## Kerala Plus One Physics Notes Chapter 6 Work, Energy and Power

Introduction:

In physics, work is done on a body or by a body. Work is done on a body when a force on the body makes it move, similarly the body spends energy when it does work. Everybody knows that any type of motion requires energy. Let us try to understand what these are, through this chapter.

Work:

Work is said to be done if the body on which the force is applied gets displaced through some distance in the direction of the force. We shall consider two cases:

1. When force acts along the direction of motion:

Work done = Force x distance moved in direction of force

2. When force makes an angle with the direction of motion:

Consider a man pulling a trolley. The direction of force is along the string. The trolley moves along the horizontal surface. Let 0 be the angle between the direction of force and the horizontal surface. Displacement is caused only by the horizontal component of force (Fcosθ) and not by the vertical component.

Work is defined as the product of displacement

W = S × F cosθ

W = Fs cos θ

Thus work done is the dot product of force and displacement vectors. Note that work done is a scalar quantity.

Dimensional formulae of work =[F]×[S]

= MLT^{2 }L = [ML^{2}T^{2 }]

SI unit of work is Joule (J). One joule of work is said to be done when a force of one newton displaces a body through a distance of one metre in the direction of force.

1 Joule = 1 Nm

Special cases:

- If θ = 0 then we get the first case where W = Fs (max. work)

eg., when a ball falls freely under gravity, the work done by gravity is positive. - If θ = 90°, W = Fs cos 90 = 0. No work is done.

eg., during circular motion, the centripetal force and displacement are perpendicular to each other and thus work done by centripetal force is zero.

- If θ< 90°, work done is positive.
- If θ= 180°, i.e., force and displacement are

in opposite direction then W = Fs cos180

=-FS

Here work done is negative,eg., when we throw a ball upwards, the gravitational force acts downards by still the motion remains upwards (θ = 180). Hence here, work done by gravitational force is negative. - If θ > 90°, work done is negative.

CAQ

Question 1.

A constant force acts on a particle displacing it from a position to a new position. Find the work done by the force.

Answer:

Energy:

Energy of a body is its ability to do work. There are different forms of energy like mechanical energy, heat energy, light energy, sound energy, nuclear energy, electric energy etc. When energy is produced by mechanical means it is called mechanical energy which is of two types namely:

- Kinetic Energy
- Potential Energy

Note: Energy is also a scalar quantify and its unit is same as that of work (joule).

Kinetic Energy:

A jumping, running man is said to be more energetic compared to a sleeping man. We consider this motion here.

Thus kinetic energy is defined as the energy possessed by a body due to its motion.

Relation between Kinetic energy and Linear momentum

CAQ

Question 2.

A light body and a heavy body have the same kinetic energy. Which one will have the greater momentum?

Answer:

Kinetic Energy

Thus the heavier body has a greater momentum than the lighter one.

Work Energy Theorem for a constant force

According to work energy theorem the change in kinetic energy of a particle is equal to the work done on it by the net force.

Consider a force F acting on a body of mass m so that its velocity changes from u to v in travelling a distance s.

Change in K.E of the body = Work done on the body by the net force.

Hence work energy theorem is proved.

Note: When work done is positive, the body’s K.E will increase and when the work done on the body is negative, its K.E decreases.

CAQ

Question 3.

A bullet weighing 10g is fired with a velocity of 800 ms^{-1}. After passing, through a mud will 1 m thick, its velocity decreases to 100 ms^{-1}. Find the average resistance offered by the mud wall.

Answer:

Mass of bullet, m = 10g = 0.01 kg Velocity of bullet before passing through mud wall,

u = 800 ms^{-1}

Velocity of bullet after passing through mud wall,

v =100 ms^{-1}

Distance covered by the bullet, s = 1 m

Let average resistance offered by the wall=F

According to work-energy theorem,

Work done by resistance offered by mud wall = Decrease in K. E. or

Work done by a Variable force

Consider a continuously varying force F acting on a body and produces a very small displacement Δx each time. As the displacement is small, the force F can be considered as constant.

Then the small work done, Δ W = FΔx which is equal to the area of rectangle abcd.

If we add the area of all such rectangles in the above figure, we get the total work

done, = total area under the graph.

Work Energy Theorem for a variable force:

Consider a variable force acting on a body of mass m and produces displacement in the same direction. Small work done,

W=K_{f}-K_{i }Hence total work done is equal to the change in Kinetic energy which proves the work energy theorem for a variable force.

Note:

When a body executes circular motion with uniform speed, there is no change in its kinetic energy. By W-E theorem the work done by the centripetal force is zero We had seer this at the stalling of the Chapter1

Conservative and Non-conservative forces:

Conservative forces:

- A force is said to be conservative, if work done by or against the force in moving a body depends only on the initial and final positions of the body, and not on the nature of path followed between the initial and the final position.
- The work done by the conservative force in a closed path is zero.
- The force will be conservative, if it can be derived from a scalar quantity.
- Gravitational force and elastic spring force are conservative forces.

Non-conservative forces :

- A force is said to be non-conservative, if the work done by or against the force in moving a body from one position to another, depends on the path followed between these positions.
- Frictional force, normal force, force of viscosity are non-conservative forces.

Potential Energy:

Potential energy is the energy stored in a body or a system by virtue of its position in a field of force or by its configuration.

Due to position:

The potential energy of water stored in great heights in dams is used to run turbines to generate electricity.

Due to configuration:

A bullet is fired with a large velocity in a loaded gun due to the potential energy of the compressed spring in it. Different types of potential energies are gravitational potential energy, elastic potential energy and electrostatic potential energy.

Note:

Potential energy is defined only for conservative forces does not exist for non conservative forces.

