Important Questions (1 mark): Work & Energy - UPSC MCQ

When a light and a heavy body have equal kinetic energy, then which one has a greater momentum?

Answer: B. Heavy body

Explanation:

1. Kinetic Energy (KE):

• Kinetic energy is the energy possessed by a body due to its motion.


• It is given by the equation KE = 1/2 * m * v^2, where m is the mass of the body and v is its velocity.

2. Momentum:

• Momentum is the product of an object's mass and its velocity.


• It is given by the equation p = m * v, where p is the momentum, m is the mass of the body, and v is its velocity.

3. Comparison:

• When a light body and a heavy body have equal kinetic energy, it means they have the same value of KE = 1/2 * m * v^2.


• Since the kinetic energy is the same, let's compare the momentum of the light body and the heavy body.


• The momentum of an object is directly proportional to its mass and velocity.


• As the light body has a smaller mass compared to the heavy body, in order to have the same kinetic energy, it must have a higher velocity to compensate for its lower mass.


• Therefore, the heavy body will have a greater momentum as it has a larger mass.

4. Conclusion:

• In the given scenario, when a light body and a heavy body have equal kinetic energy, the heavy body will have a greater momentum due to its larger mass.

Joule. As potential energy is a form of energy so it's SI unit is also Joule. The gravitational potential energy has the same units as kinetic energy, kg m2 / s2. In fact, all energy has the same units, kg m2 / s2, and is measured using the SI unit is Joule (J).

The kilowatt-hour (SI symbol: kW⋅h or kW h; commonly written as kWh) is a unit of energy equal to 3600 kilojoules (3.6 megajoules). The kilowatt-hour is commonly used as a billing unit for energy delivered to consumers by electric utilities.

Let initial speed of the car be v m/s and mass be m kg
Using the equation
K.E.= (mv²) / 2 
It can be observed that, if we increase the speed (v) by 6 times the kinetic energy will increase by 36 times.

To convert kilowatt-hours (kWh) to joules, we need to use the conversion factor that 1 kilowatt-hour is equal to a certain number of joules.
1 kilowatt-hour (kWh) is defined as the energy consumed by a 1 kilowatt (kW) load in 1 hour.
To convert from kilowatt-hours to joules, we need to multiply the number of kilowatt-hours by the conversion factor.
The conversion factor is obtained by multiplying the number of kilowatts by the number of seconds in an hour (3600 seconds) and the number of joules in a watt-second (1 joule = 1 watt-second).
Using the conversion factor, we can calculate:
1 kWh = 1 kW * 3600 s * 1 J/Ws
Simplifying further:
1 kWh = 1000 W * 3600 s * 1 J/Ws
1 kWh = 3.6 × 10^6 J
Therefore, 1 kWh is equal to 3.6 × 10^6 joules.
Answer:
C:

3.6 × 10^6 Joules

By this equation, we clearly understand that the velocity is doubled then the kinetic energy becomes 4 times.

Explanation:
When the time taken to complete a given amount of work increases, it means that the work is being done at a slower rate. This leads to the following consequences:
1. Power decreases:
- Power is defined as the rate at which work is done or energy is transferred.
- When the time taken to complete a given amount of work increases, the power decreases because the work is being done at a slower rate.
- Therefore, option B is correct.
2. Energy remains constant:
- Energy is the capacity to do work.
- The amount of work being done may increase or decrease, but the energy remains constant.
- Therefore, options C and D are incorrect.
3. Power formula:
- Power is calculated using the formula: Power = Work / Time.
- When the time taken to complete a given amount of work increases, the denominator of the formula increases.
- As a result, the value of power decreases.
- Therefore, option A is incorrect.
In conclusion, when the time taken to complete a given amount of work increases, the power decreases.

When the force applied and the displacement of the body are inclined at 90° with each other, the work done is zero. This can be explained using the following points:
1. Definition of work: Work done is defined as the product of the magnitude of the force and the displacement of the body in the direction of the force. Mathematically, work done (W) = force (F) × displacement (d) × cosθ, where θ is the angle between the force and displacement vectors.
2. When the force and displacement are inclined at 90° with each other, the angle θ becomes 90°. In this case, cosθ = cos90° = 0.
3. Since the cosine of the angle is zero, the work done becomes zero, regardless of the magnitude of the force or displacement.
4. This means that no energy is transferred to or from the body when the force and displacement are perpendicular to each other.
5. Examples of such situations include pushing a block horizontally while it moves vertically, or pulling an object vertically while moving it horizontally.
6. Therefore, the correct answer is option C: Zero.
To summarize, when the force applied and the displacement of the body are inclined at 90° with each other, the work done is zero because the cosine of the angle between them is zero.

