Principle of Conservation of energy | ICSE Class 10 Physics Notes

This note covers the principle of conservation of energy, kinetic and gravitational potential energy, the verification of constant mechanical energy during free fall, energy changes in a simple pendulum, the effects of friction and air resistance, and calculations involving energy transfers.

What does the principle of conservation of energy mean?

Energy is the capacity to do work. In the study of forces and motion, work is done when a force produces displacement with a component along the force. A force is a push or pull, and displacement is the change in position measured with direction.

Definition: The principle of conservation of energy states that energy can neither be created nor destroyed; it can change from one form into another.

An energy transformation is a change in the form of energy. The word conservation refers to the amount of energy accounted for, even when its form changes. It does not mean that every form of energy must separately remain constant.

How can energy change form?

Electrical energy is associated with the position or motion of electric charges. In a bulb, electrical energy changes into light energy. In an electric water heater, it changes into thermal energy, the energy associated with warming or heating the water.

Chemical energy is stored in the chemical bonds of substances such as food. The chemical energy in food powers muscles and changes into mechanical energy. A ringing bell converts mechanical energy into sound energy, the energy associated with vibrations.

A system is the object or collection of objects being considered. When tracking energy, distinguish energy changing form within the system from energy passing between the system and its surroundings, meaning everything outside it.

What is the mechanical form of this principle?

Mechanical energy is the sum of kinetic energy, which is energy due to motion, and potential energy, which is stored because of position or configuration. A freely falling body shows how these two contributions can change while their sum remains constant. Free fall means motion under gravity alone; here gravity is the Earth’s attraction on the body.

The general energy principle and conservation of mechanical energy must be distinguished. Friction, a force opposing relative sliding or the tendency to slide between contacting surfaces, can reduce the mechanical energy of a moving object. This does not destroy energy: other forms and the surroundings must then be included in the energy account.

Which quantities, symbols and units are needed?

Kinetic energy, represented by K, is the energy possessed by an object because of its motion. Let m represent its mass, the measure of its inertia or resistance to a change in velocity, and v its speed, the magnitude of its velocity. Velocity describes how fast position changes and in which direction.

K = ½mv²

The square applies to speed. For the same mass, doubling speed makes kinetic energy four times its earlier value. A body at rest has zero kinetic energy. Motion and rest here are described relative to the ground.

Gravitational potential energy, represented by U, is energy associated with the position of a body relative to the Earth. Let g represent acceleration due to gravity, and h the vertical height above a chosen level where potential energy is zero.

U = mgh

Acceleration is the rate of change of velocity. Near the Earth's surface, g can be treated as constant for these calculations. The expression mgh applies to heights near the Earth's surface; it should not be extended unchanged to positions far from the Earth.

The Earth is much more massive than the ball and hardly moves towards it. The stored energy of the Earth-ball system is often simply called the gravitational potential energy of the ball. This is a convenient description of the interacting system.

How are the quantities measured?

SI means the International System of Units. The SI unit of mass is the kilogram, symbol kg. The SI unit of height is the metre, symbol m. In the energy formulas, m denotes mass; after a number, m denotes metres.

The SI unit of speed is the metre per second, written m/s. The SI unit of acceleration is the metre per second squared, written m/s². The SI unit of energy is the joule, symbol J. The second, symbol s, measures time.

Let W represent work done and F represent force. The SI unit of work is the joule. The SI unit of force is the newton, symbol N. One joule is the work done when one newton displaces a body one metre along the force.

1 J = 1 N m

Let E represent total mechanical energy. Then E = U + K. All three quantities must be expressed in the same energy unit before they are added or compared. A value of speed cannot be added directly to a value of energy.

How does raising a body store gravitational potential energy?

Choose the ground as the reference level, meaning the level assigned zero gravitational potential energy. Raising an object requires work against gravity. When the object is lifted gradually, with no appreciable change in speed, the work supplied increases its gravitational potential energy.

The body's weight is the gravitational force acting on it, whose magnitude is mg. Mass and weight are different quantities: mass is measured in kilograms, whereas weight is a force measured in newtons. The lifting force balances the weight during gradual lifting.

