Work, Energy and Power

1. Work

In physics, the term 'work' has a very specific meaning that differs from its everyday usage. When a force acts on an object and causes it to move a certain distance in the direction of the force, then work is done. The work done by a force is defined as the product of the magnitude of the force and the distance moved by the object in the direction of the force.

1.1 Definition of Work

Mathematically, if a constant force 'F' acts on an object and the object undergoes a displacement 'd' in the direction of the force, then the work done 'W' is given by:

W = F × d

The unit of work in the International System of Units (SI) is the joule (J). One joule is defined as the work done when a force of one newton (N) moves an object through a distance of one meter (m) in its own direction.

If the force is not in the same direction as the displacement, we need to consider the component of the force that acts in the direction of the displacement. Let 'θ' be the angle between the force vector and the displacement vector. Then, the work done is given by:

W = F × d × cos(θ)

Here, F × cos(θ) is the component of the force in the direction of the displacement.

1.2 Types of Work

Based on the angle 'θ' between the force and displacement, work can be classified into three types:

• Positive Work: When 0° ≤ θ < 90°, cos(θ) is positive. This means the force has a component in the direction of displacement, and work done is positive. For example, when you lift a box upwards, the force you apply is upwards, and the displacement is also upwards. Here, θ = 0°, and cos(0°) = 1, so W = F × d.
• Negative Work: When 90° < θ ≤ 180°, cos(θ) is negative. This means the force has a component opposite to the direction of displacement, and work done is negative. For example, when a moving car's brakes are applied, the frictional force acts opposite to the direction of motion. If the car moves forward, the frictional force is backward, so θ = 180°, and cos(180°) = -1, resulting in negative work done by friction.
• Zero Work: When θ = 90°, cos(90°) = 0. In this case, the force is perpendicular to the displacement, and no work is done by the force. For example, if a coolie carries a load on his head and walks on a horizontal platform, the force applied by the coolie on the load is upwards (to counteract gravity), while the displacement is horizontal. The angle between these is 90°, so the work done by the coolie on the load is zero. Another example is a satellite moving in a circular orbit around the Earth. The gravitational force is always directed towards the center of the Earth, while the satellite's velocity (and thus displacement) is tangential to the orbit. Thus, the angle is 90°, and the gravitational force does no work on the satellite.

1.3 Work Done by Variable Force

When the force is not constant but varies with displacement, we can calculate the work done by integrating the force over the displacement. If the force F(x) varies with position 'x', the work done in moving an object from position x₁ to x₂ is given by:

W = ∫x₁x₂ F(x) dx

Graphically, this represents the area under the Force-Displacement curve.

1.4 Units of Work

• SI Unit: Joule (J). 1 J = 1 N·m.
• CGS Unit: erg. 1 erg = 1 dyne·cm.
• Relationship: 1 J = 10⁷ erg.
• Other Units: Kilowatt-hour (kWh) is a commercial unit of energy, often used for electrical energy. 1 kWh = 3.6 × 10⁶ J.

2. Energy

Energy is the capacity to do work. It is a fundamental property of matter and exists in various forms such as kinetic energy, potential energy, thermal energy, chemical energy, electrical energy, and nuclear energy. Energy cannot be created or destroyed, but it can be transformed from one form to another. This is known as the Law of Conservation of Energy.

2.1 Kinetic Energy (KE)

Kinetic energy is the energy possessed by an object due to its motion. An object in motion can do work on another object by virtue of its motion.

Consider an object of mass 'm' moving with a uniform velocity 'v'. Let a constant force 'F' act on it, causing a displacement 'd' and an acceleration 'a'.

From Newton's second law of motion, F = ma.

From the equations of motion, we have v² = u² + 2ad. If the object starts from rest, u = 0, so v² = 2ad, which means a = v²/2d.

The work done by the force is W = F × d.

Substituting F = ma and a = v²/2d:

W = (ma) × d = m × (v²/2d) × d = (1/2)mv²

This work done by the net force causes a change in the kinetic energy of the object. If the object starts from rest (initial kinetic energy = 0) and reaches a velocity 'v', the work done is equal to its final kinetic energy.

