NSEJS (2020) - Problems & Solution

Problem 1

Gravitational collapse is the contraction of an astronomical object under its own gravity. This draws the matter inwards towards the centre of gravity. A neutron star is an example of the collapsed core of a giant star. A certain neutron star of radius 10 km is of mass $1.5 M_{\odot}$. The acceleration due to gravity on the surface of the neutron star is nearly

(a) $2.0 \times 10^8 \mathrm{~m} / \mathrm{s}^2$
(b) $2.0 \times 10^{12} \mathrm{~m} / \mathrm{s}^2$
(c) $2.6 \times 10^{16} \mathrm{~m} / \mathrm{s}^2$
(d) $2.6 \times 10^{20} \mathrm{~m} / \mathrm{s}^2$

Problem 2

The tympanic membrane (ear drum) is a very delicate component of the human ear. Typically, its diameter is 1 cm . The maximum force the ear can withstand is 2.5 N . In case a diver has to enter sea water of density $1.05 \times 10^3 \mathrm{~kg} / \mathrm{m}^3$ without any protective gear, the maximum safe depth for the diver to go into water is about

(a) 12 m
(b) 9 m
(c) 3 m
(d) 1.5 m

Problem 3

Two illuminated point objects $\mathrm{O}_1$ and $\mathrm{O}_2$ are placed at a distance 24 cm from each other along the principal axis of a thin convex lens of focal length 9 cm such that images of both the objects are formed at the same position. Then the respective distances of the lens from $\mathrm{O}_1$ and $\mathrm{O}_2$ (in cm ) are

(a) 12 and 12
(b) 18 and 6
(c) 14 and 10
(d) 16 and 8

Problem 4

A nuclear reactor is working at $30 \%$ efficiency (i.e. conversion of nuclear energy to electrical energy). In this reactor ${ }_{92}^{235} \mathrm{U}$ nucleus undergoes fission and releases 200 MeV energy per atom. If 1000 kW of electrical power is obtained in this reactor, then the number of atoms disintegrated (undergone fission) per second in the reactor is

(a) $1.04 \times 10^{17}$
(b) $6.5 \times 10^{12}$
(c) $3.125 \times 10^{12}$
(d) $3.25 \times 10^{32}$

Problem 5

Two blocks A and B are in contact with each other and are placed on a frictionless horizontal surface. A force of 90 N is applied horizontally on block A (situation I) and the same force is applied horizontally on block B (situation II). Mass of A is 20 kg and B is 10 kg . Then the correct statement is

(a) Since both the blocks are in contact, magnitude of force by block A on B will be 90 N (situation I) and magnitude of force by block B on A will also be 90 N (situation II).
(b) Magnitude of force by block A on B is 30 N (situation I ) and magnitude of force by block B on A is 60 N (situation II).
(c) Magnitude of force by block A on B is 60 N (situation I ) and magnitude of force by block B on A is 30 N (situation II).
(d) The 90 N force will produce acceleration of different magnitudes in A and B .

Problem 6

In the adjoining circuit, $R=5 \Omega$. It is desired that the voltage across $R_x$ should be 6 V , then the value of $R_x$ should be

(a) $4 \Omega$
(b) $12 \Omega$
(c) $16 \Omega$
(d) $20 \Omega$

Problem 7

An infinitely long conductor when carrying current (I), produces a magnetic field (B) around it. If such a conductor is placed along the X-axis, then the magnitude of (B) at a distance (r) is given by the relation $B=\frac{\mu_0}{4 \pi} \frac{2 I}{r}$, (where $\frac{\mu_0}{4 \pi}=10^{-7} \mathrm{NA}^{-2}$ is a constant). The following figure shows such an infinitely long conductor placed along X -axis carrying current (I) and (B) at (S) is $2 \times 10^{-4} \mathrm{~T}$, directed into the plane of the paper at S. Given $r=1 \mathrm{~cm}$. Then, the correct statements are

(a) $I=10 \mathrm{~A}$
(b) The number of electrons transported across the cross section of the conductor during time 1 s is $6.25 \times 10^{19}$
(c) The direction of current (I) is from $X_2$ to $X_1$.
(d) The electrons will flow in the direction $\mathrm{X}_2$ to $\mathrm{X}_1$.

