## A monkey of mass 40 kg climbs on a rope (Fig. 5.20) which can stand

[question] Question. A monkey of mass 40 kg climbs on a rope (Fig. 5.20) which can stand a maximum tension of 600 N. In which of the following cases will the rope break: the monkey (a) climbs up with an acceleration of $6 \mathrm{~m} \mathrm{~s}^{-2}$ (b) climbs down with an acceleration of $4 \mathrm{~m} \mathrm{~s}^{-2}$ (c) climbs up with a uniform speed of $5 \mathrm{~m} \mathrm{~s}^{-1}$ (d) falls down the rope nearly freely under gravity? (lgnore the mass of the rope). [/question] [solutio...

## An aircraft executes a horizontal loop at a speed

[question] Question. An aircraft executes a horizontal loop at a speed of $720 \mathrm{~km} / \mathrm{h}$ with its wings banked at $15^{\circ}$. What is the radius of the loop? [/question] [solution] solution: Speed of the aircraft, $v=720 \mathrm{~km} / \mathrm{h}=720 \times \frac{5}{18}=200 \mathrm{~m} / \mathrm{s}$ Acceleration due to gravity, $g=10 \mathrm{~m} / \mathrm{s}^{2}$ Angle of banking, $\theta=15^{\circ}$ For radius $r$, of the loop, we have the relation: $\tan \theta=\frac{v^{2}}{...

## Ten one-rupee coins are put on top of each other on a table.

[question] Question. Ten one-rupee coins are put on top of each other on a table. Each coin has a mass m. Give the magnitude and direction of (a) the force on the $7^{\text {th }}$ coin (counted from the bottom) due to all the coins on its top, (b) the force on the $7^{\text {th }}$ coin by the eighth coin, (c) the reaction of the 6 th coin on the $7^{\text {th }}$ coin. [/question] [solution] solution: (a) Force on the seventh coin is exerted by the weight of the three coins on its top. Weight ...

## A helicopter of mass 1000 kg rises with a vertical acceleration

[question] Question. A helicopter of mass $1000 \mathrm{~kg}$ rises with a vertical acceleration of $15 \mathrm{~m} \mathrm{~s}^{-2}$. The crew and the passengers weigh $300 \mathrm{~kg}$. Give the magnitude and direction of the (a) force on the floor by the crew and passengers, (b) action of the rotor of the helicopter on the surrounding air, (c) force on the helicopter due to the surrounding air. [/question] [solution] solution: (a) Mass of the helicopter, $m_{\mathrm{h}}=1000 \mathrm{~kg}$ Ma...

## A stone of mass m tied to the end of a string revolves in a vertical circle of radius R.

[question] Question. A stone of mass m tied to the end of a string revolves in a vertical circle of radius R. The net forces at the lowest and highest points of the circle directed vertically downwards are: [Choose the correct alternative] [/question] $T_{1}$ and $v_{1}$ denote the tension and speed at the lowest point. $T_{2}$ and $v_{2}$ denote corresponding values at the highest point. [/question] [solution] solution: (a)The free body diagram of the stone at the lowest point is shown in the f...

## Figure 5.17 shows the position-time graph of a body of mass 0.04 kg.

[question] Question. Figure 5.17 shows the position-time graph of a body of mass 0.04 kg. Suggest a suitable physical context for this motion. What is the time between two consecutive impulses received by the body? What is the magnitude of each impulse? [/question] [solution] solution: A ball rebounding between two walls located between at $x=0$ and $x=2 \mathrm{~cm}$; after every $2 \mathrm{~s}$, the ball receives an impulse of magnitude $0.08 \times 10^{-2} \mathrm{~kg} \mathrm{~m} / \mathrm{s...

## If, in Exercise 5.21, the speed of the stone is increased

[question] Question. If, in Exercise 5.21, the speed of the stone is increased beyond the maximum permissible value, and the string breaks suddenly, which of the following correctly describes the trajectory of the stone after the string breaks: (a) the stone moves radially outwards, (b) the stone flies off tangentially from the instant the string breaks, (c) the stone flies off at an angle with the tangent whose magnitude depends on the speed of the particle ? [/question] [solution] solution: Wh...

## A batsman deflects a ball by an angle of $45^{\circ}$ without changing

[question] Question. A batsman deflects a ball by an angle of $45^{\circ}$ without changing its initial speed which is equal to $54 \mathrm{~km} / \mathrm{h}$. What is the impulse imparted to the ball? (Mass of the ball is $0.15$ kg.) [/question] [solution] solution: The given situation can be represented as shown in the following figure. Where, AO = Incident path of the ball OB = Path followed by the ball after deflection $\angle A O B=$ Angle between the incident and deflected paths of the bal...

## A shell of mass 0.020 kg is fired by a gun of mass 100 kg.

[question] Question. A shell of mass $0.020 \mathrm{~kg}$ is fired by a gun of mass $100 \mathrm{~kg}$. If the muzzle speed of the shell is $80 \mathrm{~m} \mathrm{~s}^{-1}$, what is the recoil speed of the gun? [/question] [solution] solution: Mass of the gun, M = 100 kg Mass of the shell, m = 0.020 kg Muzzle speed of the shell, v = 80 m/s Recoil speed of the gun = V Both the gun and the shell are at rest initially. Initial momentum of the system = 0 Final momentum of the system = mv – MV Here,...

