Work, Energy, and Power

• Work, energy, and power are fundamental concepts in physics, with energy and work being closely related. Progress of civilization depends on the availability of usable energy.

Objectives of Studying Work, Energy, and Power

• Define work done by a force and state its unit
• Calculate work done by an applied force
• State the work-energy theorem
• Define the power of a system
• Calculate work done by gravity when a mass moves
• Explain the meaning of energy
• Obtain expressions for gravitational and elastic potential energy
• Apply the principle of conservation of energy
• Apply the laws of conservation of momentum and energy in elastic collisions

Work

• Has a definite meaning in science, which is not always the same as the common usage of the word
• Work is the product of the magnitude of force component in the direction of displacement, and the displacement of the object
• W = F cos(θ) * d, where W is work, F is force, θ is the angle between the force and displacement, and d is displacement
• Zero displacement implies zero work
• Work is a scalar quantity
• Units of work: newton-metre (Nm) or joule (J); 1 joule is the work done by a force of one newton in displacing an object by one metre
• Dimensional formula of work is [ML²T⁻²]
• Kilowatt-hour (kWh) is another unit of work used in electrical measurements; 1 kWh = 3.6 x 10^6 J

Positive and Negative Work

• The sign of work depends on the angle θ between the force and displacement
• Positive work: θ = 0°, force and displacement are in the same direction
• Negative work: Force and displacement are in opposite directions, θ lies between 90° and 270°
• Zero work: Force and displacement are at right angles (θ = 90°)

Work Done by the Force of Gravity

• When lifting an object, work is done against gravity
• Displacement is upwards
• Work done is negative: W = -mgh
• When lowering a mass, force (mg) and displacement (d) are in the same direction
• Work done is positive: W = + mgh
• When an object is lifted up, the work done by gravitational force is negative, and the work done by the person lifting the object is positive
• When an object is lowered, the work done by gravitational force is positive, and the work done by the person lowering the object is negative

Work Done by a Variable Force

• The magnitude of the force changes with the position x of the object
• Calculated over small intervals of displacement Δx, where the force can be assumed constant
• Work done during a small displacement interval is ΔW = F(x) Δx
• The total work done by the force between positions x₁ and x₂ is the sum of all such areas
• W = ΣF(x) Δx

Work Done by a Spring

• Force exerted by a spring is a simple example of variable force
• Spring constant k is defined by |F| = kx
• The external force F causing the compression is directed towards left and the displacement x is also towards left
• The work done by the external force is positive
• A restoring force is generated in the spring is towards right
• The work done by the spring force is negative
• Magnitude of the work done is (1/2) kx²
• Average force during compression can be approximated to F/2
• Work done by the force is W = (1/2)kx²

Power

• Defined as the rate at which work is done
• Average power = Work done / time taken
• P = ΔW/Δt
• Instantaneous power is P = limit Δt->0 (ΔW/Δt) = dW/dt
• SI unit of power is watt (W), where 1 watt is 1 joule of work done in 1 second
• Common units: kilowatt (kW) and megawatt (MW)
• 1 kW = 10³ W and 1 MW = 10^6 W
• Horse power (hp) is another unit of power; 1 hp = 746 W

Work and Kinetic Energy

• The capacity to do work is called energy
• Moving objects possess kinetic energy and can do work before coming to rest
• Consider an object of mass m moving along a straight line, influenced by a constant force F
• This force causes uniform acceleration a, where F = ma
• If the object's speed changes from v₁ at time t₁ to v₂ at time t₂
• It covers a distance s during the time interval Δt = (t₂ - t₁)
• Using equations of motion: v₂² = v₁² + 2as
• Work done on the Kinetic Energy is W = K₂ - K₁
• Work-Energy Theorem: The work done by the resultant of all forces acting on a body equals the change in kinetic energy of the body

Potential Energy

• Objects possess potential energy due to their position in space
• Gravitational Potential Energy: The potential energy possessed by a body in a gravitational field
• The force must act in a similar direction as the movement of the object
• Work done W = force × distance = mgh
• At each height the Kinetic energy = 1/2mu^2 at point P
• The total kinetic energy = 1/2 * m * 2gh = MGH = mgh
• The total Energy = 1/2 MV
• i.e., total energy is conserved.

Potential energy of springs

• External force is required to compress or stretch a spring
• From Eqn.(6.11) recall that work done by external force to compress a spring is given by
• This work is stored in the spring as elastic potential energy:
• When the spring is left free, it bounces back and the elastic potential energy of the spring is converted into kinetic energy of the mass m.

Conservation of Energy

• The total energy of an isolated system always remains constant
• Energy can transform from one form to another but the total remains unchanged

Conservative and Dissipative (Non-conservative) Forces

• Conservative Forces: The work done depends only on the vertical separation between the two points
• The examples Forces are the gravitational force and electrostatic forces

Elastic and Inelastic Collisions

• Elastic Collision: If the forces of interaction between the two bodies are conservative and the total kinetic energy is conserved
• Perfectly Inelastic collision: When two colliding bodies stick together after the collision and move as one single unit
• The momentum is conserved in all types of collisions