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**1. What is the speed of an aircraft if the pilot remains in contact with the seat, even while looping in vertical plane?**

**Ans. **For the pilot to stay at rest, centripetal force should

be provided by his weight, i.e., $\frac{m v^{2}}{r}=m g$

$\therefore \quad v=\sqrt{r g}$

**Q.2 The kinetic energy of a particle moving in a horizontal circle may remain the same everywhere. Is it true for the motion in a vertical circle.?**

**Ans.** No. For the motion in a vertical circle, the speed of the particle at the bottom is $\sqrt{5 g r}$ and at the top is $\sqrt{g r}$. Therefore, $\mathrm{K} . \mathrm{E}$. of the particle in a vertical circle is maximum at the bottom and minimum at the top.

**Q.3 A stone of mass $m$ tied to the ended of a string of length $l$ is whirled in a vertical circle. What is the net force acting on the stone at the bottom and at the top of the vertical circle, if tension in the string at the bottom at the top is $T_{b}$ and $T_{t}$ respectively**

**$?$**

**Ans.** Net force acting on the stone at the bottom of

the circle $=T_{b}-m g$

Net force at the top $=T_{t}+m g$

**Q.4 Can magnitude of kinetic energy be negative?**

**Ans.** No, kinetic energy can be zero or positive only

$$\text { because } K E=\frac{1}{2} m v^{2}, \text { because } \mathrm{m} \text { and } \mathrm{v}^{2} \text { cannot }$$

be negative.

**Q.5 Is potential energy defined for non conservative force?**

**Ans.** No, potential energy is defined only for conservative force. It does not exist for non conservative force.

**Q.6 Does a body in motion always have potential energy?**

**Ans.** A body in motion may or may not have potential energy.

** Q.7 What is the significant of negative mechanical energy? **

**Ans.** Negative value of mechanical energy indicates a** **bound state electron in an atom and a satellite** **revolving around a planet both are in bound state** **having negative mechanical energy.

**Q.8 If rain drop move decreasing acceleration, does work done by gravitational force on the drop change.**

**Ans.** It should be clearly understood that whether the drop move with decreasing acceleration or with uniform speed work done by the gravitational force on the drop remains the same.

**Q.9 Can K.E. of a system be increased or decreased without applying any external force on the system. **

**Ans.** Yes, by doing work through internal force e.g. in** **explosion of a bomb.

**Q.10 Does the sun do any work on earth where earth revolves around the sun in a perfectly circular orbit?**

**Ans.** No, because angle between gravitational pull and direction of motion at any instance is $=90^{\circ}$

**Q.11 When an air bubble rises in water what happens to its potential energy?**

**Ans.** Potential energy of air bubble decreases because work is done by upthrust on the bubble.

**Q.12 Can a body have momentum without energy? Ans.** Yes, when $\mathrm{E}=\mathrm{k}+\mathrm{u}=0$ either both are zero or $\mathrm{k}$$=-\mathrm{u} .$ thus $\mathrm{K.E.}$ may or may not be zero** **As $P=\sqrt{2 m k}$** **

$\therefore P=0$ only when $\mathrm{k}=0$

and $P \neq 0$ when $\mathrm{k}=-\mathrm{u}$

**Q.13 Can a body have energy without momentum?**

**Ans.** Yes, when $P=0 \quad k=\frac{P^{2}}{2 m}=0$ But $E=k+u=u$ (potential energy) which may or may not be zero.

**Q.14 In which motion momentum change but K.E. does not?**

**Ans.** In uniform circular motion.

**Q.15 Can a body have momentum when its energy is negative?**

**Ans.** Yes, when $\mathrm{k}<\mathrm{u},$ total energy $E=k+u$ is negative, the body has the momentum $(\because k \neq 0) .$ For example in an atom, electron has momentum though its energy is negative.

**Q.17 Can work be defined at a position or at instant?**

**Ans.** No, work is not defined at a position or at an instant. Work is defined during the time interval within which the force has undergone displacement unlike velocity and acceleration, momentum and kinetic energy it is not the characteristic of moving body it is something which happen an a body which produces or destroys motion of the body.

**Q.17 A man left 10 bricks to the top of a building by going up a vertical spiral staircase in 5 mins whereas a boy lefts 10 bricks to the top of the building by going up a standing ladder in 10 mins who does greater amount of work.**

**Ans.** Both do the same amount of work, because work done does not depend on path followed and time take to complete work.

**Q018 As the bob of a pendulum swings does the tension in the string do any work on it.**

**Ans.** No, because angle between tension and displacement is $90^{\circ} .$

**Q.19 What is work done by conservative force round a closed path?**

**Ans.** Zero.

**Q.20 A man rowing boat upstream is at rest with respect to the shore is he doing work.**

**Ans.** The boat is at rest with respect to the shore but itis moving up stream with respect to water. The man is doing work relative to the stream because he is applying force to produce relative motionbetween the boat and stream, but he does no work relative to the shore as displacement relative to shore is zero.