Problem 8

The ratio of the charge of an ion or subatomic particle to its mass $(q / m)$ is called specific charge. Then the correct options are

(a) SI unit of specific charge can be written as $\mathrm{A} \cdot \mathrm{s} / \mathrm{kg}$.
(b) If all the isotopes of hydrogen are ionized then tritium will have least specific charge among them.
(c) specific charge of an $\alpha$-particle will be greater than that of an electron.
(d) specific charge ratio of an electron is $1.75 \times 10^{11} \mathrm{C} / \mathrm{kg}$.

Problem 9

A girl (G) walks into a room along the path shown by the dashed line (see figure on right). She tries to observe images of small toys numbered 1,2 , and 3 in the plane mirror on the wall. The order in which she will see images of the toys is:

(A) 3,2,1
(B) 3,2
(C) 1, 2, 3
(D) 2, 3

Problem 10

A heating element in the form of a wire with uniform circular cross sectional area has a resistance of $310 \Omega$, and can bear a maximum current of 5.0 A . The wire can be cut into pieces of equal length. The number of pieces, arranged suitably, so as to draw maximum power when connected to a constant voltage of 220 V , is

(A) 7
(B) 8
(C) 44
(D) 62

Problem 11

Consider the following two statements
Statement $S_1$: If you put 100 g ice at $0^{\circ} \mathrm{C}$ and 100 g water at $0^{\circ} \mathrm{C}$ into a freezer, which is maintained at $-10^{\circ} \mathrm{C}$, the ice will eventually lose the lager amount of heat.
Statement $S_2$ : At $0^{\circ} \mathrm{C}$, water is denser than ice
Choose the correct statement among the following.

(A) Both $S_1$and $S_2$ are true and $S_2$ is the correct explanation of $S_1$
(B) Both $S_1$ and $S_2$ are true and $S_2$ is not the correct explanation of $S_1$
(C) $S_1$ is true but $S_2$ is false
(D) $S_1$ is false but $S_2$ is true

Problem 12

Consider the paths of (1) Halley's Comet near the sun, and (2) an alpha particle scattered by a nucleus. In the figures below, the dots represent the Sun/Nuclei, and the curves with arrows mark the paths of the comet/alpha particle schematically.The correct statement about the trajectories is

(A) I represents trajectory for Halley's Comet and II for the scattering of alpha particles.
(B) III represents trajectory for Halley's Comet and II for the scattering of alpha particles
(C) II represents trajectory for Halley's Comet and I for scattering of alpha particles
(D) II represents trajectory for Halley's Comet and III for scattering for scattering of alpha particles.

NSEJS [2010] Problems & Solution

Problem 1

Which one of the following statements is INCORRECT?

(A) If the net force on a body is zero, its velocity is constant or zero
(B) If the net force on a body is zero, its acceleration is constant and
(C) If the velocity of a body is constant, its acceleration is zero
(D) A body may have a varying velocity yet a constant speed

Problem 2

Two forces each of magnitude (P) act on a body placed at a point (O) as shown. The force necessary to keep the body at rest is of magnitude.

(A) P along +X axis
(B) P along - X axis
(C) 2 P along +X axis
(D) P/2 along - X axis

Problem 3

Two spheres having masses 10 g and 25 g are projected horizontally from the same height with velocities $v_1$ and $v_2$ and they fall to the ground in time intervals $t_1$ and $t_2$ respectively. If the ratio $v_1: v_2$ is $1: 3$, the ratio $t_1: t_2$ will be

(A) $10: 25$
(B) $25: 10$
(C) $1: 1$
(D) $1: 3$

Problem 4

The SI unit of temperature is

(A) degree Fahrenheit $ ({ }^{\circ} \mathrm{F}) $
(B) degree Celsius $ ({ }^{\circ} \mathrm{C}) $
(C) degree Kelvin $ ({ }^{\circ} \mathrm{K}) $
(D) None of the above

