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State Board · Class 9 · Science and Technology

Lesson 2: Work and Energy

5 concepts · 40 practice questions with answers · Free to read

This lesson is also in: मराठी

Work

From daily life

A student pushes a heavy cupboard with all his strength but it does not move. He is tired, yet in physics he has done no work. Work needs a force and a movement in the direction of that force.

Simple definition

Work (कार्य) = force × displacement in the direction of the force: W = F s cos θ.

A force at angle theta moves a box; only the part F cos theta along the displacement does workWork = F × s × cos θs (displacement)FθF cos θ

In the figure : In the figure only the part of the force along the displacement does work.

Rule 1: Work done by a force — Work = force × displacement along the force, W = F s cos θ, where θ is the angle between the force and the displacement. If there is no displacement, the work is zero. A force acting at 90° to the displacement does zero work, for example the pull on a satellite in a circular orbit.

  • Easy : A force of 10 N moves a box 5 m in its own direction (θ = 0°, cos 0° = 1): W = 10 × 5 × 1 = 50 J.
  • Medium : A 20 N pull at 60° moves a trolley 4 m (cos 60° = 0.5): W = 20 × 4 × 0.5 = 40 J.
  • Exam-type : A man carries a bag horizontally; the force (weight support) is vertical and the displacement horizontal, so θ = 90°, cos 90° = 0 and W = 0.
Live figure

Work

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Remember

W = F s cos θ. No displacement means zero work; force at 90° does zero work.

Common mistake

Thinking effort or tiredness always means work. Work needs displacement along the force.

Shortcut

Memory clue: "No move, no work; sideways force, no work".

Spot the difference

SituationWork
θ = 0°F × s (maximum)
θ = 60°F × s × 0.5
θ = 90°Zero
No displacementZero

Practice questions

Easy
  1. 1Write the formula for work.
  2. 2When is the work done by a force zero?
  3. 3What is θ in the formula?
Medium
  1. 4Does a satellite in a circular orbit gain energy from gravity? Why?
  2. 5Find the work done by 15 N force moving a body 6 m along the force.
  3. 6A 40 N force at 60° moves a body 5 m. Find the work (cos 60° = 0.5).
Challenging
  1. 7A student pushes a wall with 200 N for a minute. How much work is done on the wall?
  2. 8A force of 30 N and displacement of 2 m give work of 30 J. Find the angle's cos θ.
Show answers (try them yourself first)
  1. 1W = F s cos θ.
  2. 2When there is no displacement, or the force is at 90° to the displacement.
  3. 3The angle between the force and the displacement.
  4. 4No; gravity is at 90° to the motion, so it does zero work.
  5. 515 × 6 = 90 J.
  6. 640 × 5 × 0.5 = 100 J.
  7. 7Zero, as the wall does not move.
  8. 8cos θ = 30 ÷ (30 × 2) = 0.5, so θ = 60°.

Units and positive/negative work

From daily life

Pushing a trolley forward is helpful work. Friction rubs against the motion and takes energy away, so we call it negative work. The sign of work tells us whether a force helps or opposes the motion.

Simple definition

The SI unit of work is the joule (जूल). 1 J = 1 N·m = 107 erg.

Positive work for angles below 90 degrees, zero at 90 and negative at 180Sign of workθ < 90°🧑‍🦽Pushing atrolley➕Positiveworkθ = 90°🛰️Satellite incircular orbit0️⃣Zeroworkθ = 180°🛷Frictionslows it➖Negativework

In the figure : In the figure when work is positive, zero or negative.

Rule 1: Unit of work — The SI unit of work is the joule. 1 joule = 1 newton × 1 metre = 107 erg.

  • Easy : 1 N moving a body 1 m is 1 J.
  • Medium : 5 J = 5 × 107 erg.
  • Exam-type : A 50 N force moves a body 0.4 m: W = 50 × 0.4 = 20 J.

Rule 2: Positive, zero and negative work — Work is positive if θ is less than 90°, zero if θ is 90° and negative if θ is 180°. Friction does negative work.

  • Easy : Pushing a cart forward: positive work.
  • Medium : Friction of 5 N acts against a 3 m movement: W = 5 × 3 × cos 180° = −15 J.
  • Exam-type : A ball thrown upward: gravity pulls down while the ball goes up, so gravity does negative work; while it falls, gravity does positive work.
Remember

1 J = 1 N·m = 107 erg. Friction: negative work.

Common mistake

Thinking negative work means no work. It means the force opposes the motion.

Shortcut

Memory clue: "0° helps (+), 90° nothing (0), 180° opposes (−)".

