Work, Energy and Power — Class 11 Physics

1. Work Done by a Constant Force

Work is done when a force produces displacement. For a constant force F making angle θ with displacement s, W = Fs cosθ. Work is a scalar quantity and can be positive, negative or zero.

If force and displacement are in the same direction, θ=0° and W=Fs. If they are opposite, θ=180° and W=−Fs. If they are perpendicular, θ=90° and work is zero.

2. Work Done by a Variable Force

When force changes with position, work is represented by the area under the force–displacement graph. In calculus notation, W = ∫F·dr. For one-dimensional motion, this becomes the area under the F-x graph.

3. Kinetic Energy

Kinetic energy is the energy associated with motion: K = ½mv². It is always non-negative for an ordinary particle because mass is positive and v² is non-negative.

4. Work-Energy Theorem

The net work done on a particle equals its change in kinetic energy: W_net = ΔK = K_f−K_i. This theorem is especially useful when forces vary or when time is not directly required.

5. Potential Energy

Potential energy is associated with configuration or position. Near Earth’s surface, gravitational potential energy relative to a chosen zero level is U=mgh. The zero level can be selected conveniently; only differences in potential energy affect physical predictions.

For a spring obeying Hooke’s law, F=kx in magnitude, and elastic potential energy is U=½kx².

6. Conservative and Non-Conservative Forces

A conservative force does work that depends only on initial and final positions, not on the path. Gravity and ideal spring force are conservative. Friction is non-conservative because its work depends on the path length.

7. Conservation of Mechanical Energy

If only conservative forces do work, the total mechanical energy remains constant: K+U=constant. When non-conservative forces such as friction act, their work changes the mechanical energy.

8. Power

Average power is work done per unit time: P_avg=W/t. Instantaneous power is P=F·v. The SI unit of power is watt (W), where 1 W = 1 J/s.

Worked Numerical 1 — Work at an Angle

Question: A 20 N force pulls a box through 5 m at an angle of 60° to the displacement. Find the work done by the force.

Solution: W=Fs cosθ =20×5×cos60° =100×½ = 50 J.

Worked Numerical 2 — Work-Energy Theorem

Question: A 2 kg object speeds up from 3 m/s to 7 m/s. Find the net work done.

W_net=½m(v²−u²)=½(2)(49−9)=40 J.

Worked Numerical 3 — Stopping Distance

Question: A 1000 kg car moving at 20 m/s is brought to rest. How much work does the net retarding force do?

W_net=K_f−K_i=0−½(1000)(20²)=−2.0×10⁵ J. The negative sign shows that the net force removes kinetic energy.

Worked Numerical 4 — Gravitational Potential Energy

Question: A 5 kg object is raised through 4 m. Take g=10 m/s². Find the increase in gravitational potential energy.

ΔU=mgΔh=5×10×4=200 J.

Worked Numerical 5 — Spring Energy

Question: A spring of force constant 200 N/m is compressed by 0.10 m. Find the elastic potential energy stored.

U=½kx²=½(200)(0.10)²=1 J.

Worked Numerical 6 — Power

Question: A machine does 6000 J of work in 20 s. Find its average power.

P=W/t=6000/20=300 W.

Worked Numerical 7 — Power from Force and Velocity

Question: A motor exerts a constant driving force of 500 N on a vehicle moving at 12 m/s in the same direction. Find instantaneous power.

P=Fv=500×12=6000 W = 6 kW.

Worked Numerical 8 — Conservation of Mechanical Energy

Question: A 1 kg ball is dropped from a height of 20 m. Ignore air resistance and take g=10 m/s². Find its speed just before reaching the ground.

Initially, E=mgh=1×10×20=200 J. At ground, U=0 and K=200 J. Thus ½mv²=200, giving v²=400 and v=20 m/s.

Explained MCQs

1. A force perpendicular to displacement does what work?
   Answer: Zero. Since cos90°=0.
2. If the net work on an object is positive, its kinetic energy:
   Answer: Increases. The work-energy theorem gives W_net=ΔK.
3. Which is a conservative force?
   Answer: Gravitational force. Its work between two points is independent of the path.
4. What happens to kinetic energy if speed doubles?
   Answer: It becomes four times. K is proportional to v².
5. The instantaneous power delivered by a force is:
   Answer: F·v. It is the rate at which the force transfers energy.
6. Can work done by a force be negative?
   Answer: Yes. Friction or a retarding force can do negative work by opposing displacement.

Competency Question

A person carries a school bag horizontally at constant height and constant speed. The upward force exerted by the person on the bag is approximately perpendicular to the bag’s horizontal displacement. Discuss the mechanical work done by this upward force and distinguish it from the metabolic energy the person expends.

HOTS

A block slides down a frictionless inclined plane from rest. Explain, without calculating time, how its speed at the bottom can be determined using conservation of mechanical energy. What changes if friction is present?

Common Mistakes

• Using W=Fs without checking the angle between force and displacement.
• Confusing net work with work done by one individual force.
• Assuming zero work means zero force; a perpendicular force can be non-zero and do zero instantaneous work.
• Forgetting that potential energy depends on the chosen reference level, while potential-energy differences are physical.
• Using P=Fv without considering the angle; the general expression is P=F·v.

Quick Revision

• W=Fs cosθ
• W_net=ΔK
• K=½mv²
• U_gravity=mgh near Earth’s surface
• U_spring=½kx²
• Mechanical energy E=K+U
• P_avg=W/t
• P=F·v
• 1 watt = 1 joule per second