The kinetic energy (KE) of an object of mass 'm' moving with velocity 'v' is given by:

KE = (1/2)mv²

The SI unit of kinetic energy is the joule (J).

Work-Energy Theorem: The work done by the net force on an object is equal to the change in its kinetic energy.

W_net = ΔKE = KEfinal - KEinitial = (1/2)mvf² - (1/2)mvi²

2.2 Potential Energy (PE)

Potential energy is the energy possessed by an object due to its position or configuration. It is stored energy.

2.2.1 Gravitational Potential Energy

This is the energy possessed by an object due to its position in a gravitational field. Consider an object of mass 'm' at a height 'h' above the ground. The force of gravity acting on it is F = mg (downwards). To lift the object to height 'h', an external force equal to 'mg' must be applied upwards.

The work done against gravity in lifting the object is:

W = Force × distance = (mg) × h

This work done is stored as potential energy in the object. The gravitational potential energy (PE) of an object of mass 'm' at height 'h' above a reference level is given by:

PE = mgh

The SI unit of potential energy is the joule (J).

Change in Potential Energy: When an object is moved from height h₁ to h₂, the change in potential energy is ΔPE = mg(h₂ - h₁). If h₂ > h₁, PE increases (positive change); if h₂ < h₁, PE decreases (negative change).

2.2.2 Elastic Potential Energy

This is the energy stored in a deformed elastic object, such as a stretched or compressed spring. When a spring is stretched or compressed by a distance 'x' from its equilibrium position, a restoring force acts on it, given by Hooke's Law: F = -kx, where 'k' is the spring constant and the negative sign indicates that the force is opposite to the displacement.

The work done in stretching or compressing the spring by a distance 'x' is:

W = (1/2)kx²

This work is stored as elastic potential energy (PE_elastic) in the spring.

PEelastic = (1/2)kx²

2.3 Conservation of Mechanical Energy

In the absence of non-conservative forces like friction and air resistance, the total mechanical energy (the sum of kinetic energy and potential energy) of a system remains constant. This is the principle of conservation of mechanical energy.

Total Mechanical Energy (E) = Kinetic Energy (KE) + Potential Energy (PE)

E = KE + PE = constant

This means that as an object moves under the influence of conservative forces, kinetic energy can be converted into potential energy, and vice versa, but their sum will always remain the same.

Example: A freely falling object. Let an object of mass 'm' be dropped from a height 'H'. At height H (just before dropping): KE = 0, PE = mgH. Total Energy E = mgH. At height H/2: Let its velocity be 'v'. PE = mg(H/2). Using v² = u² + 2gh, where u=0 and h=H/2, we get v² = 2g(H/2) = gH. So, KE = (1/2)mv² = (1/2)m(gH) = mgH/2. Total Energy E = KE + PE = mgH/2 + mgH/2 = mgH. Just before hitting the ground (height 0): PE = 0. Using v² = u² + 2gh, where u=0 and h=H, we get v² = 2gH. So, KE = (1/2)mv² = (1/2)m(2gH) = mgH. Total Energy E = KE + PE = mgH + 0 = mgH. In all cases, the total mechanical energy remains constant.

3. Power

Power is the rate at which work is done or energy is transferred. It measures how quickly work can be performed.

3.1 Definition of Power

If work 'W' is done in time 't', then the average power 'P' is given by:

P = W / t

The SI unit of power is the watt (W). One watt is defined as one joule of work done per second.

1 W = 1 J/s

If the rate of doing work is not constant, we talk about instantaneous power, which is the rate of doing work at a particular instant.

3.2 Power and Velocity

Power can also be expressed in terms of force and velocity. If a force 'F' acts on an object moving with velocity 'v' in the direction of the force, the instantaneous power delivered by the force is:

P = F × v

This can be derived from P = W/t and W = F × d. So, P = (F × d) / t. Since v = d/t (for constant velocity), P = F × v.

3.3 Units of Power

• SI Unit: Watt (W).
• Other Units:
  • Kilowatt (kW): 1 kW = 1000 W.
  • Megawatt (MW): 1 MW = 10⁶ W.
  • Horsepower (hp): A non-SI unit, commonly used for engines. 1 hp ≈ 746 W.