Spot the difference

Angle θWork
0°Positive
90°Zero
180°Negative

Practice questions

Easy
  1. 1What is the SI unit of work?
  2. 2What is 1 joule in N·m and in erg?
  3. 3When is work positive?
Medium
  1. 4When is work negative?
  2. 5Is the work done by friction positive or negative?
  3. 6Convert 3 J into erg.
Challenging
  1. 7A 8 N friction acts on a block moving 2.5 m. Find the work done by friction.
  2. 8Explain why gravity does negative work on a ball going up but positive work when it comes down.
Show answers (try them yourself first)
  1. 1Joule.
  2. 21 N·m = 107 erg.
  3. 3When θ is less than 90°.
  4. 4When θ = 180°, as with friction.
  5. 5Negative.
  6. 63 × 107 erg.
  7. 7W = 8 × 2.5 × (−1) = −20 J.
  8. 8Going up, gravity is opposite to the motion (θ = 180°); coming down it is in the same direction (θ = 0°).

Kinetic energy

From daily life

A cricket ball rolling slowly can be stopped with your foot, but a fast ball can hurt. Moving things carry energy, and faster things carry much more.

Simple definition

Kinetic energy (गतिज ऊर्जा) is the energy of a moving body: KE = ½ m v².

Bars show kinetic energy rising 1, 4 and 9 times when speed is v, 2v and 3vKE = ½mv²: double the speed, four times the KEKE × 1speed 1vKE × 4speed 2vKE × 9speed 3v

In the figure : In the figure kinetic energy grows with the square of speed.

Rule 1: Kinetic energy — KE = ½ m v². Doubling the speed gives four times the KE. Work-energy theorem: the net work done on a body equals the change in its kinetic energy.

  • Easy : m = 2 kg and v = 6 m/s: KE = ½ × 2 × 36 = 36 J.
  • Medium : At 12 m/s (double) the KE is ½ × 2 × 144 = 144 J, which is 4 × 36 J.
  • Exam-type : A 2 kg body at rest gets 100 J of net work. Its KE is 100 J, so ½ × 2 × v² = 100, v² = 100, v = 10 m/s.
Live figure

Kinetic energy

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Remember

KE = ½ m v². Net work = change in KE.

Common mistake

Forgetting to square the speed. v is squared, not doubled.

Shortcut

Memory clue: "Speed squared, half mass".

Spot the difference

ChangeEffect on KE
Speed × 2× 4
Speed × 3× 9
Mass × 2× 2

Practice questions

Easy
  1. 1Write the formula for kinetic energy.
  2. 2What happens to KE if speed is doubled?
  3. 3State the work-energy theorem.
Medium
  1. 4Find the KE of a 4 kg body moving at 3 m/s.
  2. 5What happens to KE if the speed becomes three times?
  3. 6A 1 kg body at 4 m/s speeds up to 10 m/s. Find the work done.
Challenging
  1. 7A 3 kg body has KE of 96 J. Find its speed.
  2. 8Why is a speeding truck more dangerous at double speed than double the load?
Show answers (try them yourself first)
  1. 1KE = ½ m v².
  2. 2It becomes four times.
  3. 3Net work done on a body equals the change in its kinetic energy.
  4. 4½ × 4 × 9 = 18 J.
  5. 5It becomes 9 times.
  6. 6½ × 1 × (100 − 16) = 42 J.
  7. 7½ × 3 × v² = 96, v² = 64, v = 8 m/s.
  8. 8Double speed makes the KE four times, while double load makes it only two times.

Potential energy and transformation

From daily life

A brick on the roof can hurt when it falls, although it lies still up there. It stores energy because of its height. When it falls, that stored energy turns into motion.

Simple definition

Potential energy (स्थितिज ऊर्जा) is stored energy due to position: PE = m g h.

Light becomes electrical, then kinetic, then heat energyEnergy changes form🌞Lightenergy🔋Electricalenergy🌀Kineticenergy (fan)🔥Heatenergy

In the figure : In the figure energy changing from one form to another.

Rule 1: Potential energy — PE = m g h, measured relative to a chosen reference level. The height h is measured from that level.

  • Easy : m = 5 kg, h = 4 m, g = 9.8 m/s²: PE = 5 × 9.8 × 4 = 196 J.
  • Medium : The same body at 8 m has PE = 5 × 9.8 × 8 = 392 J, double the height gives double the PE.
  • Exam-type : A 2 kg book moves from a 1 m shelf to a 3 m shelf. Gain in PE = 2 × 9.8 × (3 − 1) = 39.2 J.
Live figure

Potential energy and transformation

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Rule 2: Transformation of energy — Energy changes form: chemical, electrical, heat, light, sound, nuclear, kinetic and potential.

  • Easy : A torch changes chemical energy of the cell into light and heat.
  • Medium : A fan changes electrical energy into kinetic energy (and a little heat and sound).
  • Exam-type : A dam: water at height (potential) → falls (kinetic) → turbine turns → generator makes electrical energy.
Remember

PE = m g h. Energy changes from one form to another.

Common mistake

Measuring h from the wrong level. Always use the reference level you chose.