The kinetic energy (KE) of a body is given by the equation KE = 0.5 * mass * speed^2.
To maximize the KE of a body, we need to consider the factors that directly affect it.
The options given are mass, weight, speed, and density. Let's analyze each option to determine which one would double the KE of the body the most.
A. Mass:
- Doubling the mass will result in a proportional increase in KE according to the equation.
- KE ∝ mass
- Therefore, doubling the mass will double the KE.
B. Weight:
- Weight is the force exerted by gravity on an object and is given by the equation weight = mass * gravity.
- Doubling the weight does not directly affect the speed or velocity of the body.
- KE ∝ speed^2
- Therefore, doubling the weight will not double the KE.
C. Speed:
- The KE is directly proportional to the square of the speed.
- KE ∝ speed^2
- Doubling the speed will quadruple the KE.
D. Density:
- The density of an object does not directly affect the KE.
- KE ∝ mass and speed^2
- Doubling the density will not double the KE.
Therefore, the correct answer is C. Speed. Doubling the speed of a body will result in the largest increase in its kinetic energy.

To calculate the power, we need to use the formula:
Power = Force x Velocity
Let's break down the solution into steps:
Step 1: Identify the given values:
- Force (F)
- Velocity (v)
Step 2: Apply the formula:
Power = Force x Velocity
Step 3: Substitute the given values into the formula:
Power = F x v
Step 4: Simplify the expression:
Power = F x v
Therefore, the correct answer is A: F x v.

Work done by a centripetal force
The work done by a centripetal force depends on several factors, including the radius of the circle and the mass of the body. Let's break it down:
1. Work done by a centripetal force and the radius of the circle:
- The work done by a centripetal force is given by the formula W = F * d * cosθ, where W is the work done, F is the force, d is the displacement, and θ is the angle between the force and the displacement.
- In the case of a centripetal force, the force is always perpendicular to the displacement because it acts towards the center of the circle.
- Since cosθ = 0 when θ = 90 degrees, the work done by a centripetal force is always zero. This means that the radius of the circle does not affect the work done by the centripetal force.
2. Work done by a centripetal force and the mass of the body:
- The centripetal force is given by the formula F = (mv²) / r, where m is the mass of the body, v is the velocity, and r is the radius of the circle.
- Substituting this equation into the work formula, we get W = [(mv²) / r] * d * cosθ.
- The mass of the body does not affect the work done by the centripetal force because it cancels out in the equation. Therefore, increasing the mass of the body does not increase the work done by the centripetal force.
Conclusion:
- The work done by a centripetal force is always zero, regardless of the radius of the circle or the mass of the body.
- This is because the force is always perpendicular to the displacement, resulting in a zero angle between them.
- Therefore, the correct answer is D: the work done by a centripetal force is always zero.

Mechanical energy is the energy that is possessed by an object due to its motion or due to its position. Mechanical energy can be either kinetic energy (energy of motion) or potential energy (stored energy of position). 

We know, work done :- 
W = FS cos theta 
Here, F = 100 ,  S = 10  , theta = 60°
So,   W =  100 * 10 * cos 60° 
W   =  500 J

Unit of Energy: kWh
Explanation:
The unit kWh (kilowatt-hour) is a unit of energy. Energy is the capacity to do work or the ability to cause change. It is a scalar quantity and is measured in various units depending on the system of measurement used. In the International System of Units (SI), the unit of energy is the joule (J). However, in practical applications, especially in the context of electricity, the kilowatt-hour (kWh) is commonly used.
- Energy is defined as the amount of work done or the amount of heat transferred. It can exist in various forms such as mechanical energy, thermal energy, electrical energy, etc.
- The kWh is a derived unit of energy commonly used to measure the consumption or production of electrical energy over a period of time.
- It is equal to the amount of energy transferred or consumed when one kilowatt (kW) of power is used for one hour.
- The kWh is particularly useful for measuring electricity usage in households, businesses, and industries.
- It is also used for billing purposes by electric utilities.
- To calculate the energy consumption in kWh, one needs to multiply the power in kilowatts by the time in hours.
In summary, the unit kWh is used to measure energy, specifically electrical energy, and is widely used in practical applications such as electricity billing and consumption tracking.

