tension at angular displacement 'q' is given by T = 3mg (1+Cos )

• When the body is rotated at a constant speed, 'v'

a) Tension in the string at the lowest point
T = (mv²/r) + mg

b) Tension in the string at the highest point T = (mv²/r) - mg

c) Tension in the string at the horizontal position T = (mv²/r)

d) Time period of revolution if v =is
T = 2

• In the following cases, the minimum velocity of the body at the highest point should be

a) To rotate a can full of water, without water spilling out even at the highest position

b) In circus, for a motor cyclist to drive the cycle in a vertical circle inside a cage.

c) For the pilot of an aeroplane who is not tied to his seat, not to fall down while looping a vertical circle.

• Maximum safe speed of a car going on a convex bridge to travel in contact with the bridge is .
• A ball of mass 'm' is allowed to slide down from rest, from the top of a incline of height 'h'. For the ball to loop in a loop of radius 'r'

a) Minimum height of smooth incline h =

b) 'h' is independent of mass of the ball

• A ball of mass 'M' is suspended vertically by a string of length 'l'. A bullet of mass 'm' is fired horizontally with a velocity 'u' on to the ball sticks to it. For the system to complete the vertical circle, the minimum value of 'u' is given by u =
• A body is projected with a velocity 'u' at the lowest point

a) Height at which velocity u = 0. is h = u²/2g

b) Height at which Tension T = 0 is h =

c) Angle with vertical at which velocity v = 0. is Cos = 1-

d) Angle with vertical at which the tension T = 0 is Cos = 2/3 - u2/3 gr

e) Tension in the string at an angular displacement with vertical is
T = mu²/r - mg (2-3 cos )





WHEN A BODY IS ROTATED IN A HORIZONTAL CIRCLE :

• The body can be rotated with uniform speed or non-uniform speed.
• When the body is moving round with uniform speed.

i) It has centripetal force of constant magnitude equal to .
ii) It has tension of constant magnitude

iii) Its KE remains constant.
iv) It has centripetal acceleration of constant magnitude or .

• As the weight of the body is acting vertically downwards it does not effect the centripetal force and hence tension.
• When the body is whirled with increasing speed or decreasing speed,

i) its centripetal force is changing in both magnitude and direction.

ii) it has tangential acceleration aTand centripetal acceleration or radial acceleration a_r.
iii) its acceleration 'a' is equal to

iv) its acceleration changes both in magnitude and direction.

• When the body is rotating such that the centripetal force is greater than breaking force of the string it snaps and the body flies off in a tangential direction.