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Work and energy: Class 11 Physics
Work as a dot product, kinetic and potential energy, the work energy theorem, conservative forces, conservation of energy, power, efficiency, and sources of energy.
Key points
In physics, work is done when a force moves its point of application. The work done by a constant force F over a displacement d is W = F·d = Fd cos θ, where θ is the angle between the force and the displacement. Work is a scalar. Its SI unit is the joule (J): 1 J = 1 N × 1 m.
The sign of work depends on the angle θ:
- θ = 0°: work is positive and equals Fd (a force pushing a trolley forward).
- θ = 90°: work is zero. The force of a satellite's centripetal pull or the normal reaction does no work.
- θ = 180°: work is negative, as with friction that opposes the motion.
- No displacement means no work, even if the force is large. A person holding a heavy bag still does no work on it.
When the force changes with position, the work equals the area under the force displacement graph. For a spring that obeys Hooke's law, F = kx, so the work to stretch it by x is the area of a triangle: W = ½kx². Here k is the spring constant in N/m.
Energy is the capacity to do work. Its unit is also the joule. Mechanical energy is of two kinds, kinetic and potential. Other forms are heat, light, sound, chemical, electrical and nuclear energy.
Kinetic energy is the energy a body has because of its motion: KE = ½mv². It depends on the square of the speed, so doubling the speed makes the kinetic energy four times larger. The work energy theorem says that the net work done on a body equals the change in its kinetic energy: W = ½mv² − ½mu². It gives a quick way to solve many problems without finding the acceleration.
Potential energy is the energy stored in a body because of its position or state. Gravitational potential energy near the Earth is PE = mgh, where h is the height above a chosen reference level. The work done against gravity in lifting a mass depends only on the vertical height, not on the path. Elastic potential energy stored in a stretched or compressed spring is ½kx².
Conservative and non-conservative forces:
- A force is conservative if the work done by it depends only on the starting and end points, not on the path. Gravity and the spring force are examples.
- The work done by a conservative force over a closed path is zero.
- Friction and air resistance are non-conservative. The work they do depends on the path, and mechanical energy is changed into heat.
- Potential energy can be defined only for conservative forces.
The law of conservation of energy: energy can neither be created nor destroyed. It changes from one form to another, and the total remains constant. When only conservative forces act, the mechanical energy KE + PE is constant. A body falling from a height h loses potential energy mgh and gains the same amount of kinetic energy. Just before it reaches the ground, ½mv² = mgh, so v = √(2gh), independent of the mass. For a pendulum, energy swaps between kinetic and potential at each swing.
Power is the rate of doing work: P = W/t. Its SI unit is the watt (W), equal to 1 J/s. Larger units are the kilowatt (1 kW = 1000 W) and horsepower (1 hp is about 746 W). When a force moves a body at velocity v, P = Fv. Electrical energy is billed in kilowatt hours: 1 kWh = 3.6 × 10⁶ J.
Efficiency is the ratio of useful output energy (or power) to the total input: efficiency = (output / input) × 100%. No machine is 100% efficient, because some energy is always lost as heat, sound or friction.
Sources of energy: fossil fuels (coal, oil and gas) are non-renewable and release carbon dioxide. Renewable sources include hydro, solar, wind and geothermal energy. Pakistan makes much of its electricity from hydro and thermal power stations, and is increasing its use of solar and wind power. A hydroelectric station changes the gravitational potential energy of stored water into kinetic energy, then into electrical energy through turbines and generators.
Study tip: Choose between Newton's laws and the energy method. If a question gives speeds and heights but no time, the energy method is usually shorter. Write the energy equation in words first, such as loss of PE equals gain of KE, then insert numbers with units.
Definitions and flashcards
Say the meaning first, then open the card to check.
Work
The product of the force and the displacement in the direction of the force.
Joule
The work done when a force of 1 N moves a body 1 m in its own direction.
Kinetic energy
Energy possessed by a body due to its motion.
Potential energy
Energy stored due to position or state.
Conservative force
A force whose work does not depend on the path taken.
Power
The rate of doing work.
Watt
A power of one joule per second.
Efficiency
The ratio of useful output to total input, usually as a percentage.