Derivation: Gravitational potential energy

1. Take a body of mass m initially at the ground reference level, where its gravitational potential energy is zero.
2. Lift it gradually through a vertical height h. The required upward force has magnitude F = mg, balancing its downward weight.
3. Work done by the lifting force equals force multiplied by displacement along that force. Thus W = Fh = mgh.
4. With no appreciable change in kinetic energy, this supplied work becomes the gain in gravitational potential energy of the Earth-body system.

U = mgh gives the gravitational potential energy at height h above the selected zero level.

The drawing shows a block near the ground and at a higher position. A vertical height h separates the levels, and an upward force arrow is labelled F = mg.

See Fig. 7.18 in your NCERT textbook

Why does the reference level matter?

The choice of zero potential energy is a convenience, but it must be used consistently. If the ground is the zero level at the start of a calculation, retain that level at every later position in that calculation.

The gain in potential energy depends on the change in vertical height. Reaching the same upper level by a staircase or by a vertical lift gives the same gravitational potential-energy gain for the same mass and g. The route does not alter this gain.

At a fixed height above the same reference level, moving horizontally does not change gravitational potential energy. During gradual upward motion, potential energy increases. These statements concern gravitational potential energy; they do not by themselves specify every other energy transfer involved.

How is conservation of mechanical energy proved for free fall?

Consider a body of mass m released from rest at point A, a height h above the ground. Let B be an intermediate point at height h′, read as “h prime”, and C the ground level. The distance fallen from A to B is h − h′.

Let u be initial velocity and t be the elapsed time after release. At release, u = 0. Take downward as the positive direction for the motion equations. Assume gravity alone acts and g remains constant; ignore air resistance, the force exerted by air opposing motion.

The drawing places A above B and C. It labels total height h, remaining height h′ and fallen distance h − h′. The body has mass m, with downward weight mg and times marked at A and B.

See Fig. 7.19 in your NCERT textbook

Derivation: Constant energy at an intermediate position

1. At A, the body is at rest. Its kinetic energy is zero and its gravitational potential energy is mgh. Its initial mechanical energy is therefore mgh.
2. After time t, the downward velocity is v = u + gt = gt. The distance fallen from rest is ½gt², so the remaining height is h′ = h − ½gt².
3. At B, potential energy is U = mgh′ = mgh − ½mg²t². Kinetic energy is K = ½mv² = ½mg²t².
4. Adding the energies gives U + K = mgh − ½mg²t² + ½mg²t². The equal negative and positive terms cancel, leaving the initial value mgh.

E = mgh remains constant throughout this free fall. The height h in this expression is the original release height, not the body's changing height.

What do the separate terms tell us?

v = gt shows that the downward speed increases with time in this model. Meanwhile, h′ = h − ½gt² shows that the remaining height decreases. Consequently, kinetic energy rises while gravitational potential energy falls.

More explicitly, K = ½mg²t² is the kinetic energy gained after time t. The potential energy is U = mgh − ½mg²t². The amount subtracted from potential energy is exactly the amount appearing as kinetic energy.

This is a theoretical verification: the equations of motion and the energy expressions establish the result at an arbitrary intermediate time. Checking only the starting point and ground would not show the changing contributions as clearly.

Note: The claim is that U + K remains constant under the stated free-fall conditions. Neither U nor K separately remains constant. If air resistance is significant, include the energy transferred out of mechanical form.

How do energy and speed change at different heights?

For the same falling body, retain h as its release height and h′ as its current height. The reference level is still the ground. The mechanical energy at every point before contact with the ground equals the initial value mgh under the ideal free-fall assumptions.

How can speed be obtained from energy?

At an intermediate height, the loss in potential energy is mg(h − h′). This equals the gain in kinetic energy. Therefore, ½mv² = mg(h − h′). Cancelling the common mass and rearranging gives v² = 2g(h − h′).

Just before ground contact, the remaining height is zero. The result becomes v² = 2gh, or v = √(2gh), where √ means the positive square root. The positive root gives speed, which is a magnitude.

