Class 9th Science: Chapter 11 Energy and Work


Work

If a force displaces an object, the work is said to be done. (Here force is the net force and there should be a net displacement of the object).

Magnitude of work
The amount of work done by a force is equal to the product of the force and the displacement of the point of application of the force in the direction of force. Mathematically work is expressed as the following
W = F × S,
Where
W = Work done on an object
  F = Net force on the object
  S = Displacement of the object

If the displacement of the body, S⃗   is in a direction making an angle, θ with the direction of force, F⃗   the amount of work done is equal to the product of

i. Magnitude of force
ii. Magnitude of displacement and
iii. Cosine of the angle between the directions of force F and the displacement. i.e.,
                                  W = ∣∣F⃗ ∣∣ ×∣∣S  −→∣∣×cos θ

Since force, F⃗   and displacement, S⃗   are the vectors and work is a scalar quantity, work done is expressed as the following in the vector form

W = F.→ S⃗

Thus work is expressed as a scalar product (or the dot product) of force and displacement vectors.

Note:The Scalar product of two vecctors is a scalar.

The work done is measured in joule in the SI system after the scientist James Prescott Joule.
erg is the CGS unit of work.

Conditions for work to be done:
•  A net force should act on an object.
•  The object must be displaced in the direction of the net force.
•  The angle between the net force acting and the displacement of the object should not be perpendiculatr to each other.

Work is said to be done when an object is displaced on applying a certain force.
     W = F × s, Where
     W = Work done on an object
      F = Net force on the object
      s = Displacement of the object

Joule is defined as the work done when the net force of one newton acts on a body and displaces it in the direction of the force by one metre.

Work done could be either positive or negative.

•  When both the force and the displacement are in the same direction, positive work is done.
•  When force acts in a direction opposite to the direction of displacement, the work done is negative.

Energy

The energy of an object is its ability to do work. Energy is the cause and work is its effect. Therefore both work and energy have the same unit, which is joule (J) in the SI system and erg in the CGS system. Energy is  a scalar quantity. Energy exists in many forms.

Forms of  Energy
Mechanical energy (mechanical energy is either in the form of potential energy or kinetic energy or a combination of the both), electrical energy, light energy, thermal energy, nuclear energy and sound energy etc.

Potential Energy
Potential energy is the energy possessed by a body by virtue of its state of rest or deformed state i.e, the energy of an object due to its position or arrangement in a system is called potential energy.

It is further classified into gravitational potential energy (GPE) and elastic potential energy (EPE). GPE is by virtue of height of a body from a reference level. The gravitational potential energy of an object is the work done in raising it from the ground to a certain point against gravity.

It can be expressed as  GPE = mgh (where, m is mass of the body, g is the acceleration due to gravity and h is  the height of the body from the reference level)

Gravitational potential energy, P.E = mgh

If the height, H of a body is considered from the ground, then the gravitational potential energy of the body, P.E = mgH

If the height of the body is raised from a  height, h  to the other height, H the gravitational potential energy of the body is  P.E = mg(H-h)

EPE of a body is by virtue of its stretched state.

Kinetic Energy
Kinetic energy (KE) is the energy possessed by a body by virtue of its motion and is given by,
K.E = ½mv2.

•  The work done on a body is equal to the change in its kinetic energy.
•  The kinetic energy of a body is given by  K.E = ½mv2.
•  Energy can be converted from one form into another.

The Relation Between Kinetic Energy and Momentum
Momentum is the quantity of motion of a moving body, its magnitude is equal to the producct of its mass and velocity of the body at a particular time.

If mass of the body = m and the velocity = v
its momentum (linear) p = mv
           p = mv
Kinetic energy is defined as the energy possessed by a body because of its motion.
If mass of the boody = m
Velocity = v
Kinetic energy = ½ x mass x velocity2
⇒ K.E = ½ mv2
⇒ K.E =  (½ mv) x v
but mv = p
⇒ K.E = ½ p x v
⇒ p = 2K.E/v
Or  
Kinetic Energy = ½ mass x velocity2
⇒ K.E = ½ mv2

On multiplying and dividing the above equation with m
⇒ K.E =  (½ mv )x(v) x m/m
⇒ K.E =  ½ (mv x mv )/m
⇒ K.E =  ½ (mv)2/m   
⇒ K.E =  ½ p2/m

Law of Conservation of Energy
The law of conservation of energy is the fundamental law, law of conservation of energy says that the energy can neither be created nor destroyed, the sum total energy existing in all forms in the universe remains constant. Energy can only be transformed from one form to another.

Electrical Energy
Electrical energy commercially is measured in the units of kilowatt hour (kW h). Power is defined as the rate of doing work. Power is measured in watt which is equal to joule per second. Power can also be measured as the product of force and velocity of an object. Energy can be expressed in terms of product of power and time.

1 kW h = 3.6 x 106 J.

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Kinetic Energy:
The relation connecting the initial velocity (u) and final velocity (v) of an object moving with a uniform acceleration a, and the displacement, S is
                            v2 - u2 = 2aS
This gives  
                            S =v2-u22a

We know F = ma. Thus using above equations, we can write the workdone by the force, F as

                           W = ma×(v2-u22a)
                              or
                            W =12m(v2-u2)


If object is starting from its stationary position, that is, u = 0, then
                           W =12mv2

It is clear that the work done is equal to the change in the kinetic energy of an object.

If u = 0, the work done will be  W =12mv2 .

Thus, the kinetic energy possessed by an object of mass, m and moving with a uniform velocity, v is Ek =  ½  mv2

Potential Energy:
Let the work done on the object against gravity be W. That is,
                Work done, W = force × displacement
                                      = mg × h
                                      = mgh
Since work done on the object is equal to mgh, an energy equal  to mgh units is gained by the object . This is the potential energy (Ep) of the object.
                             Ep = mgh
  
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