Problem 5

A convex lens $L_1$ forms an image of the same size as that of the object at a distance of 24 cm . If the lens $\mathrm{L}_1$ is replaced by another convex lens $\mathrm{L}_2$, the image formed is magnified and erect. Therefore, the focal lenght of $L_2$ is

(A) less than 12 cm
(B) 12 cm
(C) between 12 cm and 24 cm
(D) more than 24 cm

Problem 6

A wooden ball of density $0.8 \mathrm{~g} / \mathrm{cm}^3$ is placed in water. The ratio of the volume above the water surface to that below the water surface is

(A) 0.25
(B) 0.20
(C) 2.0
(D) 4.0

Problem 7

A stone is released from an elevator moving upwards with an acceleration (a). The acceleration of the stone after the release is

(A) (a) upwards
(B) ($\mathrm{g}-a)$ upwards
(C) ($\mathrm{g}-a)$ downwards
(D) g downwards

Problem 8

A converging beam of light falls on a convex mirror of radius of curvature 20 cm , the point of convergence being 10 cm behind the mirror. The image is

(A) virtual and formed 10 cm in front of the mirror
(B) real and formed in front of the mirror
(C) formed at infinity
(D) virtual and formed 10 cm behind the mirror

Problem 9

When a sound wave moves through air along (X) axis, there is a variation in density of air in this direction. The graphical representation of this variation for two sound waves A and B is as shown. Which of the following statements is correct?

(A) Frequency of A is greater than that of B
(B) Velocity of B is greater than that of A
(C) Wavelength of B is greater than that of A
(D) Loudness of A is greater than that of B

Problem 10

In the figure shown below, each of the lenses has a focal length of 10 cm. Therefore, the image formed by the combination of lenses is

(A) virtual, erect and magnified
(B) virtual, inverted and diminished
(C) virtual, erect and diminished
(D) real, erect and diminished

Problem 11

A convex mirror used as the rear view mirror of a motor vehicle has a warning written on it - 'Objective in this mirror are nearer than they appear'. The reason for this warning is that

(A) the image is diminished
(B) the image distance is less than the focal length of the mirror
(C) the image distance is less than the object distance
(D) the image distance is more than the object distance

Problem 12

A uniform wire of resistance 36 ohm is bent into a circle. A battery is connected between points (A) and (B) as shown. The effective resistance between (A) and (B) is

(A) 36 ohm
(B) 30 ohm
(C) 6 ohm
(D) 5 ohm

Problem 13

On North Pole, when the surface of sea gets frozen due to cold weather, eskimos can still fish by cutting a portion of ice at the surface to find water underneath. This is possible because water

(A) has low thermal conductivity
(B) has high specific heat
(C) has high surface tension
(D) shows anomalous behaviour

Problem 14

A, DC current flows through a vertical wire in the downward direction. For an observer looking at the wire, the direction of magnetic field at a point between him and the wire is

(A) upward
(B) to the right
(C) to the left
(D) downward

Problem 15

A number of electric bulbs of rating 220 volt, 100 watt are to be connected in parallel to a 220 volt supply. If a 5 A fuse wire is used for this arrangement to bulbs, the maximum number of bulbs that can be included in the arrangement will be

(A) 10
(B) 11
(C) 22
(D) 44

Problem 16

In the circuit given below, AB is a thick copper wire and S is a switch. When the switch is closed, the effective resistance of the circuit will be

(A) 5 ohm
(B) $6 / 5 \mathrm{ohm}$
(C) 3 ohm
(D) zero

Problem 17

In figure (1) ammeter reads $I_1$ and voltmeter reads $V_1$. Similarly, in figure (2) ammeter reads $I_2$ while voltmeter reads $\mathrm{V}_2$. Then which of the following statements is correct?