The mass cancels because both the original potential energy and the final kinetic energy are proportional to the same mass. Thus the ideal result for speed depends on release height and g, not on mass. This conclusion assumes the body started from rest.

What does “just before” exclude?

Just before ground contact describes the final instant of free fall. It does not describe the body after striking the ground. During impact, forces other than gravity act, and mechanical energy may change into other forms.

The same energy reasoning gives the speed of a child starting from rest at the top of a slide of vertical height h. Neglecting friction, potential energy becomes kinetic energy at the bottom. Under those conditions the shape of the slide does not alter the final speed.

These energy equations compare positions without finding the time taken between them. They are particularly useful when the required answer is a speed or energy at a specified height, rather than a time of travel.

How does a simple pendulum demonstrate energy conservation?

A simple pendulum has a small suspended mass, called its bob, hanging from a fixed support by a string and free to swing. Its repeated to-and-fro movement is called oscillation. The lowest position is the position occupied by the stationary hanging bob.

Let P and R name the two extreme positions, meaning the turning points at either side, and Q name the lowest position. For this discussion choose Q as the zero level of gravitational potential energy. The pendulum's treatment here concerns energy changes qualitatively.

The drawing shows a bob at two raised side positions labelled R and P and at the lowest central position Q. The side labels indicate potential energy with no kinetic energy; the bottom label indicates kinetic energy with no potential energy.

See Fig. 7.20 in your NCERT textbook

What happens during an ideal swing?

1. Hold the bob at the raised position P and release it without pushing. At release it is at rest, so its kinetic energy is zero and its energy is gravitational potential energy.
2. As it moves down towards Q, its height falls and its speed rises. Gravitational potential energy changes into kinetic energy.
3. At Q, potential energy is zero relative to the chosen lowest level. The bob's kinetic energy and speed are greatest during that swing.
4. As it rises towards R, its speed falls. Kinetic energy changes back into gravitational potential energy until it comes momentarily to rest at the turning point.

For an ideal pendulum, meaning one whose energy losses are neglected, the opposite extreme is at the same height as the release point. The equal heights follow from equal potential energies at the two turning points, where kinetic energy is zero.

What does the actual observation show?

Place a horizontal reference line behind the pendulum at the release height. Release the bob from that level and observe the extreme positions during the first couple of oscillations. The bob reaches almost the same height from which it started.

The word almost matters: the observation is not an exact claim about a real apparatus. Friction at the support and air resistance reduce the bob's mechanical energy. With time the real pendulum slows and eventually stops.

Does friction contradict the conservation of energy?

Friction is a force that opposes relative sliding or the tendency to slide between contacting surfaces. Air resistance opposes a body's motion through air. Both can reduce the mechanical energy available to sustain motion.

In an ideal free-fall calculation, air resistance is ignored. In an ideal pendulum description, air resistance and friction at the support are ignored. These assumptions make it possible to treat the exchange between kinetic and potential energy without a reduction in their sum.

Where does the missing mechanical energy go?

When friction matters, energy is transferred from mechanical form into other forms, including thermal energy. The decrease in a body's mechanical energy is therefore not evidence that energy has ceased to exist. The energy account must include these other forms and the surroundings.

A decrease in mechanical energy and a violation of energy conservation are different claims. The former can occur in ordinary motion with resistance. The latter would mean energy had actually been created or destroyed.

For a real pendulum, decreasing turning heights show that less gravitational potential energy is available at successive extremes. At an extreme the bob is momentarily at rest, so the energy associated with its height provides a useful indication of its remaining mechanical energy.

How should ideal and actual statements be separated?

State the condition together with the conclusion. Write that mechanical energy remains constant provided energy losses are neglected. When discussing the actual pendulum, retain the observation that it reaches almost the same starting height during its early swings.

Likewise, a rising ball may reach a lower height than predicted by converting all its initial kinetic energy into potential energy. If air resistance acts, part of that initial mechanical energy has been transferred elsewhere before the ball reaches its highest point.

The highest point is still an instant of zero upward speed. It does not imply that gravity has stopped acting. Zero speed at a turning point must not be confused with zero gravitational force or zero acceleration.

How can energy calculations be organised and checked?

