Work energy and power

Work
School of Biotechnology, KIIT University
Mass = m
Vi Vf
W = +ve when energy is transferred into the system.
W = -ve when energy is transferred from the system.
The work W done on an object by an agent exerting a constant force on the object is the
product of the component of the force in the direction of the displacement and the
magnitude of the displacement: W = Fd Cos = F.d
This relationship applies only when F is constant in magnitude and direction.

When an object is displaced on a frictionless, horizontal,
surface, the normal force n and the force of gravity mg do
no work on the object. In the situation shown here, F is the
only force doing work on the object.

A man cleaning a floor pulls a vacuum cleaner with a force of magnitude F = 50N at an
angle of 30° with the horizontal. Calculate the work done by the force on the vacuum
cleaner as the vacuum cleaner is displaced 3.00 m to the right.

Work done by a varying force
Let a particle being displaced along the x-axis under the action of a varying force.
The particle is displaced in the direction of increasing x from x = xi to x = xf
If more than one force acts on a particle, the total work done is just the work done
by the resultant force.

A force acting on a particle varies with x. Calculate the work done by the force as the
particle moves from x 0 to x = 6.0 m.

Work done by a spring force
X = 0, F =0
If the spring is either stretched or compressed a small distance from its unstretched
(equilibrium) configuration, it exerts on the block a force of magnitude.
Where x is the displacement of the block from its unstretched (x = 0) position and k is a positive constant
called the force constant of the spring.
F
The negative sign signifies that the force exerted by the spring is always directed opposite
the displacement.

The force exerted by a spring on a block varies
with the block’s displacement x from the
equilibrium position x = 0.
(a) When x is positive (stretched spring), the
spring force is directed to the left.
(b) When x is zero (natural length of the spring),
the spring force is zero.
(c) When x is negative (compressed spring), the
spring force is directed to the right.
(A)
(B)
(C)

If the spring is compressed until the block is at the point -xmax and is then released, the
block moves from -xmax through zero to xmax , then calculate the net work done by the spring
force.
Calculate the work done on the spring by the applied that stretches the spring from xi = 0
to xf = xmax .

The spring is hung vertically and an object of mass m is attached to its lower end. Under
the action of the “load” mg, the spring stretches a distance ‘d’ from its equilibrium position.
Calculate the value of spring constant, k.

s
F F

s
F F
F
s
F
Identify the cases where the work done is positive, negative and zero.
A)
B)
C)
When the force has a component in
the same direction as the
displacement, then work is positive.
Work is negative, when displacement
is in opposite direction to the
component of force.

A student lifts a book (mass = m) from to a vertical distance h and then he
walks in horizontal direction a distance s. Calculate i) Work done by him on
the box. Ii) Work done by the force of gravity on the box.
Example: Work done by the Gravitational force.

Work and Kinetic Energy Theorem for Constant net Force
d
m
vi vf
= Kf - Ki
A particle of mass m moving to the right under the action of a constant net force F.
The change in kinetic energy is equal to the net work done.

Work-energy theorem in case of variable force
Thus, we conclude that the net work done on a particle by the net force acting on it is
equal to the change in the kinetic energy of the particle.

Force is given by ,
F = x i + y2 j + z2 k
The displacement vector is given by,
ds = dx i + dy j + dz k
Calculate the work done, W by the variable force, F in moving from the point (0,0,0)
to (1,1,1)

Work done by Gravitational force
A particle of mass, m is thrown upward with a velocity, vi .
Mass = m
Velocity = vi
Gravitational force = Fg
Work done by the Gravitational force when the particle
rises,
Wg = Fg.d = Fgd Cos () = -mgd
Work done by the Gravitational force when the particle
falls back,
Wg = Fg.d = Fgd Cos () = mgd

Question_1:
A 5.0 kg block initially at rest is pulled to the right along a horizontal, frictionless
surface by a constant horizontal force of 10N. (a) Find the speed of block after it has
moved 2.0 m. (b) Find the acceleration of the block.