Total height, h = 6 x 3.5 = 21 m
So total work done, W = mgh = 1000 x 9.8 x 21 = 205800J
Power, P = W/t = 205800 / 6 = 3.43 x 10⁴ watt

The work done by a body is directly proportional to:
A: Force acting on the body
- The force acting on a body is one of the factors that determine the amount of work done.
- When a force is applied to a body, it causes the body to move or be displaced.
- The magnitude of the force directly affects the amount of work done on the body.
B: Displacement produced in the body
- The displacement of a body is another factor that affects the work done.
- Displacement refers to the change in position of the body.
- When a body is displaced, work is done on the body.
- The greater the displacement, the more work is done.
C: Mass of the body
- The mass of the body does not directly affect the work done.
- Work is a measure of the transfer of energy, and the mass of the body is not directly related to the transfer of energy.
D: Both (A) and (B)
- The work done by a body is directly proportional to both the force acting on the body and the displacement produced in the body.
- Both factors play a role in determining the amount of work done.
- If either the force or the displacement is increased, the work done will also increase.
In conclusion, the work done by a body is directly proportional to both the force acting on the body and the displacement produced in the body. The mass of the body does not have a direct impact on the work done.

Work done is said to be positive when a force causes displacement:
To understand why work done is said to be positive when a force causes displacement, let's break down the concept and consider each option given.
A: In its own direction
- This means that the force and displacement are in the same direction.
- When a force is applied in the same direction as the displacement, work is done.
- Example: Pushing a box forward along a horizontal surface.
B: In the direction opposite to the applied force
- This means that the force and displacement are in opposite directions.
- When a force is applied in the opposite direction to the displacement, work is done, but it is considered negative.
- Example: Slowing down a moving object by applying a force in the opposite direction to its motion.
C: In the direction at right angles to the direction of applied force
- This means that the force and displacement are perpendicular to each other.
- When a force is applied at a right angle to the displacement, no work is done because there is no component of the force in the direction of displacement.
- Example: Pushing a box sideways without causing it to move forward or backward.
D: None of the above
- This option implies that work done can occur in a different scenario not covered by options A, B, or C.
- However, this is not the case as work done is only considered positive when the force and displacement are in the same direction.
Conclusion:
The correct answer is option A: In its own direction. When a force causes displacement in the same direction, work is done and it is considered positive.

The work done on an object does not depend on:

A: Displacement
- The work done on an object is defined as the product of the force applied on the object and the displacement of the object in the direction of the force.
- However, the magnitude of the displacement does not affect the work done on the object.
- Whether the object is displaced a large distance or a small distance, the work done on the object remains the same as long as the force and the displacement are in the same direction.

B: Angle between force and displacement
- The work done on an object also does not depend on the angle between the force applied and the displacement of the object.
- The work done is only affected by the component of the force that is parallel to the displacement.
- If the force and displacement are perpendicular to each other, the work done is zero.
- If the force and displacement are at an angle less than 90 degrees, the work done is positive.
- If the force and displacement are at an angle greater than 90 degrees, the work done is negative.

C: Force applied
- The work done on an object is directly proportional to the force applied.
- If the force applied is doubled, the work done on the object is also doubled.
- If the force applied is tripled, the work done on the object is also tripled.
- Therefore, the force applied does affect the work done on the object.

D: Initial velocity of the object
- The initial velocity of the object does not affect the work done on the object.
- Work is only dependent on the force applied and the displacement of the object.
- The initial velocity of the object only affects the kinetic energy of the object, not the work done on it.
In conclusion, the work done on an object does not depend on its displacement, the angle between force and displacement, or the initial velocity of the object. It only depends on the force applied to the object.

Case in which work is not done:

A: A girl swimming in a pond

- When the girl swims in the pond, she exerts force against the water to move herself forward, so work is done in this case.

B: A windmill lifting water from a well

- The windmill uses its blades to rotate and lift the water from the well. It requires energy to do this work, so work is done in this case.

C: A standing man holding a suitcase in his hand

- The man is not exerting any force or doing any work on the suitcase as he is not moving it. The force is balanced and no displacement is involved, so no work is done in this case.

D: A sailboat moving in the direction of the wind

- The sailboat utilizes the wind's force to move forward. The wind exerts force on the sail, causing displacement of the boat. Work is done in this case.
Summary:
In the given options, the case in which work is not done is option C: A standing man holding a suitcase in his hand.