Gravitational Potential Energy:

The work done to raise the mass m to a height h against the gravitational force of attraction,

W = F.s = mg × h

=mgh

This work done is stored in it as potential energy.

∴U = mgh

Note:

When a body is released from height h. it comes down with an increasing speed- i The velocity y with which it hits the ground is calculated, as we know that the gravitational P.E of the body at height h transforms itself as K.E of the body on reaching the ground.

Mechanical Energy:

The mechanical energy (E) of a body is the sum of kinetic energy (K) and potential en-ergy (U) of the body,

ie., E = K + U

Principle of conservation of mechanical energy:

It states that if only the conservative forces are doing work on a body, then its total mechanical energy (K+U) remains constant.

Proof:

Suppose a body undergoes a small displacement ∆x under the action of a conservative force, F. According to work-energy i theorem, change in K.E = work done

∆K = F(x)∆x

As the force is conservative the change in j potential energy is given by

∆U = -F (x) ∆x (negative of work done)

∴ ∆K = -∆U

∆K + ∆U = 0 or ∆(K+U) = 0

⇒ K + U = constant

K_{i} + U_{i}– K_{t}+ U_{t }Although the kinetic energy K and potential energy U may change from one state to another, but their sum (total mechanical energy) remains constant under a conservative force.

Conservation of mechanical energy for a freely falling body:

Consider a body of mass ‘m’ at a height h from the ground.

ie., The total energy of the body during free fall remains constant at all positions. Note that the form of energy (K.E and P.E) keeps on changing.

The Potential Energy of a Spring

Consider an elastic spring with negligibly small mass with one end attached to a rigid support. Fig (1) shows the spring in its relaxed state ie., neither compressed nor extended.

If we stretch the spring by pulling the block to the right as in fig (2) the spring pulls on the block toward the left.

If we compress the spring by pushing the block to the left as in fig (3), the spring now pushes on the block toward the right.

The force trying to bring the spring back to its relaxed state (original configuration) is called restoring force or spring force.

For small stretch or compression, spring obeys Hooke’s law, Restoring force a stretch or compression

– Fα × or -F = kx or F= -kx

where k is called the spring constant.

Note:

The negative sign shows the force acts in the opposite direction of x. Also smaller the length of the spring, greater will be tile spring constant.

Work done by the spring force for the small extension is dx is dw = -Fdx = kx dx

If the block is moved from an initial displacement x_{i} to the final displacement x_{f} the work done by the spring force is

This work done will be equal to the increase in P.E of the spring and is given by,

If the block is pulled from x. and allowed to return to x , then

If we take the potential energy U of the spring to be zero when the block is in equilibrium position, the P.E of the spring for an extension x will be

Maximum Speed:

Note:

Total energy in all positions is same.The variation in K.E and PE with distance is as follows.

CAQ

Question 4.

Springs A and B are identical except that A is stiffer than B, i.e., force constant k_{A}> kg. In which spring is more work expended, If they are stretched by the same amount?

Answer:

Work done in stretching a spring of force constant k through a distance x

The Principle of Conservation of Energy

‘Energy cannot be created or destroyed it can be transformed from one form to another.”

Power:

Power is defined as the time rate at which work is done.

ie., power can be expressed as the dot product of force and velocity.

SI unit of power is watt (W). 1 W = 1 Js^{-1 }Dimensional formula [ML^{2}T^{3}]

CAQ

Question 5.

A man weighing 60 kg climbs up a stair case carrying a load of 20 kg on his head. The stair case has 20 steps each of height 0.2 m. If he takes 10 s to climb, find his power.

Answer:

Collisions:

When two bodies move, sometimes they can ; hit. Let us consider two bodies of masses m_{1 }and m_{2}. m_{1} is moving with initial velocity u_{1} and m_{2} is at rest, initially. After the collision m, moves with v_{1} and m_{2} moves with v_{2 }

In all types of collisions, momentum is conserved. But in all cases the kinetic energy is not conserved. In a collision, sound may be ; produced. That is energy. This energy is lost from the kinetic energy. That is why the kinetic energy is not conserved. A collision in which initial K.E equals final K.E(K.E is conserved) is called an elastic collision. If K.E it is inelastic collision.

Collisions in one Dimension:

If the initial velocities and final velocities of both the bodies are along the straight line, then it is called one dimensional motion

Consider two bodies fo masses m_{1} and m_{2 }moving with velocities u_{1} and u_{2} in the same direction and in the same line. If u, > u_{2} they will collide. After collision let v_{1} and v_{2} be their velocities.

By conservation of linear momentum:_{ }

If masses of A and B are equal, m_{1} = m_{2} = m

Inelastic colllision in one-dimension

As the total linear momentum of the system reamins constant. Total momentum before collision = total momentum after the collision

m_{1}u_{1} + m_{2}u_{2} = m_{1}v_{1} + m_{2}v_{2}

Collision in Two Dimensions:

Consider two bodies of masses m_{1 }and m_{2} moving with velocities u_{1} and u_{2} along parallel lines. If u, > u_{2}, they will collide. Let v_{1} and v_{2} be their velocities after collision along directions θ_{1} and θ_{2}. v_{1} and v_{2} can be resolved into v_{1} cos θ_{1} v_{2}cos θ_{2} parallel to x axis and v_{1} sin θ_{1}, and v_{2}sin θ_{2} parallel to Y-axis. .

By conservation of momentum parallel to X-axis, m_{1}u_{1}+m_{2}u_{2}= m_{1}v_{1}cosθ_{1}+m_{2}v_{2}cosθ_{2 }By conservation of momentum parallel to Y = axis. m_{1}v_{1}sinθ_{1}, + m_{2}v_{2}sinθ_{2} = 0+0 = 0

By conservation of energy,

## Leave a Reply