**Q.21 Why a metal ball rebounds better than a rubber ball?**

**Ans.** When rubber ball hits a massive object say earth the ball is distorted. A large amount of heat is generated in the ball by the rubbing of the rubber molecules against each other this effect is essentially absent in a band material so metal ball would often loseless energy upon collision than would a rubber ball.

**Q.22 The outer casing of a rocket is burnt due to friction between casing and air, who supplies the energy necessary for burning. The rocket or the atmosphere?**

**Ans.** Due to friction with the air the speed hence the kinetic energy of the rocket decreases. This decrease in the kinetic energy appears in the form of heat energy, thus energy necessary for burning is supplied by the rocket itself.

**Q.23 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.**

**Ans.** As a satellite comes closer to the earth its potential energy decrease. According to the law of conservation of energy the total energy of the system must remain constant. Now total energy $=\mathrm{KE}+\mathrm{PE}+$ heat energy. Because the heat loss due to friction is very small a decrease in potential energy result in a net increase in kinetic energy. Hence speed of a satellite progressively increase as it comes closer to the earth.

**Q.24 In a tug-of-war one team is slowly giving way to the other is work being done on the losing team. How about the wining team.**

**Ans.** Work is done on the losing team because that team is being dragged in the direction of the force applied by the wining team. This work done is said to be positive work, work is also done on the wining team because the wining team is being displaced opposite to the direction of the force applied by the losing team. This work done is said to be negative work.

**Q.25 In a circus, the diameter of globe of death is 30 m. From what minimum height must a cyclist start in order to roll down the inclined and go round the globe successfully?**

**Ans.** Diameter of globe $=30 \mathrm{m}$

Radius of globe, $r=15 \mathrm{m}$

Let $h$ be the minimum height from which the

cyclist after rolling down an incline will acquire velocity $=\sqrt{2 g h}$

For looping the loop, the minimum velocity at the lowest point should be $\sqrt{5 g r}$.

$\therefore \sqrt{5 g r}=\sqrt{2 g h}$

$\Rightarrow h=\frac{5 r}{2}=\frac{5 \times 15}{2}=37.5 \mathrm{m}$

**Q.26 Two inclined frictionless tracks one gradual and the other steep meet at A from where two stones are allowed to slide down from rest one on. Each track well the two stones reach the bottom with the same speed. Will they reach the bottom at the same time.**

**Ans.** Let $m_{1}$ and $m_{2}$ be the masses of the stones sliding down respectively on the track $A B$ and $A C$ of same height $A D=h$ but of different inclination $\theta_{1}$ and $\theta_{2}$ where $\theta_{2}>\theta_{1}$. In the sliding down of the stones their potential energies at a will be converted into their kinetic energies at $\mathrm{B}$ and $\mathrm{C}$ that is $m_{1} g h=\frac{1}{2} m_{1} v_{1}^{2}$ and $m_{2} g h=\frac{1}{2} m_{2} v_{2}^{2}$

Then, $v_{1}=v_{2}=\sqrt{2 g h}$

The stones will reach the bottom with same speed. As clear from the figure the accelerations of the stones along the inclined planes are

$a_{1}=g \sin \theta_{1}$ and $a_{2}=g \sin \theta_{2}$

$a_{2}>a_{1}$ because $\theta_{2}>\theta_{1}$

From the relation $v=u$ tat $(\text { here } u=0)$ the time taken by the stones in reaching the bottom are $f_{1}=\frac{v_{1}}{a_{1}}, f_{2}=\frac{v_{2}}{a_{2}}$ since $v_{1}=v_{2}$ and $a_{2}>a_{1}$ we have $f_{1}>f_{2}$

The stones will reach the bottom at different times that on the steep plane will reach earlier.

**Q.27. Given below $P Q R S T$ is a channel in the vertical plane, part $Q R S T$ being circular arc of radius $r . \mathrm{A}$ ball is released from $P$, and slides without friction, and without rolling. Show that it will complete the loop path if $h$ is greater than $\frac{5 \mathrm{r}}{2}$.**

**Sol.** When the ball ( 0 f mass $m$ ) reaches $\theta,$ its $\mathrm{P.E.}$** ** changes into kinetic energy such that** **$$m g h=\frac{1}{2} m v_{1}^{2}$$

**Q.28 A small washer is placed on the top of a hemisphere of radius $r$. What minimum horizontal velocity should be imparted to the washer to detach it from the hemisphere at the initial point of motion $?$**

**Sol.**

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