Shortcut

Memory clue: "PE = m g h : Mass, Gravity, Height".

Spot the difference

DeviceEnergy change
Electric bulbElectrical → light and heat
LoudspeakerElectrical → sound
BatteryChemical → electrical
Falling bodyPotential → kinetic

Practice questions

Easy
  1. 1What is potential energy?
  2. 2Write the formula for PE.
  3. 3Name four forms of energy.
Medium
  1. 4What energy change takes place in a battery?
  2. 5Find the PE of a 3 kg body at 10 m (g = 9.8 m/s²).
  3. 6A body has PE of 147 J at 5 m height. Find its mass (g = 9.8 m/s²).
Challenging
  1. 7Describe the energy changes in a hydroelectric dam.
  2. 8Why is PE always stated "relative to a level"?
Show answers (try them yourself first)
  1. 1Stored energy due to the position of a body.
  2. 2PE = m g h.
  3. 3Chemical, electrical, heat, light (also sound, nuclear, kinetic, potential).
  4. 4Chemical energy into electrical energy.
  5. 53 × 9.8 × 10 = 294 J.
  6. 6m = 147 ÷ (9.8 × 5) = 3 kg.
  7. 7Potential energy of water → kinetic energy → turbine rotation → electrical energy.
  8. 8Height h depends on which level we take as zero, so the PE value also depends on it.

Conservation, free fall and power

From daily life

A swing at its highest point stops for an instant and has all its energy as height. At the lowest point it moves fastest. The energy has changed form, but the total is not lost.

Simple definition

The law of conservation of energy says energy is neither created nor destroyed. Power (शक्ती) = work ÷ time.

A falling ball loses potential energy and gains the same kinetic energy; their sum stays constantFree fall: PE + KE stays the sametop= mghmiddle= mghground= mghPEKE

In the figure : In the figure PE falls and KE rises; the total stays the same.

Rule 1: Conservation of energy and free fall — Energy is neither created nor destroyed. In free fall, PE + KE = constant; the PE lost equals the KE gained, which is m g h.

  • Easy : A 1 kg ball dropped from 10 m: PE = 1 × 9.8 × 10 = 98 J at the top.
  • Medium : Just before it hits the ground, KE = 98 J, so ½ × 1 × v² = 98, v² = 196, v = 14 m/s.
  • Exam-type : Halfway down (5 m) the ball has PE = 49 J and KE = 49 J, and 49 + 49 = 98 J, the same total.
Live figure

Conservation, free fall and power

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1000 watts for 3600 seconds is 3.6 million joules or 1 kilowatt hourOne unit of electricity1000 W(1 kW)× 3600 s(1 hour)= 3.6 × 10⁶ J= 1 kWh

In the figure : In the figure one kilowatt-hour in joules.

Rule 2: Power and units — Power = W ÷ t. The unit is the watt. 1 hp = 746 W. 1 kWh = 3.6 × 106 J.

  • Easy : 600 J of work in 20 s: P = 600 ÷ 20 = 30 W.
  • Medium : A 100 W bulb for 10 hours uses 100 × 10 = 1000 Wh = 1 kWh.
  • Exam-type : A 2 kW heater for 3 hours uses 6 kWh = 6 × 3.6 × 106 = 2.16 × 107 J.
Remember

PE + KE = constant in free fall. P = W ÷ t. 1 kWh = 3.6 × 106 J.

Common mistake

Thinking kWh is a unit of power. It is a unit of energy.

Shortcut

Memory clue: "kW = rate, kWh = amount".

Spot the difference

QuantityUnit
Work, energyjoule
Powerwatt (1 hp = 746 W)
Electrical energykWh (3.6 × 106 J)

Practice questions

Easy
  1. 1State the law of conservation of energy.
  2. 2What stays constant in free fall?
  3. 3Write the formula for power.
Medium
  1. 4What is the unit of power?
  2. 5How many watts make 1 horse power?
  3. 6A body of 2 kg falls from rest through 5 m. Find its KE at the bottom (g = 9.8 m/s²).
Challenging
  1. 7A motor of 2 hp runs for 1 s. How many joules of work does it do?
  2. 8Find the cost of running a 1.5 kW geyser for 2 hours at 8 rupees per unit (kWh).
Show answers (try them yourself first)
  1. 1Energy is neither created nor destroyed; it only changes form.
  2. 2PE + KE.
  3. 3P = W ÷ t.
  4. 4Watt.
  5. 5746 W.
  6. 6KE = PE lost = 2 × 9.8 × 5 = 98 J.
  7. 72 × 746 = 1492 W, so 1492 J.
  8. 8Energy = 1.5 × 2 = 3 kWh; cost = 3 × 8 = 24 rupees.

Quick revision

  • W = F s cos θ; KE = ½mv2; PE = mgh; P = W/t.
  • Total mechanical energy is conserved in free fall.