(A) $\mathrm{V}_1>\mathrm{V}_2$ and $\mathrm{I}_1>\mathrm{I}_2$
(B) $\mathrm{V}_1>\mathrm{V}_2$ and $\mathrm{I}_1<\mathrm{I}_2$
(C)$\mathrm{V}_1<\mathrm{V}_2$ and $\mathrm{I}_1>\mathrm{I}_2$
(D) $\mathrm{V}_1<\mathrm{V}_2$ and $\mathrm{I}_1<\mathrm{I}_2$

Problem 18

The latent heat of fusion of a solid is the quantity of heat in joules required to convert

(A) 1 mg of the solid to liquid, without any change in temperature.
(B) 1 g of the solid to liquid, without any change in temperature.
(C) 100 g of the solid to liquid, without any change in temperature.
(D) 1000 g of the solid to liquid, without any change in temperature.

Problem 19

If the pressure of a given mass of a gas is reduced to half and temperature is doubled simultaneously, the volume will be-

(A) the same as above
(B) twice as before
(B) four times as before
(D) one forth as before

Problem 20

While picking up a pair of eye glasses dropped by a friend, you notice that they form an inverted image of the background and that the image is stretched horizontally as well. Your friend suffers from

(A) only myopia
(B) only hypermetropia
(C) only astigmatism
(D) hypermetropia as well as astigmatism

NSEJS – 2023 - Problems & Solution

Problem 1

Two blocks A and B of masses 1 kg and 4 kg respectively are moving with equal kinetic energies. Read the following statements $S_1$ and $S_2$
Statement $S_1$ : Ratio of speed of the block A to that of B is (1: 2)
Statement $S_2$ : Ratio of magnitude of linear momentum of (A) to that of (B) is (1: 2)
Now choose the correct option:

(a) Both $\mathrm{S}_1$ and $\mathrm{S}_2$ are true
(b) Both $\mathrm{S}_1$ and $\mathrm{S}_2$ are false
(c) $S_1$ is true, $S_2$ is false
(d) $\mathrm{S}_1$ is false, $\mathrm{S}_2$ is true

Problem 2

The mass of a straight copper wire is 20.95 g and its electrical resistance is $0.065 \Omega$. If the density and resistivity of copper are $\mathrm{d}=8900 \mathrm{~kg} / \mathrm{m}^3$ and $\rho=1.7 \times 10^{-8}$ ohm-meter respectively, the length of the copper wire is

(a) 3 m
(b) 6 m
(c) 12 m
(d) date is insufficient

Problem 3

It is known that the speed of sound in a gas is directly proportional to square root of its absolute temperature T measured in Kelvin i.e. $v \propto \sqrt{T}$ Speed of sound in air at $0^{\circ} \mathrm{C}$ is $332 \mathrm{~m} / \mathrm{s}$. On a hot day, the speed of sound was measured $360 \mathrm{~m} / \mathrm{s}$ in NCR Delhi, the temperature of air in Delhi on that very day must have been close to

(a) $40^{\circ} \mathrm{C}$
(b) $42^{\circ} \mathrm{C}$
(c) $44^{\circ} \mathrm{C}$
(d) $48^{\circ} \mathrm{C}$

Problem 4

A small bar magnet is allowed to fall vertically through a metal ring lying in a horizontal plane. During its fall, the acceleration of the magnet in the region close to the ring must be ( g is the acceleration due to gravity)

(a) equal to (g)
(b) less than (g) and uniform
(c) less than g and non-uniform
(d) greater than (g) and uniform

Problem 5

A U-tube of uniform cross section contains two different liquids in its limbs namely water (density $1.0 \times 10^3 \mathrm{~kg} / \mathrm{m}^3) $ and Mercury (density $13.6 \times 10^3 \mathrm{~kg} / \mathrm{m}^3) $ as shown in figure. The difference of height of mercury column in two limbs of the tube is $\mathrm{H}=1.5 \mathrm{~cm}$. The height h of the water column in the left limb above the Mercury column must be nearly (Neglect surface tension effects)

(a) 13.6 cm
(b) 20.4 cm
(c) 27.0 cm
(d) 9.0 cm

Problem 6

An object pin is placed at a distance 10 cm from first focus of a thin convex lens on its principal axis, the lens forms a real and inverted image of this object pin at a distance 40 cm beyond the second focus. The focal length of the lens is