Start with the physical conditions: whether the body starts from rest, whether resistance can be neglected, and which level is assigned zero potential energy. Then identify the quantities supplied. Do not silently assume a value of g when one is needed.

1. List mass, height, speed and the supplied acceleration due to gravity. Convert masses to kilograms and express lengths and speeds in compatible SI units.
2. Choose a single zero level for gravitational potential energy. Distinguish the current height from the distance already fallen.
3. Calculate kinetic and potential energies at the required positions. Add them to obtain mechanical energy where necessary.
4. For ideal motion, equate initial and final mechanical energy. If resistance is specified, calculate the mechanical-energy decrease and explain its meaning.
5. Give the answer with its unit and check whether it agrees with the stated conditions: in free fall from rest, kinetic energy increases as gravitational potential energy decreases.

How are height and mass used?

Worked example 1. A cricket ball of mass 200 g is thrown to a maximum height of about 10 m above the ground. Here g after 200 denotes grams. Take acceleration due to gravity as 10 m/s². Find its gravitational potential energy relative to the ground.

Formula: m = mass in grams ÷ 1000; U = mgh. Substitute: m = 200 ÷ 1000 = 0.2 kg; U = 0.2 × 10 × 10.

Answer: U = 20 J, using the stated height approximation. Convert the mass before multiplying; 200 g is not 200 kg.

Worked example 2. A student of mass 50 kg is lifted slowly in an elevator from ground level through a vertical height of 72.5 m. Take g = 10 m/s². Find the gain in gravitational potential energy.

Formula: U = mgh. Substitute: U = 50 × 10 × 72.5.

Answer: The gain is 36,250 J. This is the change in gravitational potential energy between the given lower and upper levels.

Worked example 3. A student of mass 50 kg climbs stairs from ground level to a height of 72.5 m. Take g = 10 m/s². Find the gravitational potential-energy gain and compare it with an elevator journey through the same height.

Formula: U = mgh. Substitute: U = 50 × 10 × 72.5.

Answer: The gain is 36,250 J, equal to the elevator result. The comparison concerns gravitational potential energy, which depends on vertical height gained, not the route followed.

How can a rising ball reveal energy transferred by resistance?

The next three worked examples examine successive parts of one situation. A ball is thrown vertically upwards. Calculating its initial kinetic energy, its potential energy at the highest point, and the difference makes the complete energy account visible.

Worked example 4. A ball of mass 2 kg is thrown vertically upwards from a level chosen to have zero potential energy. Its initial speed is 20 m/s. Find its initial kinetic energy and initial mechanical energy.

Formula: K = ½mv²; E = U + K. Substitute: K = ½ × 2 × 20² = 400 J; E = 0 + 400.

Answer: Initial kinetic energy is 400 J and initial mechanical energy is also 400 J. The equality follows from the stated zero potential-energy level.

Worked example 5. A ball of mass 2 kg, thrown vertically upwards, reaches a highest point 19.4 m above its release level. Take g = 10 m/s² and zero potential energy at release. Find its potential and mechanical energies at the highest point.

Formula: U = mgh; E = U + K. Substitute: U = 2 × 10 × 19.4 = 388 J. At the highest point the speed and hence kinetic energy are zero.

Answer: Potential energy is 388 J and mechanical energy is 388 J. Gravity still acts even though the ball is momentarily at rest.

Worked example 6. A 2 kg ball is thrown vertically upwards at 20 m/s and reaches 19.4 m above its release point. Take g = 10 m/s², assign zero potential energy at release, and treat air resistance as the cause of mechanical-energy loss. Find that loss and the work done by air resistance.

Formula: K = ½mv²; U = mgh. Substitute: initial mechanical energy = ½ × 2 × 20² = 400 J; final mechanical energy = 2 × 10 × 19.4 = 388 J.

Answer: Mechanical-energy loss = 400 − 388 = 12 J. Work done by air resistance = final mechanical energy − initial mechanical energy = −12 J. The negative sign indicates removal of mechanical energy.

These calculations separate a positive amount of energy lost from the negative work done by a resistive force. Use the requested quantity carefully. A decrease in mechanical energy does not mean the total energy of the full physical process is destroyed.

