Exercise Solutions

Q.1 Multiple Choice Questions (Choose the correct option):

1. The work done by a force is zero if:
   a) The force is very large.
   b) The object moves at a constant velocity.
   c) There is no displacement.
   d) The force is very small.
   Solution: Work done = Force × Displacement. If displacement (s) = 0, then Work = 0.
   Correct Option: c) There is no displacement.
2. The energy possessed by a moving object is:
   a) Potential energy
   b) Chemical energy
   c) Kinetic energy
   d) Heat energy
   Solution: Kinetic energy is the energy of motion.
   Correct Option: c) Kinetic energy
3. The SI unit of power is:
   a) Joule
   b) Newton
   c) Watt
   d) Horsepower
   Solution: The Watt is the standard unit for power.
   Correct Option: c) Watt
4. Which of the following is a non-renewable source of energy?
   a) Solar energy
   b) Wind energy
   c) Natural gas
   d) Hydroelectric energy
   Solution: Natural gas is a fossil fuel, which is a finite resource.
   Correct Option: c) Natural gas
5. If the velocity of an object is halved, its kinetic energy becomes:
   a) Half
   b) Double
   c) Four times
   d) One-fourth
   Solution: Kinetic Energy (KE) = ½mv². If 'v' becomes v/2, then KE' = ½m(v/2)² = ½m(v²/4) = ¼(½mv²). So, kinetic energy becomes one-fourth.
   Correct Option: d) One-fourth

Q.2 Fill in the blanks:

1. Work is done when a force causes a displacement in the direction of the force.
2. The capacity to do work is called energy.
3. The energy possessed by an object due to its position is called potential energy.
4. Power is the rate of doing work.
5. Energy can neither be created nor destroyed, it can only be transformed from one form to another.

Q.3 Match the pairs:

(Note: As an AI, I will provide the correct pairs.)

• 1. Work - c. Joule
• 2. Kinetic Energy - a. ½mv²
• 3. Potential Energy - d. mgh
• 4. Power - b. Watt

Q.4 True or False. If false, write the correct statement:

1. Work is a vector quantity.
   False. Work is a scalar quantity.
2. A coolie carrying a load on his head and walking on a horizontal platform does work against gravity.
   False. A coolie carrying a load on his head and walking on a horizontal platform does *zero* work against gravity, because the force of gravity is vertical and the displacement is horizontal (they are perpendicular).
3. The energy of a stretched spring is kinetic energy.
   False. The energy of a stretched spring is elastic potential energy.
4. Solar energy is a non-renewable source of energy.
   False. Solar energy is a renewable source of energy.
5. The total energy in an isolated system remains constant.
   True.

Q.5 Answer the following questions:

1. Define work. Give two examples where work is done and two examples where work is not done.
   Answer: Work is done when a force causes a displacement of an object in the direction of the force.
   Examples where work is done:
   1. Pushing a book across a table.
   2. Lifting a bucket of water from a well.
   Examples where work is not done:
   1. Pushing against a stationary wall.
   2. Holding a heavy bag without moving it.
2. What is energy? Explain kinetic energy and potential energy with examples.
   Answer: Energy is the capacity of an object to do work.
   Kinetic Energy: Energy possessed by an object due to its motion.
   Example: A moving car, a flying bird, flowing water.
   Potential Energy: Energy possessed by an object due to its position or configuration.
   Example: A book kept on a high shelf (gravitational potential energy), a stretched bow (elastic potential energy).
3. Derive the formula for kinetic energy ($KE = \frac{1}{2}mv^2$).
   Answer: (Refer to Q7 in 'Important Questions and Answers' section above for the derivation.)
4. Derive the formula for gravitational potential energy ($PE = mgh$).
   Answer: (Refer to Q9 in 'Important Questions and Answers' section above for the derivation.)
5. State the Law of Conservation of Energy. Explain with an example of a falling object.
   Answer: The Law of Conservation of Energy states that energy can neither be created nor destroyed; it can only be transformed from one form to another. The total energy in an isolated system remains constant.
   Example (Falling Object): When an object is dropped from a certain height, it initially has maximum gravitational potential energy and zero kinetic energy (since it's at rest). As it falls, its height decreases, so its potential energy decreases. Simultaneously, its speed increases, so its kinetic energy increases. Just before hitting the ground, its potential energy is minimum (or zero), and its kinetic energy is maximum. Throughout the fall (ignoring air resistance), the sum of its potential and kinetic energy (total mechanical energy) remains constant.
6. Differentiate between renewable and non-renewable sources of energy.
   Answer: (Refer to Q13 in 'Important Questions and Answers' section above for the differentiation.)
7. Explain different forms of energy.
   Answer: (Refer to Q12 in 'Important Questions and Answers' section above for the explanation of different forms of energy.)