(a) 16 cm
(b) 20 cm
(c) 25 cm
(d) 40 cm

Problem 7

A bullet of mass 0.25 kg moving horizontally with velocity $v(\mathrm{~m} / \mathrm{s})$ strikes a stationary block of mass 1.00 kg suspended by a long inextensible string of negligible mass and length $\ell$. The bullet gets embedded in the block and the system rises up to maximum height $\mathrm{h}=19.6 \mathrm{~cm}$ (as shown in the figure. The string still remains taut). The value of initial speed (v) of the bullet is

(a) $5.9 \mathrm{~m} / \mathrm{s}$
(b) $7.8 \mathrm{~m} / \mathrm{s}$
(c) $9.8 \mathrm{~m} / \mathrm{s}$
(d) $11.8 \mathrm{~m} / \mathrm{s}$

Problem 8

The equivalent resistance between points A and B in the following electrical network is

(a) $\frac{3}{4} \Omega$
(b) $\frac{4}{3} \Omega$
(c) $\frac{2}{5} \Omega$
(d) $\frac{9}{14} \Omega$

Problem 9

The order of magnitude of the pressure (in pascal) exerted by an adult human on the Earth when he stands bare footed on the Earth on both of his legs, is

(a) $10^2$
(b) $10^4$
(c) $10^7$
(d) $10^9$

Problem 10

On the board of an experiment, three bulbs $\mathrm{B}_1(100 \mathrm{~W}, 200 \mathrm{~V})$, $\mathrm{B}_2(60 \mathrm{~W}, 200 \mathrm{~V})$ and $\mathrm{B}_3(40 \mathrm{~W}, 200 \mathrm{~V})$ are connected to a 200 V fluctuating supply with a fuse in series as shown in the figure. The electric current rating of the fuse required in the circuit to protect all the three bulbs must be

(a) 0.2 Amp
(b) 0.3 Amp
(c) 0.5 Amp
(d) 1.0 Amp

Problem 11

An ant is sitting on the principal axis of a convex mirror of focal length (f), at a distance (2 f) from the pole in front of the mirror. It starts moving on principal axis towards the mirror. During the course of motion, the distance between the ant and its image

(a) throughout increases
(b) throughout decreases
(c) first increases, then decreases
(d) first decreases, then increases

Problem 12

You are given three resistance of values $2 \Omega, 4 \Omega$ and $6 \Omega$. Which of the following values of equivalent resistance is not possible to get by using/arranging these three resistors in any circuit?

(a) Less than $2 \Omega$
(b) Equal to $4.4 \Omega$
(c) Equal to $5.5 \Omega$
(d) Equal to $7.6 \Omega$

Problem 13

ABC is a 0.8 meter long curved wire track in a vertical plane. A bead of mass 3 g is released from rest at A . It slides along the wire and comes to rest at C . The average frictional force opposing the motion in a single trip from A to C is

(a) $18.40 \times 10^{-3} \mathrm{~N}$
(b) $29.4 \times 10^{-3} \mathrm{~N}$
(c) $11.04 \times 10^{-3} \mathrm{~N}$
(d) $7.36 \times 10^{-3} \mathrm{~N}$

Problem 14

Two long straight conductors 1 and 2 , carrying parallel currents $I_1$ and $I_2$ in the same direction, are lying parallel and close to each other, as shown in the figure. $F_e$ and $F_m$ respectively represent the electric and the magnetic forces, applied by conductor 1 on conductor 2. Choose the correct alternative regarding nature of $\mathrm{F_e}$ and $\mathrm{F_m}$

(a) $\mathrm{F_e}$ is repulsive while $\mathrm{F_m}$ is attractive
(b) $\mathrm{F_e}$ is repulsive and $\mathrm{F_m}$ is repulsive too
(c) $F_e$ is zero and $F_m$ is repulsive
(d) $\mathrm{F_e}$ is zero and $\mathrm{F_m}$ is attractive

Problem 15

A doctor measures the temperature of a patient by a digital thermometer as $37.3^{\circ} \mathrm{C}$. As a Physics student you will record his temperature in Kelvin as