Glossary

• Energy — The capacity to do work, which can appear in different forms.
• Work — Energy transferred when a force produces displacement with a component along that force.
• Kinetic energy — Energy possessed by a body because of its motion.
• Gravitational potential energy — Energy associated with the position of a body relative to the Earth.
• Mechanical energy — The sum of a body's kinetic energy and potential energy.
• Conservation of energy — The principle that energy cannot be created or destroyed, although its form can change.
• Free fall — Motion of a body under gravity alone, with air resistance neglected.
• Reference level — The selected level at which gravitational potential energy is assigned zero.
• Simple pendulum — A small suspended mass free to swing about a fixed support.
• Bob — The suspended mass of a pendulum that moves during its oscillations.
• Extreme position — A turning point where the swinging bob is momentarily at rest.
• Air resistance — The force exerted by air that opposes a body's motion through it.

Common errors and misconceptions

• Misconception: Conservation means kinetic energy stays constant during free fall. Correct: Kinetic energy increases while potential energy decreases; their sum stays constant under ideal conditions.
• Misconception: A body with zero speed has no energy. Correct: It has zero kinetic energy but may have gravitational potential energy above the chosen reference level.
• Misconception: Height in potential energy means the distance already fallen. Correct: It means the height remaining above the selected zero level.
• Misconception: Every real pendulum returns to exactly the release height. Correct: It reaches almost the same height in early swings; friction and air resistance reduce its mechanical energy.
• Misconception: A pendulum's stopping proves energy destruction. Correct: Mechanical energy changes into other forms and is transferred to the surroundings.
• Misconception: A ball at its highest point has no gravitational force acting on it. Correct: Gravity still acts even though its speed is momentarily zero.
• Misconception: Ground-level kinetic energy describes the body after impact. Correct: The free-fall result refers to just before impact, before contact forces alter the motion.

Exam-style questions with model answers

Q1. State the principle of conservation of energy and explain what an energy transformation means. [2 marks]

1. Energy can neither be created nor destroyed; it can change from one form into another.
2. An energy transformation is such a change of form, as when gravitational potential energy becomes kinetic energy during free fall.

Q2. A cricket ball of mass 200 g reaches a maximum height of about 10 m above the ground. Taking acceleration due to gravity as 10 m/s² and zero potential energy at ground level, calculate its gravitational potential energy. [3 marks]

1. Let m be the mass, g the acceleration due to gravity and h the height above the reference level. Convert the mass: m = 200 g = 0.2 kg.
2. Gravitational potential energy, U, is given by U = mgh. Substitute the supplied values: U = 0.2 × 10 × 10.
3. The potential energy is 20 J relative to the ground, using the stated approximate height.

Q3. A body of mass m is released from rest at height h above the ground. It falls under gravity alone with constant gravitational acceleration g. After time t its downward speed is v = gt and its remaining height is h′ = h − ½gt². Taking ground potential energy as zero, verify conservation of mechanical energy. [5 marks]

1. At release, the body's kinetic energy K is zero because it starts from rest. Its gravitational potential energy U is mgh, so its initial mechanical energy, the sum U + K, is mgh.
2. At the intermediate position, substitute the supplied remaining height into U = mgh′. This gives U = mgh − ½mg²t².
3. Using the supplied speed v = gt in K = ½mv² gives K = ½mg²t² at that same position.
4. Add the two expressions: U + K = mgh − ½mg²t² + ½mg²t² = mgh. The changing terms cancel exactly.
5. The sum equals its initial value at any time before impact. Thus the gain in kinetic energy equals the loss in potential energy under the stated free-fall conditions.

Q4. An ideal simple pendulum is released from rest at a raised extreme position P. Q is its lowest point and R its opposite extreme. Neglect friction and air resistance and choose Q as zero potential-energy level. Explain the energy changes from P through Q to R. [5 marks]

1. At P, the bob is at rest and therefore has zero kinetic energy, meaning energy due to motion. Its mechanical energy is gravitational potential energy because it is above Q.
2. During descent from P towards Q, its height decreases while its speed increases. Gravitational potential energy is converted into kinetic energy.
3. At Q, gravitational potential energy is zero relative to the selected level. Kinetic energy and speed are greatest during the swing.
4. As the bob rises from Q towards R, its kinetic energy decreases and its gravitational potential energy increases. Its speed falls as it approaches R.
5. At R it is momentarily at rest. With losses neglected, its mechanical energy equals the initial value, so it reaches the same height as P.