A block of mass 1.5 kg is attached to a horizontal spring that has a force constant of 2
N/m. The spring is compressed 2.0 cm and is then released from rest. Calculate the
speed of the block as it passes through the equilibrium position x = 0 if the surface is
frictionless.

A force, F = (3x2 i + 4 j) N acts on a particle which changes the kinetic energy of the
particle. How much work is done on the particle as it moves from coordinates (2m, 3m) to
(3m, 0m) ?.

Question_1:
m = 0.5 kg
V = 1 m/s
A block of mass 0.5 kg moving with a speed of 1 m/s on a frictionless surface
compresses the spring (of spring constant = 650 N/m) and slows down its speed.
When the block is momentarily stopped by the spring, calculate the distance d,
for which the spring is compressed.

Potential energy and Conservation of Energy

Potential energy (U) is associated with the arrangement of a system of objects that exert
forces on each other. If the arrangement of the system changes, then the potential
energy of the system changes. It is energy stored in the system.
Examples:Examples:
GravitationalGravitational potentialpotential energyenergy isis associatedassociated withwith thethe statestate ofof separationseparation betweenbetween
objectsobjects whichwhich cancan attractattract oneone anotheranother viavia thethe gravitationalgravitational forceforce..
ElasticElastic potentialpotential energyenergy isis associatedassociated withwith thethe statestate ofof compressioncompression oror
extensionextension ofof anan elasticelastic objectobject..

Examples of potential energy
Gravitational potential energy
Work done on the any object by the gravitational force is
equal to the –ve of the change of potential energy.
W = - dU = (x) dx

Potential energy is the energy associated with the arrangement of a
system of objects that exert forces on each other. If the arrangement of
the system changes, then the potential energy of the system changes.
Increasing concentration

Dr. Priti S. Mohanty, 2nd Floor , Email: psm_mohanty@yahoo.co.in
1. Work done by the force in moving the particle from one point
to another point is independent of the path taken. It depends
only on the initial and final position of the particle.
Conservative force:
A
B
1
2
2. The work done by a conservative force on a particle moving
through any closed path is zero.

Gravitational force
A
mg
B
mg
dh
Work done by a conservative force is
independent of the path taken.
Spring force
It depends on the initial and final
co-ordinate of the block.

The total mechanical energy of a system remains constant in any isolated system of
objects that interact only through conservative forces.
E = K + U
Conservation of Mechanical energy :
Example: Simple Pendulum

Relationship between conservative forces, potential energy and work
That is, any conservative force acting on an object within a system equals the
negative derivative of the potential energy of the system with respect to x.
The potential energy function for a system is given by, U(x) = - x3 + 2x2 + 3x.
(a) Calculate the force Fx as a function of x.
(b) For what values of x is the force equal to zero?.
(c) Plot U(x) versus x and Fx versus x, and indicate points of stable and unstable
equilibrium
Potential energy function for a two-dimensional force is given by, U (x,y) = 3 x3y – 7x.
Calculate the force that acts at the point (x, y).
W = - dU = (x) dx

Potential Energy Diagram & the Equilibrium of a System
Us = 1/2 kx2Potential energy function for a block–spring system is given by,
The x = 0 position for a block–spring system is one of stable equilibrium. That is, any
movement away from this position results in a force directed back toward.
In general, positions of stable equilibrium correspond to points for which U(x) is a
minimum
Stable
Or
Equilibrium
position
Unstable position
Unstable position

A man throw a ball of 0.5 kg and gives the ball an initial velocity of 10 m/s. What is
the maximum height the ball goes ?.

A ball of mass , m is dropped from a height, h above the ground as shown in the figure.
Neglecting the air resistance, determine the speed of the ball, when it is at a height, y
above the ground.
Use conservation of mechanical energy.
Determine the speed of the ball at y if at the instant of release it already has an initial
speed vi at the initial altitude h.

0
Y
-Y
X-X
Question:

Conservative force and their Properties
1. Work done by the force in moving the particle from one point to another
point is independent of the path taken. It only depends on the initial and final
points.
2. For an closed path, total work done is zero.
3. It is the difference between the initial and final value of the energy function.
4. It is reversible.
5. Total mechanical energy , E = K+U is constant.