## Two billiard balls each of mass 0.05 kg moving in opposite directions

[question] Question. Iwo billiard balls each of mass $0.05 \mathrm{~kg}$ moving in opposite directions with speed $6 \mathrm{~m} \mathrm{~s}^{-1}$ collide and rebound with the same speed. What is the impulse imparted to each ball due to the other? [/question] [solution] solution: Mass of each ball = 0.05 kg Initial velocity of each ball = 6 m/s Magnitude of the initial momentum of each ball, $p_{i}=0.3 \mathrm{~kg} \mathrm{~m} / \mathrm{s}$ After collision, the balls change their directions of m...

## A nucleus is at rest in the laboratory frame of reference.

[question] Question. A nucleus is at rest in the laboratory frame of reference. Show that if it disintegrates into two smaller nuclei the products must move in opposite directions. [/question] [solution] solution: Let $m, m_{1}$, and $m_{2}$ be the respective masses of the parent nucleus and the two daughter nuclei. The parent nucleus is at rest. Initial momentum of the system (parent nucleus) $=0$ Let $v_{1}$ and $v_{2}$ be the respective velocities of the daughter nuclei having masses $m_{1}$ ...

## A man of mass 70 kg stands on a weighing scale in a lift which is moving

[question] Question. A man of mass 70 kg stands on a weighing scale in a lift which is moving (a) upwards with a uniform speed of $10 \mathrm{~m} \mathrm{~s}^{-1}$, (b) downwards with a uniform acceleration of $5 \mathrm{~m} \mathrm{~s}^{-2}$, (c) upwards with a uniform acceleration of $5 \mathrm{~m} \mathrm{~s}^{-2}$. What would be the readings on the scale in each case? (d) What would be the reading if the lift mechanism failed and it hurtled down freely under gravity? [/question] [solution] s...

## A rocket with a lift-off mass 20,000 kg is blasted upwards

[question] Question. A rocket with a lift-off mass $20,000 \mathrm{~kg}$ is blasted upwards with an initial acceleration of $5.0 \mathrm{~m} \mathrm{~s}^{-2}$. Calculate the initial thrust (force) of the blast. [/question] [solution] solution: Mass of the rocket, m = 20,000 kg Initial acceleration, $a=5 \mathrm{~m} / \mathrm{s}^{2}$ Acceleration due to gravity, $g=10 \mathrm{~m} / \mathrm{s}^{2}$ Using Newton’s second law of motion, the net force (thrust) acting on the rocket is given by the rel...

## A body of mass 5 kg is acted upon by two perpendicular forces 8 N and 6 N.

[question] Question. A body of mass 5 kg is acted upon by two perpendicular forces 8 N and 6 N. Give the magnitude and direction of the acceleration of the body. [/question] [solution] solution: $2 \mathrm{~m} / \mathrm{s}^{2}$, at an angle of $37^{\circ}$ with a force of $8 \mathrm{~N}$ Mass of the body, $m=5 \mathrm{~kg}$ The given situation can be represented as follows: The resultant of two forces is given as: $R=\sqrt{(8)^{2}+(-6)^{2}}=\sqrt{64+36}=10 \mathrm{~N}$ $\theta$ is the angle made...

## A constant retarding force of 50 N is applied to a body of mass

[question] Question. A constant retarding force of $50 \mathrm{~N}$ is applied to a body of mass $20 \mathrm{~kg}$ moving initially with a speed of $15 \mathrm{~ms}^{-1}$. How long does the body take to stop? [/question] [solution] solution: Retarding force, $F=-50 \mathrm{~N}$ Mass of the body, $m=20 \mathrm{~kg}$ Initial velocity of the body, $u=15 \mathrm{~m} / \mathrm{s}$ Final velocity of the body, $v=0$ Using Newton's second law of motion, the acceleration (a) produced in the body can be c...

## One end of a string of length l is connected to a particle

[question] Question. One end of a string of length l is connected to a particle of mass m and the other to a small peg on a smooth horizontal table. If the particle moves in a circle with speed v the net force on the particle (directed towards the centre) is: (i) $T$, (ii) $T-\frac{m v^{2}}{l}$ (iii) $T+\frac{m v^{2}}{l}$ (iv) 0 T is the tension in the string. [Choose the correct alternative]. [/question] [solution] solution: Answer: (i) When a particle connected to a string revolves in a circul...

## Give the magnitude and direction of the net force acting on a stone of mass 0.1 kg,

[question] Question. Give the magnitude and direction of the net force acting on a stone of mass 0.1 kg, (a) just after it is dropped from the window of a stationary train, (b) just after it is dropped from the window of a train running at a constant velocity of 36 km/h, (c) just after it is dropped from the window of a train accelerating with $1 \mathrm{~m} \mathrm{~s}^{-2}$, (d) Iying on the floor of a train which is accelerating with $1 \mathrm{~m} \mathrm{~s}^{-2}$, the stone being at rest r...

## A pebble of mass 0.05 kg is thrown vertically upwards.

[question] Question. A pebble of mass 0.05 kg is thrown vertically upwards. Give the direction and magnitude of the net force on the pebble, (a) during its upward motion, (b) during its downward motion, (c) at the highest point where it is momentarily at rest. Do your answers change if the pebble was thrown at an angle of $45^{\circ}$ with the horizontal direction? Ignore air resistance. [/question] [solution] solution: 0.5 N, in vertically downward direction, in all cases Acceleration due to gr...

## Give the magnitude and direction of the net force acting on

[question] Question. Give the magnitude and direction of the net force acting on (a) a drop of rain falling down with a constant speed, (b) a cork of mass 10 g floating on water, (c) a kite skillfully held stationary in the sky, (d) a car moving with a constant velocity of 30 km/h on a rough road, (e) a high-speed electron in space far from all material objects, and free of electric and magnetic fields. [/question] [solution] solution: (a) Zero net force The rain drop is falling with a constant ...