Q5. A 2 kg ball is thrown vertically upwards at 20 m/s. Air resistance acts, and it reaches 19.4 m above release. Take gravitational acceleration as 10 m/s² and zero potential energy at release. Calculate initial mechanical energy, final mechanical energy, energy lost and work done by air resistance. [4 marks]

1. Initially potential energy is zero. Kinetic energy, given by half the mass multiplied by speed squared, is ½ × 2 × 20² = 400 J. Initial mechanical energy is therefore 400 J.
2. At the highest point, speed is zero. Final mechanical energy equals gravitational potential energy: 2 × 10 × 19.4 = 388 J.
3. The positive amount of mechanical energy lost is 400 − 388 = 12 J.
4. Work done by air resistance equals final minus initial mechanical energy: 388 − 400 = −12 J.

Q6. A real pendulum eventually stops because of friction at its support and air resistance. Does this contradict conservation of energy? Explain. [2 marks]

1. No. Friction and air resistance reduce the pendulum's mechanical energy, the sum of its kinetic and potential energies.
2. Energy changes into other forms, including thermal energy, and is transferred to the surroundings. It is not destroyed.

Q7. A student of mass 50 kg reaches a floor 72.5 m above ground, first by an elevator and then by stairs on a separate journey starting from ground level. Take gravitational acceleration as 10 m/s². Calculate and compare the gravitational potential-energy gains. [3 marks]

1. Potential-energy gain equals mass multiplied by gravitational acceleration and vertical height gained. For the elevator journey, this is 50 × 10 × 72.5 = 36,250 J.
2. The staircase journey has the same mass, gravitational acceleration and vertical height gain. Its potential-energy gain is also 50 × 10 × 72.5 = 36,250 J.
3. The gains are equal because gravitational potential-energy change depends on the vertical height difference, not the route taken between the two levels.

Key takeaways

• Energy can change form, but it cannot be created or destroyed during a physical process.
• Mechanical energy is the sum of kinetic energy and potential energy, expressed in a common energy unit.
• During ideal free fall from rest, decreasing gravitational potential energy is matched by increasing kinetic energy.
• Choose a zero potential-energy level and use it consistently throughout every stage of a calculation.
• The free-fall energy result at ground level describes the body just before impact, when contact forces have not acted.
• An ideal pendulum exchanges potential and kinetic energy and reaches equal heights at its two extremes.
• A real pendulum reaches almost the same height initially, then loses mechanical energy through friction and air resistance.
• A reduction in mechanical energy requires accounting for other energy forms and the surroundings, rather than assuming energy destruction.

Test yourself

What two contributions make up mechanical energy?

Mechanical energy is the sum of kinetic energy and potential energy.

Why does a released body's kinetic energy increase during ideal free fall?

Its gravitational potential energy decreases as its height falls, and the lost potential energy becomes kinetic energy.

What does height mean in the expression for gravitational potential energy?

It is the vertical height above the chosen level of zero potential energy.

Where is an ideal pendulum's speed greatest during a swing?

At the lowest point, where its kinetic energy is greatest during that swing.

Does momentary rest at a pendulum's extreme mean that it has no mechanical energy?

No. Its kinetic energy is zero, but it has gravitational potential energy relative to the lowest point.

Why should “almost the same height” be retained for an observed pendulum?

Friction and air resistance act in the real apparatus, so an exact return to the release height is not the observation.

What happens to the kinetic energy of the same vehicle when its speed doubles?

Its kinetic energy becomes four times the original value because kinetic energy depends on speed squared.

Why can two routes to the same upper floor give equal gravitational potential-energy gains?

For the same mass and gravitational acceleration, the gain depends on vertical height change, not the route.