Work

Work is said to be done if certain amount of force applied on a body displace it by a certain distance along the direction of force. Mathematically,

W = .

It means work is scalar product of force and displacement.

i.e., W =

when,  = 0^o.

W = FS

When,  = 90^o, W = 0.

Unit of work is Joule in SI unit and erg in CGS unit so that, 1 Joule = 10⁷ erg.

Workdone by variable force

Let us consider, a body moving due to the application of variable force. In certain time, the body transit from P to Q to find workdone in this transition. Let us divide the section PQ into large number of rectangular strips with infinitely small thickness .

Now, the small workdone by the body moving A to B is:

W = , where, i = workdone on moving i^th strip [AD = area of i^th strip]

Now, the total workdone by a body from P to Q,

W =

                = Area of region bounded by curve SR on x-axis from P to Q.

In force is continuous,

W =  = Area bounded by the curve.

It means the workdone by variable force is equivalent to the area under the graph obtained from force vs. displacement.

Energy:

The capacity of doing work is called Energy. It is measured in the unit of Joule. There are two types of mechanical energy:

1. Kinetic Energy.
2. Potential Energy.

Principle of conservation of energy:

Statement: “Energy neither be created nor be destroyed but it can be changed from one form to other easily.”

Proof:

The total mechanical energy of a freely falling body under the influence of gravity is conserved.

To verify the principle of conservation of energy of a freely falling body, let us consider, a body of mass ‘m’ falling from a certain height ‘h’. Let, the position of the object when it is at height ‘h’ is A. After travelling the distance, x vertically downward, it comes to a position B, at that time it is at a height of ‘h-x’. After travelling distance h it falls to the ground i.e. the position C. Now, the total mechanical energy of a body,

E_T = K.E. + P.E. …………….. (i)

At position A,

KE = 0.

PE = mgh ……… (ii)

From (i) and (ii),

E_T = mgh ……… (iii)

At position B,

KE =  …………….. (iv)

P.E. = mg(h-x) ………………. (v)

From (iv), (v) and (i), we get,

E_T = mgx + mgh – mgx

                = mgh …………. (vii)

At position C,

KE =  m x 2gh = mgh ……………. (vii)

P.E. = 0.

From eq^n. (vii) and (i), we get,

E_T = mgh ………….. (viii)

From eq^n. (iii), (vi), (viii), it is seen that, “the total mechanical energy of a body, at every position is same so that it proves the principle of conservation of energy”.

Graphically,

Conservative and non-conservative force

The force is said to be conservative if workdone by a body by the application of force independent of path followed by it. Due to the conservative nature of force, work done by a body on moving one complete revolution on circle is zero.

The force is said to be non-conservative force if the work done by a body moving from one point to other is different on following different path. The frictional force and viscous force are considered to be non-conservative forces.

Collision

When a body strikes to other, this activity is called as collision in general but in physics, it is not necessary to strike the body for collision”. There are two types of collision:

• Elastic collision
• Inelastic Collision

(i) Elastic collision:

The collision between two bodies is said to be elastic if both kinetic energy and linear momentum of bodies conserve during the collision. The collision between nuclear particles is said to be elastic.

Let us consider, a body of mass m₁ moving with u₁ collides with m₂ moving with velocity u₂ so that after collision, the body m₁ moves with velocity v₁ and that of m₂ with v₂ as shown in figure.

In this type of collision,

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ ……………………. (i)

m₁u₁² + m₂u₂² = m₁v₁² + m₂v² …………………………….. (ii)

From (i),

m₁(u₁-v₁) = m₂(v₂-u₂)…………… (iii)

From (ii),

m₁(u₁²-v₁²) = m₂(v₂²-u₂²)

m₁(u₁+v₁)(u₁-v₁) = m₂(v₂-u₂)(v₂+u₂)………………. (iv)

Diving (iv) by (iii), we get,

Or, v₁ + u₁ = v₂ + u₂

Or, v₁ – v₂ = u₂ – u₁

∴ u₁ – u₂ = v₂ –v₁ ……………. (v)

From eq^n. (v), it is clear that during the elastic collision between two bodies in one dimension, the relative velocity of approaching of two bodies before collision is equal to relative velocity of separation of two bodies after collision.

