Work, Energy, and Simple Machines — NCERT Solutions

From the chapter:

• Between A and B, the car moves at constant speed if the motion is still uniform before braking. - The brakes do negative work on the car. - The car’s kinetic energy is converted into thermal energy.

For a frictionless track, mechanical energy is conserved. So only the method and total energy can be stated from the chapter source.

Using conservation of mechanical energy, the coconut’s potential energy at height $h$ becomes kinetic energy just before impact:

$$mgh=\frac12 mv^2$$

So,

$$v=\sqrt{2gh}=\sqrt{2\times 10\times 10}=\sqrt{200}\approx 14.1\,\text{m s}^{-1}$$

On impact, the coconut comes to rest and its energy is used to make the depression in the sand. If the sand exerts an average resistive force of $3000\,\text{N}$, then

$$\text{work done} = F\times d = mgh = 1.5\times 10\times 10 = 150\,\text{J}$$

So,

$$d=\frac{150}{3000}=0.05\,\text{m}=5\,\text{cm}$$

The statements are:

1. Work is said to be done when a force is applied, even if the object does not move. → False because work requires displacement.
2. Lifting a bucket vertically upward results in positive work done on the bucket. → True because force and displacement are in the same direction.
3. The SI unit for both work and energy is joule (J). → True.
4. A motionless stretched rubber band has kinetic energy. → False because it has potential energy, not kinetic energy.
5. Energy can change from one form to another. → True.

The blanks are filled as follows:

1. Work done $=$ force × displacement in the direction of force.
2. 1 joule of work is done when a force of 1 newton displaces an object by 1 metre in the direction of the force.
3. The expression for kinetic energy of a body of mass $m$ and velocity $v$ is $\frac12 mv^2$.
4. The potential energy of an object of mass $m$ at a small height $h$ from the Earth's surface is $mgh$.
5. Power is defined as the rate at which work is done.

The energy transformations are:

1. Truck moving uphill → kinetic energy → potential energy
2. Unwinding of a watch spring → potential energy of the spring → kinetic energy
3. Photosynthesis in green leaves → light energy → chemical energy
4. Water flowing from a dam → potential energy → kinetic energy
5. Burning of a matchstick → chemical energy → heat and light energy
6. Explosion of a fire cracker → chemical energy → heat, light, sound and kinetic energy
7. Speaking into a microphone → sound energy → electrical energy
8. A glowing electric bulb → electrical energy → light and heat energy
9. A solar panel → light energy → electrical energy

No. While holding the barbell steady, the barbell does not move, so its displacement is zero. Since work done requires displacement in the direction of the force, the work done on the barbell is zero.

The force of friction acts opposite to the direction of motion of the coins. Since the force and displacement are in opposite directions, the work done by friction is negative.

As you pedal a bicycle on a flat road, the muscular energy of your body appears in several forms:

• mechanical energy for moving the bicycle,
• heat energy due to internal energy used by muscles,
• and partly sound energy.

So, muscular energy is converted mainly into mechanical energy, with some loss as heat and sound.

For equal kinetic energy,

$$\frac12 m v_A^2 = \frac12 (4m) v_B^2$$

Cancel $\frac12 m$:

$$v_A^2 = 4v_B^2$$

Taking square root:

$$v_A = 2v_B$$

So the ratio of their speeds is $2:1$.

If an object moves with constant velocity, its speed does not change. Therefore its kinetic energy also remains constant and does not change with position.

If the object moves with constant velocity in the horizontal direction near the Earth's surface, its height does not change, so its potential energy does not change.

If the object is gradually raised in the vertical direction, its height increases, so its potential energy increases according to $U=mgh$.

At the top of the fall, the object has potential energy $mgh$ and zero kinetic energy. As it falls, potential energy decreases and kinetic energy increases, but their sum remains constant.

Just before it hits the ground:

• potential energy $\approx 0$
• kinetic energy $= mgh$

So mechanical energy is

$$\text{PE} + \text{KE} = 0 + mgh = mgh$$

Thus, even just before impact, the mechanical energy remains $mgh$.

At the highest point A, the ball has maximum potential energy and minimum or nearly zero kinetic energy.

At point B, some of the potential energy has been converted into kinetic energy, so potential energy decreases and kinetic energy increases.

At point C at the lowest position, kinetic energy is maximum and potential energy is minimum.

The later points such as C, D and E usually have lower heights than the earlier ones because some energy is lost to friction and air resistance. Therefore, the ball cannot rise to the same height again.

Roads on hills are built as gentle winding slopes because an inclined path reduces the force needed to reach the top. The same gain in height is achieved by applying the force over a longer distance, so the task becomes easier. Also, vehicles can move more safely on a gentler slope.

An inclined ladder is easier to climb because it is like an inclined plane. You gain the same height with a smaller force acting over a longer distance. A vertical ladder requires you to lift your body almost directly against gravity, so the effort is greater.

It is easier to open the lid of a can with a spoon because the spoon acts as a lever. By using the spoon as a lever, a small effort applied at a larger distance from the fulcrum produces a larger force on the lid, so the lid can be opened more easily.

You push the object closer to the scissors fulcrum so that the load arm becomes smaller. This increases the mechanical advantage of the scissors, meaning a smaller effort can produce a larger cutting force on the hard object.

Real machines eventually slow down and stop because some of the work done by them is used up in overcoming friction and other resistive forces. Part of the useful mechanical energy gets converted into heat, sound, and other non-useful forms, so the machine loses energy continuously. Since no machine can supply energy forever by itself, it cannot keep moving indefinitely.

Initial speed from $K=\frac12 mv^2$:

$$\frac12 \times 2 \times 20^2 = 400\,\text{J}$$

At the highest point, potential energy gained is

$$mgh = 2 \times 10 \times 19.4 = 388\,\text{J}$$

So the loss of mechanical energy is

$$400 - 388 = 12\,\text{J}$$

This lost energy is the work done by air resistance, and it is negative because it opposes motion.

• Work done by gravity during upward motion: negative
• Work done by gravity during downward motion: positive
• Work done by air resistance: $-12\,\text{J}$

From A to B, the car moves with constant speed because it has not yet started braking significantly.

The kinetic energy at A is

$$K=\frac12 mv^2$$

From the figure, the speed at A is $20\,\text{m s}^{-1}$, so

$$K=\frac12 \times 1000 \times 20^2 = 2.0\times 10^5\,\text{J}$$

Between B and C, the brakes bring the car to rest, so the work done by the brakes is the change in kinetic energy:

$$W = 0 - 2.0\times10^5 = -2.0\times10^5\,\text{J}$$

The kinetic energy transforms mainly into heat energy due to friction in the brakes and tyres, and also a little sound.

(i) Just before hitting the sand, the coconut’s gravitational potential energy becomes kinetic energy:

$$mgh=\frac12 mv^2$$

$$v=\sqrt{2gh}=\sqrt{2\times 10\times 10}=\sqrt{200}\approx 14.1\,\text{m s}^{-1}$$

(ii) The coconut’s energy at impact is

$$mgh = 1.5\times 10\times 10 = 150\,\text{J}$$

If all this energy is used to make the depression, then

$$F\times d = 150$$

$$3000\times d = 150$$

$$d = \frac{150}{3000}=0.05\,\text{m}=5\,\text{cm}$$