Work, Energy
Work
The work is done when a body is displaced by applied force. The product of force
and displacement along the direction of the force is called work.
Mathematically, the above statement is expressed as follows:
W = F. d = Fdcosθ
Where,
W is the work done by the force.
F is the force, d is the displacement caused by the force
θ is the angle between the force vector and the displacement vector
The dimension of work is the same as that of energy and is given as, [ML2T–2].
The unit of work or energy is the Newton meter or Joule, which is defined as the
amount work done by 1 Newton force in displacing an object 1 meter in the direction
of the force.
Work is scaler product of two vector quantities. The dot product of vector quantities
is always scalar which means it is has only magnitude and no direction, that is why
work is a scalar quantity.
Positive work
When force and displacement are in the same direction, the work performed on an
object is said to be positive work. In this case θ has values ranges from (00 ≤ θ <
900).
Example: When a ball is thrown upwards, work done by this force is positive work.
More examples: pushing or moving a table, kicking a football, throwing a stone.
Negative work
Negative work is performed if the displacement is opposite to the direction of the
force applied. In this case θ has values ranges from (900 < θ ≤ 1800).
Example: Work done by the gravitational force of the earth on the ball which is
thrown upward. When we walk friction force act opposite to the displacement,
frictional force does negative work.
Zero work
When force and displacement are perpendicular to each other, or when force or
displacement is zero. In this case θ = 900.
Example: Work done by the centripetal force.
Energy:
If a body can do work, we say it has energy. The ability of doing work of a body
or system is called energy. Work is transfer of energy from one system to another.
Energy is measured by total work done, so the magnitude of energy and work is
same. The dimension of energy and work is same that is [ML2T–2]. Unit is also same
that is joule (J).
When a body has some energy due to it’s motion and position, the energy is called
mechanical energy has two forms: Kinetic energy and potential energy.
Kinetic energy:
The ability of doing work of a moving body due to it’s motion is called kinetic
energy. An object has kinetic energy if it has mass and if it is moving. For an
object traveling at a speed v and with a mass m, the kinetic energy is given by:
1 P2
K.E = m v2 =
2 2m
Potential energy:
A body can store energy due to it’s physical condition or position. The ability of
doing work gained by a body due to it’s change in normal state is called potential
energy. It is divided into two categories.
i. Gravitational Potential energy
ii. Elastic potential energy
Gravitational Potential energy:
Gravitational potential energy is a function of the position of the object in a
gravitational field, force of gravity at that point and mass of the object. The formula
is as follows:
Gravitational potential energy, P.E. = mgh
Where m is the mass, g is the acceleration due to gravity (9.8 ms-2) and h is the
height above the Earth surface.
Elastic potential energy:
Elastic potential energy is the potential energy observed due to stretching or
compression by an external force to a given elastic object. The Elastic Potential
Energy Formula for the stretched spring is given by,
1
P.E. = = k x2
2
Where k is the spring constant, x is the displacement.
Work- Energy Theorem:
The work done on a body by an applied force is equal to the change of it’s kinetic energy.
W = K f − K i = ∆K
Derivation of Work-Energy Theorem using a Constant Net Force:
Let us consider a constant force F is applied on a body of mass m moving with velocity vo along
+X- axis. Due to this the body attains a velocity v and cover a distance x in the direction of
applied force.
Hence, the work done by the force is,
W= Fx
From Newton’s second law, F ma
So, W = max -----------------(1)
From equation of motion,
v2 = vo2 + 2ax
or, 2ax = v2 – vo2
1
or, ax = 2 (v2 – vo2)
Using this value, in equation (1), we obtain,
1
W = m 2 (v2 – vo2)
1
or, W = 2 (mv2 – mvo2)
1 1
But 2 mv2 is the initial kinetic energy Ki and 2 mv2 is the final kinetic energy Kf.
So, W = K f − K i = ∆K
Work done by the constant force is equal to the change in kinetic energy.