(a) 310.30 K
(b) 310.45 K
(c) 310.46 K
(d) 310.31 K

Problem 16

Two planets $P_1$ and $P_2$ are moving around the Sun, in circular orbits of radii $10^{13} \mathrm{~m}$ and $10^{12} \mathrm{~m}$ respectively. The ratio of the orbital speeds of planets $P_1$ and $P_2$ in their respective orbits is

(a) $\sqrt{10}$
(b) 10
(c) $10 \sqrt{10}$
(d) $\frac{1}{\sqrt{10}}$

Problem 17

Crane A and crane B take 1 minute and 2 minute respectively to lift a car of mass 2 ton ( 2000 kg ) upward through a vertical height $\mathrm{h}=3$ meter. If the efficiencies of the engines (defined as the ratio of work output to fuel energy input) of both the cranes are equal, your inference is that

(a) the power supplied by crane B is 1000 kW
(b) the crane A and the crane B consume equal amount of fuel
(c) the power supplied by crane A is more than the power supplied by crane B
(d) the crane A consumes more fuel in lifting the car than the crane B

Problem 18

Two tungsten filament bulbs with rating 100 watt, 200 volt and 60 watt, 200 volt are connected in series with a variable supply of $0-400 \mathrm{~V}$ range, as shown. The supply voltage is gradually increased from 0 to 400 V . Choose the correct statement(s).

(a) When supply voltage is 200 volt, 60 W bulb glows brighter
(b) When supply voltage is 200 volt, total power dissipated in both the bulbs is greater than 37.5 W
(c) When the supply voltage is 400 V , the 100 W bulb gets fused
(d) When supply voltage becomes 400 V , none of the bulbs glow

Problem 19

A solid sphere of radius $R=10 \mathrm{~cm}$ floats in water with $60 \%$ of its volume submerged. In an oil, this sphere floats with $80 \%$ of its volume submerged. If the density of water is $1000 \mathrm{~kg} / \mathrm{m}^3$. The correct statement(s) is/are that

(a) the density of the material of sphere is $600 \mathrm{~kg} / \mathrm{m}^3$
(b) the density of the oil is $750 \mathrm{~kg} / \mathrm{m}^3$
(c) the weight of the sphere in air is close to 24.64 N
(d) the loss in weight of the sphere when floating in oil is close to 30.82 N

Problem 20

A particle starts moving from origin O along (x) axis. The velocity-time graph of motion of particle is given below. The positive values of (v) refer to direction of motion along (+x) axis, the negative values of (v) refer to direction of motion along (-x) direction. Choose the correct statement(s).

(a) Initial acceleration of the particle is $4 \mathrm{~m} / \mathrm{s}^2$
(b) The displacement of particle from origin is 130 m after 16 second
(c) Average speed of the moving particle during (0-16) second is $11.88 \mathrm{~m} / \mathrm{s}$
(d) Somewhere during the motion for (0-16) second, the retardation of the particle is $10 \mathrm{~m} / \mathrm{s}^2$

NSEP 2015 Problem 12 | Periodic Motion due to Electrostatic Force

Try this problem on periodic motion due to electrostatic force from NSEP 2015 Problem 12.

NSEP 2015-16 Problem 12

A particle of mass m and charge $-q$ moves along a diameter of a uniform spherical distribution of radius R with total charge $+Q$. The angular frequency of the periodic motion performed by the particle is

a) $\sqrt{\frac{2\pi qQ}{\epsilon_0 mR^3}}$ b) $\sqrt{\frac{qQ}{2\pi \epsilon_0 mR^3}}$ c) $\sqrt{\frac{qQ}{\epsilon_0 mR^3}}$ d) $\sqrt{\frac{qQ}{4\pi \epsilon_0 mR^3}}$

Key Concepts

Periodic Motion

Gauss's Law of Electrostatic

Suggested Book | Source | Answer

Concept of Physics H.C. Verma

University Physics by H. D. Young and R.A. Freedman

Fundamental of Physics D. Halliday, J. Walker and R. Resnick

National Standard Examination in Physics(NSEP) 2015-2016

Option-(d) $\sqrt{\frac{qQ}{4\pi \epsilon_0 mR^3}}$

Try with Hints

The problem describe a structure something like that,

The whole blue region is charged and the black particle oscillates between A and B. We have to find the angular velocity of this oscillation.