Now,

v₂ = u₁ –u₂+v₁

using v₂ in eq^n. (i),

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

m₁u₁ + m₂u₂ = m₁v₁ + m₂(u₁ –u₂+v₁)

Or, m₁u₁ + m₂u₂ = m₁v₁ + m₂u₁ –m₂u₂+m₂v₁

Or, u₁ (m₁-m₂) + 2m₂u₂ = v₁(m₂+m₁)

If, m₁ = m₂. Let, m₁ = m₂ = m.

2mu₂ = v₁.2m

u₂ = v₁

Similarly,

v₂₌ u₁

From this result, it is clear that during the elastic collision between two bodies of equal mass, in one dimension, they exchange their velocities before and after collision in elastic collision.

(ii) In-elastic collision

The collision between two bodies is said to be inelastic collision if linear momentum of body is conserved but not kinetic energy. Almost all the collision in our daily life is in-elastic type.

Let us consider, a body of mass m₁ moving with velocity u₁ collides with m₂ at rest in-elastically. After the collision, two bodies combine to form a single body and moves with common velocity v.

Now,

From principle of conservation of linear momentum,

m₁u₁ = m₁v + m₂v

u₁ =

Now, K.E. before collision,

K.E._i =

                = ………….. (i)

K.E. after collision,

K.E._f = …………….. (ii)

Now,

⇒ KE_i > KE_f

Work Energy Theorem:

Statement:

Work done on a moving body is equal to its change in kinetic energy.

ΔW = ΔK. E

Proof:

Suppose, a constant force ‘F’ is acting on a body of mass ‘m’ moving with initial velocity ‘u’ so that it travels a distance ‘s’ and obtain the final velocity ‘v’ then, we have,

v² = u² + 2as

or, a =

                =  …………… (i)

We have,

ΔW = F x S

                = ma x s

Substituting the value of ‘a’ from eq^n. (i), we get,

ΔW = m x  x s

                =

ΔW = K.E._f – K.E._i

ΔW = ΔK.E.

Proved.

Conservative forces:

A force that offers an opportunity of two-way conversion between kinetic energy and potential energy is called conservative forces.

Eg. Spring or string force, electrostatic force, magnetic force, gravitational force, etc.

An important feature of conservative forces is their workdone on the body is independent on the path depend only on initial and final position. Another important aspect of conservative forces is their work done is always reversible. Any amount of energy deposited on the energy bank can be withdrawn later on without any loss of energy.

Another important property of conservative forces is if starting and the ending points are the same, then, their work done is zero.

The workdone by a conservative force always has four properties:

i) It is independent of the path of the body and depends only on starting and ending point.

ii) It is reversible.

iii) When the starting and ending points are the same, then, the total workdone is zero.

iv) It can be expressed as the difference between the initial and final values of potential energy function.

Statement:

When the only forces that do work are conservative forces. The total mechanical energy,

E = K.E. + P.E. is constant.

Non – Conservative forces:

A force that doesn’t offer an opportunity of two-way conversion between kinetic and potential energy.

Eg. Frictional force.

Properties of non – conservative force:

i) If the starting and the ending point are the same, the workdone by non – conservative force will not be zero.

Suppose, a body is displaced from A to B so that its displacement if ‘s’ as shown in fig. (a). Then, workdone by frictional force is:

W_AB = -F_f x S ……………. (i)

Now, the body is displaced from B to A from the same path so that the starting and the ending point is the same. Now, the workdone by the frictional force is given by:

W_BA = -F_f x S …………… (ii)

Now,

W_AB – W_BA = -F_f x S – F_f x S

                = – 2 F_f x S                 ≠ 0