Answer the following: (a) The casing of a rocket in flight burns up due to friction. At whose expense is the h

Exercises · 5.5

Q. Answer the following:

(a) The casing of a rocket in flight burns up due to friction. At whose expense is the heat energy required for burning obtained? The rocket or the atmosphere?

(b) Comets move around the sun in highly elliptical orbits. The gravitational force on the comet due to the sun is not normal to the comet's velocity in general. Yet the work done by the gravitational force over every complete orbit of the comet is zero. Why?

(c) An artificial satellite orbiting the earth in very thin atmosphere loses its energy gradually due to dissipation against atmospheric resistance, however small. Why then does its speed increase progressively as it comes closer and closer to the earth?

(d) In Fig. 5.13(i) the man walks

2\ \text{m}

carrying a mass of

15\ \text{kg}

on his hands. In Fig. 5.13(ii), he walks the same distance pulling the rope behind him. The rope goes over a pulley, and a mass of

15\ \text{kg}

hangs at its other end. In which case is the work done greater?

Rajasthan RbseTextbookSubjective· 5mImportance★★★★★est

20% · 17/83 Questions

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

Concept understanding — Conservation Of Mechanical Energy

Conservation of Mechanical Energy

The Intuition First

Imagine you're holding a heavy stone at shoulder height. Your arm is tired. That stone has potential energy — energy stored because of its position. Now let it go. As it falls, it speeds up. The potential energy is turning into kinetic energy — the energy of motion. Just before it hits the ground, all the original potential energy has become kinetic energy.

Now imagine the reverse: you throw a ball straight up. It leaves your hand fast (lots of kinetic energy), rises, slows down, stops for an instant at the top (zero kinetic energy), then falls back. At the top, all the kinetic energy you gave it has turned back into potential energy.

This back-and-forth transformation — potential ↔ kinetic — is the heart of the idea. Energy doesn't disappear; it just changes form. That's conservation.

The Precise Statement

Important

Conservation of Mechanical Energy: In an isolated system where only conservative forces (like gravity or an ideal spring) do work, the total mechanical energy of the system remains constant.

Total mechanical energy is the sum of kinetic energy ( $K$ ) and potential energy ( $U$ ):

E_{\text{mech}} = K + U

The law says:

K_{\text{initial}} + U_{\text{initial}} = K_{\text{final}} + U_{\text{final}}

Or, in symbols:

E_{\text{mech, initial}} = E_{\text{mech, final}}

What This Means in Practice

Let's go back to the falling stone. Suppose you hold it 5 metres above the ground. Its mass is 2 kg. Take $g = 10\ \text{m/s}^2$ .

At the top (initial):

$K_i = 0$ (not moving)

$U_i = mgh = 2 \times 10 \times 5 = 100\ \text{J}$

$E_{\text{mech}} = 0 + 100 = 100\ \text{J}$
Just before hitting ground (final):

$U_f = 0$ (height = 0)

$K_f = \frac{1}{2}mv^2$

Conservation says $K_f = 100\ \text{J}$ , so $\frac{1}{2} \times 2 \times v^2 = 100$ , giving $v = 10\ \text{m/s}$ .

You never needed to know the time of fall or acceleration. Energy conservation gave you the speed directly.

The Two Critical Conditions

Watch out

Mechanical energy is not always conserved. It is conserved only when:

No non-conservative forces (like friction, air resistance, or applied pushes/pulls) do work.
The system is isolated — no external forces transfer energy in or out.

If friction is present, some mechanical energy turns into heat (thermal energy). The total energy of the universe is still conserved, but mechanical energy alone is not.

A Simple Example to Cement It …

Start your 14-day free trial to unlock the full solution →

Each part tests a different nuance of energy conservation and work. (a) The rocket's kinetic energy provides the heat. (b) Gravity is conservative — net work over a closed orbit is zero. (c) Loss of total energy shrinks the orbit, increasing speed. (d) Work is greater in case (ii) because the man must lift the mass against gravity.

(a) The burning casing — whose energy?

The key idea: energy is conserved. The heat that burns the casing must come from somewhere — it cannot be created from nothing.

The rocket is moving at high speed through the atmosphere. Friction (air resistance) does negative work on the rocket, converting its kinetic energy into thermal energy. That thermal energy heats the casing to the point of burning.

Watch out

A common mistake is to think the atmosphere "gives" the energy. But the atmosphere is not a fuel source — it merely provides the force that dissipates the rocket's own kinetic energy.

So the heat energy comes at the expense of the rocket's kinetic energy (and hence its total mechanical energy). The atmosphere is just the medium that causes the dissipation.

(b) Zero work over a complete orbit — why?

Gravitational force is a conservative force. For any conservative force, the work done over a closed path is zero. That's the short answer.

But let's see why that's true for a comet. The gravitational force on the comet is always directed toward the Sun. Over one complete elliptical orbit, the comet returns to the same point with the same speed (by conservation of mechanical energy). The work-energy theorem says:

W_{\text{gravity}} = \Delta K

Since the comet's speed (and therefore kinetic energy) is the same at the start and end of a full orbit, $\Delta K = 0$ , so $W_{\text{gravity}} = 0$ .

Tip

Even though the force is not perpendicular to velocity at every instant, the net work over the whole orbit is zero because gravity is conservative. The work done while the comet moves toward the Sun (gaining speed) is exactly cancelled by the work done while it moves away (losing speed).

(c) Satellite losing energy but gaining speed — paradox?

This seems contradictory: if the satellite loses energy, shouldn't it slow down? The catch is that total mechanical energy $E = K + U$ is decreasing, but the satellite is also falling to a lower orbit.

For a satellite in a circular orbit of radius $r$ :

K = \frac{GMm}{2r}, \quad U = -\frac{GMm}{r}, \quad E = -\frac{GMm}{2r}

As energy $E$ becomes less negative (i.e., the satellite loses energy), $r$ decreases. But look at the kinetic energy: $K = \frac{GMm}{2r}$ — as $r$ gets smaller, $K$ increases. So the satellite speeds up.

Note

The lost energy goes into heating the atmosphere (dissipation). The satellite's potential energy decreases more than its kinetic energy increases, so total energy drops — but the kinetic energy itself rises.

--- …

Start your 14-day free trial to unlock the full solution →

←Previous question5.4 — The potential energy function for a particle executing linear simple h…→Next question5.6 — Underline the correct alternative: (a) When a conservative force does …

Q.Answer the following:

Concept understanding — Conservation Of Mechanical Energy

Conservation of Mechanical Energy

The Intuition First

The Precise Statement

What This Means in Practice

The Two Critical Conditions

A Simple Example to Cement It …

(a) Source of heat energy for the burning casing

(a) The burning casing — whose energy?

(b) Zero work over a complete orbit — why?

(c) Satellite losing energy but gaining speed — paradox?

Concept: Energy Conservation and Work in Everyday Situations

Q.Answer the following:

(a) The burning casing — whose energy?

(b) Zero work over a complete orbit — why?

(c) Satellite losing energy but gaining speed — paradox?

🎓Unlock everything free for 14 days

Unlock everything free for 14 days