3.4 Relationship between Work, Energy, and Power

Power is the bridge between work and time, and since work is a measure of energy transfer, power is also the rate of energy transfer.

Power = Energy Transferred / Time Taken

P = E / t

For example, a 100 W light bulb converts 100 joules of electrical energy into light and heat energy every second.

4. Examples and Applications

Understanding work, energy, and power is crucial in many fields of science and engineering.

4.1 Lifting Weights

A weightlifter lifts a barbell of mass 100 kg to a height of 2 meters.

• Work done: W = F × d = (mg) × h = (100 kg × 9.8 m/s²) × 2 m = 1960 J.
• If the lifter takes 2 seconds to lift the barbell, the average power is P = W / t = 1960 J / 2 s = 980 W.
• If they lift it in 1 second, the power is P = 1960 J / 1 s = 1960 W. More power means the work is done faster.

4.2 Moving a Car

A car of mass 1000 kg is moving at 20 m/s.

• Its kinetic energy is KE = (1/2)mv² = (1/2) × 1000 kg × (20 m/s)² = 500 × 400 = 200,000 J.
• If the brakes are applied and the car stops in 10 seconds, the frictional force does negative work. The work done by friction is equal to the change in kinetic energy: Wfriction = ΔKE = 0 - 200,000 J = -200,000 J.
• The average power dissipated by friction is P = |Wfriction| / t = 200,000 J / 10 s = 20,000 W = 20 kW.

4.3 Inclined Plane

Consider pushing a box of mass 50 kg up an inclined plane of length 10 m and height 3 m. The force required to push it up is 200 N.

• Work done: The displacement along the plane is 10 m. W = F × d = 200 N × 10 m = 2000 J.
• The work done against gravity is Wgravity = mgh = 50 kg × 9.8 m/s² × 3 m = 1470 J. The difference (2000 J - 1470 J = 530 J) is likely due to friction or the force used to accelerate the box.
• If the push takes 5 seconds, the power is P = W / t = 2000 J / 5 s = 400 W.

4.4 Electrical Appliances

The power rating of electrical appliances tells us how quickly they consume energy.

• A 1000 W heater converts 1000 J of electrical energy into heat energy every second.
• A 60 W light bulb converts 60 J of electrical energy into light and heat energy every second.
• The energy consumed by an appliance is given by Energy = Power × Time. For example, a 1000 W heater used for 1 hour consumes 1000 W × 3600 s = 3,600,000 J, which is equal to 1 kWh (kilowatt-hour).

4.5 Biological Systems

Living organisms continuously perform work and utilize energy. Muscle contractions involve converting chemical energy into mechanical work. The metabolic rate of an organism is a measure of its power consumption. For instance, a resting human might consume energy at a rate of about 100 W, while during intense exercise, this rate can increase to over 1000 W.

The efficiency of energy transfer is also a key concept. For example, when a car engine burns fuel, only a fraction of the chemical energy is converted into useful mechanical work; the rest is lost as heat. Similarly, biological systems have efficiencies that vary depending on the process.

Summary of Concepts

Concept	Formula	SI Unit	Description
Work (Constant Force)	W = Fd cos(θ)	Joule (J)	Force applied over a distance in the direction of the force.
Kinetic Energy	KE = (1/2)mv²	Joule (J)	Energy due to motion.
Gravitational Potential Energy	PE = mgh	Joule (J)	Energy due to position in a gravitational field.
Elastic Potential Energy	PE = (1/2)kx²	Joule (J)	Energy stored in a deformed spring.
Work-Energy Theorem	Wnet = ΔKE	Joule (J)	Net work done equals change in kinetic energy.
Conservation of Mechanical Energy	KE + PE = Constant (in absence of non-conservative forces)	Joule (J)	Total mechanical energy remains constant.
Average Power	P = W / t or P = ΔE / t	Watt (W)	Rate at which work is done or energy is transferred.
Instantaneous Power	P = Fv	Watt (W)	Power at a specific moment.