Case where work is done:
A trolley rolling down a slope:
- When a trolley rolls down a slope, it moves due to the force of gravity.
- The force of gravity acts in the direction of the slope, causing the trolley to move.
- As the trolley moves, it covers a distance and displaces from its initial position to a final position.
- Work is done when a force acts on an object and causes it to move over a distance.
- In this case, the force of gravity is doing work on the trolley as it moves down the slope.
- The work done can be calculated using the formula: Work = Force x Distance.
- The force of gravity and the distance covered by the trolley are both non-zero values, so work is done.
Other cases where work is not done:
A green plant carrying out photosynthesis:
- Photosynthesis is a biological process that occurs in green plants.
- It involves the conversion of light energy into chemical energy to produce glucose.
- While photosynthesis is an important process for plants, it does not involve the movement of objects over a distance.
- Therefore, no work is done in this case.
A porter standing at a place and carrying a heavy load on his head:
- In this case, the porter is exerting a force to hold the load on his head.
- However, since the porter is not moving or displacing the load over a distance, no work is done.
Drying of food grains in the sun:
- When food grains are dried in the sun, the heat from the sun causes the moisture in the grains to evaporate.
- While the drying process involves a change in the state of the grains, no work is done as there is no displacement of objects over a distance.
In conclusion, work is done when a force acts on an object and causes it to move over a distance. In the given cases, work is only done when a trolley rolls down a slope.

Explanation:
When a stone is tied to a string and whirled in a circular path, the work done by the stone can be determined by considering the forces acting on the stone and the displacement of the stone.
1. Forces acting on the stone:
- Tension force in the string: This force acts towards the center of the circular path and is responsible for keeping the stone moving in a circular path.
- Gravitational force: This force acts vertically downwards and is responsible for the weight of the stone.
2. Displacement of the stone:
- The displacement of the stone is along the circular path, which is perpendicular to the tension force and the gravitational force.
3. Work done:
- Work done is defined as the product of the force applied on an object and the displacement of the object in the direction of the force.
- Since the tension force and the gravitational force are both perpendicular to the displacement of the stone, the work done by these forces is zero.
- Therefore, the total work done by the stone is zero.
Conclusion:
The work done by the stone when whirled in a circular path is zero (option B) since both the tension force and the gravitational force are perpendicular to the displacement of the stone.

Introduction:
When it comes to doing work, energy plays a crucial role. Energy is required to perform tasks and bring about changes in the physical world. In order for work to be done, energy must be transferred or converted from one form to another.
Explanation:
To further understand how energy is involved in work, let's break it down into several key points:
1. Energy transfer:
- Work involves the transfer of energy from one object or system to another.
- Energy can be transferred in different forms such as mechanical, thermal, electrical, or chemical energy.
- For example, when a person lifts a heavy object, the energy from their muscles is transferred to the object, causing it to move.
2. Energy conversion:
- Energy can also be converted from one form to another during the process of doing work.
- Different types of work may require different forms of energy.
- For instance, electrical energy can be converted into light energy in a light bulb, or chemical energy can be converted into mechanical energy in a car engine.
3. Conservation of energy:
- Energy is not "used up" in the process of doing work.
- According to the law of conservation of energy, energy cannot be created or destroyed; it can only be transferred or converted from one form to another.
- This means that the total amount of energy in a closed system remains constant.
4. Efficiency:
- While energy is not lost during work, it can be lost in the form of heat or other forms of waste energy.
- This loss of energy reduces the efficiency of the work being done.
- Efforts are made to minimize energy losses and improve the overall efficiency of systems.
Conclusion:
In conclusion, in order to do work, energy is transferred or converted from one form to another. It is not used up or lost but rather conserved in accordance with the law of conservation of energy. Understanding the role of energy in work is essential for various fields such as physics, engineering, and everyday life applications.

As the rocket moves in the direction of force generated by fuel, therefore it is positive work. As the force of gravity acts in the direction opposite to the displacement. Therefore, it is negative work.

Explanation:
To find the distance through which the force acts, we can use the formula for work:
Work = Force x Distance
We are given that one joule of work is done, which means the work is equal to 1 J. We are also given that the force is 1 newton. We need to find the distance.
Using the formula for work, we can rearrange it to solve for distance:
Distance = Work / Force
Plugging in the given values:
Distance = 1 J / 1 N
Simplifying, we find that the distance is equal to 1 meter.
Therefore, the correct answer is C: 1 m.
Summary:
The distance through which a force of one newton acts to do one joule of work is 1 meter.

Work is the product of time and power:
Explanation:
- Work is a measure of the energy transfer that occurs when an object is moved by a force applied to it.
- The equation for work is given by W = F * d, where W is the work done, F is the force applied, and d is the distance over which the force is applied.
- Power, on the other hand, is the rate at which work is done or the amount of work done per unit time.
- The equation for power is given by P = W / t, where P is the power, W is the work done, and t is the time taken to do the work.
- By rearranging the equation for power, we can find that W = P * t, which shows that work is the product of power and time.
- Therefore, the correct answer is A: power.
Summary:
- Work is the product of time and power.
- The equation for work is W = F * d, and the equation for power is P = W / t.
- Work can also be calculated as the product of power and time, W = P * t.
- Therefore, the correct answer is A: power.