Work energy theorem
Net work done equals the change in kinetic energy.
Formula sheet
| Name | Formula | What it means |
|---|---|---|
| Work | W = Fd cos θ | θ is the angle between force and displacement. |
| Kinetic energy | KE = ½mv² | Unit joule. |
| Gravitational PE | PE = mgh | h is height above the reference level. |
| Spring energy | E = ½kx² | k is the spring constant. |
| Work energy theorem | W = ½mv² − ½mu² | Net work equals change in KE. |
| Power | P = W/t = Fv | Unit watt. |
| Efficiency | η = (output/input) × 100% | Always less than 100%. |
Short questions with model answers
These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.
Define work and state its SI unit. (2 marks)
Work is the scalar product of force and displacement, W = Fd cos θ. Its SI unit is the joule, which is the work done when a force of one newton moves a body one metre in the direction of the force.
When is the work done by a force zero? Give two cases. (3 marks)
Work is zero when the displacement is zero, for example pushing against a fixed wall. It is also zero when the force is perpendicular to the displacement (θ = 90°), for example the normal reaction on a body sliding on a table, or the centripetal force on a body moving in a circle.
State the work energy theorem and prove it for a constant force. (5 marks)
The work done by the net force on a body equals the change in its kinetic energy. Proof: let a body of mass m change its velocity from u to v under a constant net force F over a displacement s. Then F = ma and v² = u² + 2as, so as = (v² − u²)/2. Work W = Fs = mas = m(v² − u²)/2 = ½mv² − ½mu². So W = KE final − KE initial.
Why is friction called a non-conservative force? (2 marks)
The work done by friction depends on the length of the path, not just on the end points, and it changes mechanical energy into heat. The work over a closed path is not zero.
Show that the speed of a freely falling body at the ground is independent of its mass. (3 marks)
A body of mass m falls from height h. By conservation of energy, the loss of potential energy equals the gain in kinetic energy: mgh = ½mv². The mass m cancels on both sides, so v = √(2gh). Hence the speed depends only on g and h.
Define power and derive P = Fv. (3 marks)
Power is the rate of doing work, P = W/t. If a constant force F moves a body through a distance s in its own direction in time t, then W = Fs. So P = Fs/t. Since s/t is the velocity v, P = Fv.
Numericals with worked steps
A force of 100 N pulls a box 5 m along the floor. The force makes an angle of 60° with the floor. Find the work done.
- W = Fd cos θ.
- W = 100 × 5 × cos 60° = 500 × 0.5 = 250.
Answer: 250 J.
A car of mass 1200 kg moves at 72 km/h. Find its kinetic energy.
- Convert the speed: 72 km/h = 72 × 1000/3600 = 20 m/s.
- KE = ½mv² = ½ × 1200 × 20² = 600 × 400 = 240000 J.
Answer: 2.4 × 10⁵ J.
A 2 kg stone falls from rest through 5 m. Find its potential energy loss and its speed on reaching the ground. Take g = 9.8 m/s².
- Loss of PE = mgh = 2 × 9.8 × 5 = 98 J.
- This becomes kinetic energy: ½ × 2 × v² = 98, so v² = 98.
- v = √98 = 9.90 m/s.
Answer: 98 J and 9.9 m/s.
A motor lifts a 50 kg load through 10 m in 20 s. Find the power developed. Take g = 9.8 m/s².
- Work = mgh = 50 × 9.8 × 10 = 4900 J.
- P = W/t = 4900/20 = 245 W.
Answer: 245 W.
A machine takes in 500 W and gives out 400 W of useful power. Find its efficiency.
- Efficiency = output/input × 100%.
- = 400/500 × 100% = 0.8 × 100% = 80%.
Answer: 80%.
Common mistakes
- Using the total distance instead of the displacement in the direction of the force, or forgetting the cos θ factor.
- Saying a person holding a load does work on it. With no displacement, no work is done.
- Forgetting to change km/h to m/s before calculating kinetic energy.
- Thinking that kinetic energy doubles when the speed doubles. It becomes four times larger.
- Using conservation of mechanical energy when friction acts. Include the work done against friction.