Q.6 Give reasons:

1. Work done by centripetal force is zero.
   Reason: Work done is zero when the force applied is perpendicular to the direction of displacement. In uniform circular motion, the centripetal force always acts towards the center of the circular path (radially inward), while the displacement of the object at any instant is tangential to the path. Since the force and displacement are always perpendicular to each other, the work done by the centripetal force is zero.
2. A stretched rubber band has potential energy.
   Reason: A stretched rubber band has potential energy because work is done on it to change its shape (configuration) from its relaxed state. This work is stored within the rubber band as elastic potential energy. This stored energy gives the rubber band the capacity to do work when it returns to its original shape (e.g., when it is released, it can propel an object).
3. A moving car possesses kinetic energy.
   Reason: A moving car possesses kinetic energy because it is in motion. Kinetic energy is defined as the energy an object has due to its motion. The car's motion means it has the capacity to do work, for example, by colliding with another object and causing its displacement.
4. We use renewable sources of energy.
   Reason: We use renewable sources of energy because they are sustainable and environmentally friendly. Unlike fossil fuels, which are finite and cause pollution, renewable sources like solar, wind, and hydro are naturally replenished, do not deplete over time, and produce significantly fewer greenhouse gas emissions, helping to combat climate change and ensure long-term energy security.
5. An object kept at a height has potential energy.
   Reason: An object kept at a height has potential energy because work was done against the force of gravity to lift it to that position. This work is stored as gravitational potential energy. This stored energy gives the object the potential to do work if it falls (e.g., a hammer held above a nail can drive the nail into wood when dropped).

Q.7 Solve the following numerical problems:

1. A force of 15 N displaces an object by 8 m in the direction of the force. Calculate the work done.
   Solution: Given: Force ($F$) = $15 \text{ N}$ Displacement ($s$) = $8 \text{ m}$ Work done ($W$) = $F \times s$ $W = 15 \text{ N} \times 8 \text{ m}$ $W = 120 \text{ J}$
   The work done is $120 \text{ J}$.
2. A car of mass 1500 kg accelerates from 10 m/s to 20 m/s. Calculate the change in its kinetic energy.
   Solution: Given: Mass ($m$) = $1500 \text{ kg}$ Initial velocity ($u$) = $10 \text{ m/s}$ Final velocity ($v$) = $20 \text{ m/s}$ Initial Kinetic Energy ($KE_i$) = $\frac{1}{2}mu^2 = \frac{1}{2} \times 1500 \text{ kg} \times (10 \text{ m/s})^2$ $KE_i = 750 \text{ kg} \times 100 \text{ m}^2/\text{s}^2 = 75000 \text{ J}$ Final Kinetic Energy ($KE_f$) = $\frac{1}{2}mv^2 = \frac{1}{2} \times 1500 \text{ kg} \times (20 \text{ m/s})^2$ $KE_f = 750 \text{ kg} \times 400 \text{ m}^2/\text{s}^2 = 300000 \text{ J}$ Change in Kinetic Energy ($\Delta KE$) = $KE_f - KE_i$ $\Delta KE = 300000 \text{ J} - 75000 \text{ J} = 225000 \text{ J}$
   The change in kinetic energy is $225000 \text{ J}$.
3. A boy of mass 40 kg climbs a staircase of 50 steps. Each step is 15 cm high. Calculate the work done by the boy. (g = 10 m/s²)
   Solution: Given: Mass of boy ($m$) = $40 \text{ kg}$ Number of steps = $50$ Height of each step = $15 \text{ cm} = 0.15 \text{ m}$ Total height climbed ($h$) = Number of steps $\times$ Height of each step $h = 50 \times 0.15 \text{ m} = 7.5 \text{ m}$ Acceleration due to gravity ($g$) = $10 \text{ m/s}^2$ Work done ($W$) = Work done against gravity = Potential Energy gained = $mgh$ $W = 40 \text{ kg} \times 10 \text{ m/s}^2 \times 7.5 \text{ m}$ $W = 400 \times 7.5 \text{ J}$ $W = 3000 \text{ J}$
   The work done by the boy is $3000 \text{ J}$.
4. A motor pump of 200 W power is used to lift water. How much water can it lift to a height of 10 m in 1 minute? (g = 10 m/s²)
   Solution: Given: Power ($P$) = $200 \text{ W}$ Height ($h$) = $10 \text{ m}$ Time ($t$) = $1 \text{ minute} = 60 \text{ s}$ Acceleration due to gravity ($g$) = $10 \text{ m/s}^2$ We need to find the mass of water ($m$) that can be lifted. We know that Power ($P$) = $\frac{\text{Work Done}}{\text{Time Taken}}$ Work Done ($W$) = Potential Energy gained by water = $mgh$ So, $P = \frac{mgh}{t}$ $200 \text{ W} = \frac{m \times 10 \text{ m/s}^2 \times 10 \text{ m}}{60 \text{ s}}$ $200 = \frac{100m}{60}$ $200 \times 60 = 100m$ $12000 = 100m$ $m = \frac{12000}{100} = 120 \text{ kg}$
   The motor pump can lift $120 \text{ kg}$ of water.
5. An object of mass 2 kg is dropped from a height of 5 m. Calculate its kinetic energy just before it hits the ground. (g = 9.8 m/s²)
   Solution: Given: Mass ($m$) = $2 \text{ kg}$ Height ($h$) = $5 \text{ m}$ Acceleration due to gravity ($g$) = $9.8 \text{ m/s}^2$ Using the Law of Conservation of Energy: When the object is dropped, its initial kinetic energy is zero. All its energy is potential energy. Just before hitting the ground, all its potential energy is converted into kinetic energy (ignoring air resistance). So, Kinetic Energy just before hitting ground = Initial Potential Energy $KE_{final} = PE_{initial} = mgh$ $KE_{final} = 2 \text{ kg} \times 9.8 \text{ m/s}^2 \times 5 \text{ m}$ $KE_{final} = 98 \text{ J}$
   The kinetic energy of the object just before it hits the ground is $98 \text{ J}$.