The potential energy function for a system is given by, U(x) = - x3 + 2x2 + 3x.
(a) Calculate the force Fx as a function of x.
(b) For what values of x is the force equal to zero?.
(c) Plot U(x) versus x and Fx versus x, and indicate points of stable and unstable equilibrium
Potential energy function for a two-dimensional force is given by, U (x,y) = 3 x3y – 7x.
Calculate the force that acts at the point (x, y).

Total mechanical energy of the a system does not remain constant. It simply can not
be described by kinetic and potential energy .
Non-conservative force
Examples: Friction force
fk
vi vf
F
d
Example: In the presence of frictional force. (Under an external force)
When a body is in motion either on a surface or in a viscous medium such as air or water,
there is resistance to the motion because the body interacts with its surroundings. Such
resistance is called as a force of friction

fs F
mg
n n = normal force mg
fs = s n
s is the coefficient of static friction.
Static friction : Arises between two relatively smooth, flat objects in contact that
are at rest with respect to each other.

Static friction : Arises between two relatively smooth, flat objects in contact that are
at rest with respect to each other.
Properties of Friction Force
Kinetic friction: It arises between two relatively smooth, flat objects in contact that are
in motion with respect to each other. The direction of fk is opposite to the direction of
object motion.
s and k are the coefficient of static and kinetic friction, n is the normal force.
It is a measure of the degree to which two surfaces resist moving with respect to
each other.
fs = sn
Frictional force between two surface is given by,
fk = kn
s and k depends on nature of surfaces, but NOT on the surface area.
Experimentally, k < s when the object is in motion.

fs = fs,max = sn = Object is on the verge of moving.
F = fk : Move with constant speed.

Forces on a Stationary Block
FN
Fg
Frictional force =0
F
Fg
fs
Frictional force = fs
(Balanced by the applied force)
F overcome the fs
(Starts accelerating)
Fg
Ffs
a
Fg
Ffk
v
Moves with constant
speed, balanced with fk

Coefficient of static friction and kinetic friction
Static Case:
(1)
(2)
From equation, (2)
mg = n/cos
Put this equation in equation (1)
fs = sn = n tan s
s= tan s
Once the block starts to move at   s , then
The kinetic friction coefficient, k = tan k (k > s )

A force is non-conservative if it causes a change in mechanical energy E, which we define
as the sum of kinetic and potential energies
fk
vi vf
F
d
Non-conservative force
Examples: Friction force

1. Work and kinetic theorem for constant force and variable
force.
2. Potential energy ( Gravitational, Spring force).
3. Conservative system
4. Force and potential energy.
5. Non-conservative force (adding friction)
Summary

A 2.0 kg box starts from rest and slides down a distance of 1.0m on a plane that is
inclined at an angle of 30.0°. While sliding down, the box experiences a constant
frictional force of magnitude 5.0 N. When the box reach at the bottom , it has a speed
vf . Calculate this speed, vf .

A child of mass m rides on an irregularly curved slide of height, h =2m. The child starts
from rest at the top. (a) Determine his speed at the bottom, assuming no friction is
present. b) If a force of kinetic friction acts on the child, how much mechanical energy
does the system lose ?. Assume that the final velocity is 3 m/s and mass, m = 20 kg.

A block having a mass of 0.80 kg is given an initial velocity 1.2 m/s to the right and
collides with a spring of negligible mass and force constant 50 N/m. Assuming the surface
to be frictionless, a) calculate the maximum compression of the spring after the collision.
b) There is constant force of kinetic friction of k = 0.5 acts between the block and the
surface. If the speed of the block at the moment it collides with the spring is 1.0 m/s,
what is the maximum compression in the spring?

Two blocks , m1 and m2 are connected by a string which passes over a
pulley. m1 is connected to a spring of force constant k. Under un-stretched
condition, the block is released from rest. If the block m2 falls a distance h
before it comes to rest, then calculate the coefficient of kinetic friction,  k
between the block m1 and the surface.
f k =  kmg

Work energy and power

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