To solve this problem we have to find what will be the Electric force inside the sphere. Using the gauss's law we can solve that.

Using the gauss's law, if we consider a Gaussian surface (sphere) of radius $r$, then charge inside that region is $Q_{in} = \frac{Q}{\frac{4}{3}\pi R^3}\frac{4}{3}\pi r^3 = Q\cdot \frac{r^3}{R^3}$

$$E\cdot 4\pi r^2 = \frac{Q_{in}}{\epsilon_0}$$

This means the particle will get a force of $F = -q E = -\frac{qQ}{4\pi \epsilon}\frac{r}{R^3}$

Arranging this in a equation,

$$ F = m\frac{d^2r}{dt^2} = -\frac{qQr}{4\pi \epsilon R^3}$$

This reduces to ,

$$ \frac{d^2r}{dt^2} + \frac{qQ}{4\pi \epsilon mR^3}r =0$$

We know that $w^2 = \frac{qQ}{4\pi \epsilon mR^3}$. This gives $$ w = \sqrt{\frac{qQ}{4\pi \epsilon_0 mR^3}} $$

NSEP 2015 Problem 9 | Pulley problem

Try this problem on pulley problem on inclined plane from NSEP 2015 Problem 9.

NSEP 2015-16 ~ Problem 9

Maases $m_1$ and $m_2$ are connected to a string passing over a pulley as shown. Mass $m_2$ starts from rest and falls through a distance $d$ in time t. Now, by interchanging the masses the time required for $m_1$ to fall through the same distance is $2t$. Therefore, the ratio of masses $m_2 : m_1$

a) $\frac{2}{3}$ b) $\frac{3}{2}$ c) $\frac{5}{2}$ d) $\frac{4}{3}$

$m_1$ and $m_2$ are interchanged from real problem

Key Concepts

Newton's Laws of Motion

Idea of accelerations, velocity and displacement

Suggested Book | Source | Answer

Concept of Physics H.C. Verma

University Physics by H. D. Young and R.A. Freedman

Fundamental of Physics D. Halliday, J. Walker and R. Resnick

National Standard Examination in Physics(NSEP) 2015-2016

Option-(b) $\frac{3}{2}$

Try with Hints

We know at the beginning the blocks have zero velocities. Using the relation $s= ut+\frac{1}{2}at^2$, we can find the relation between the accelerations for two cases (i.e., when they are interchanged).

Knowing the accelerations we can now use the second law of newton to find the ratio of masses.

From the first hint,

$$ \frac{1}{2}a_1t^2 = \frac{1}{2}a_2 (2t)^2 \to a_1 =4 a_2 $$

Now, we find the value of $a_1$ and $a_2$ using $a = \frac{F}{M}$

$$ \frac{m_2 g - m_1g \sin(30)}{m1+m_2} = 4 \frac{m_1 g - m_2g \sin(30)}{m1+m_2} $$

Rearranging this expression and using $\sin(30) = \frac{1}{2}$,

This gives, $\frac{m_2}{m_1} = \frac{3}{2}$

NSEP 2015 Problem 8 | One Dimensional Motion

Try out this problem on one dimensional motion(Kinematics) from National Standard Examination in Physics 2015-2016.

NSEP 2015-16 ~ Problem 8

A car moving along a straight road at a speed of u m/s applies brakes at $t=0$ seconds. The ratio of distances travelled by the car during $3^{rd}$ and $8^{th}$ seconds is $15:13$. The car covers a distance of $0.25m$ in the last second of it's travel. Therefore, the acceleration a (in $m/s^2$) and the speeed u($m/s$) of the car are respectively,

a) $-0.1,16$

b) $-0.2,12$

c) $-0.5,20$

d) $-0.1,16$

Key Concepts

Basic Equation of motions

Idea of accelerations, velocity and displacement

Suggested Book | Source | Answer

Concept of Physics H.C. Verma

University Physics by H. D. Young and R.A. Freedman

Fundamental of Physics D. Halliday, J. Walker and R. Resnick

National Standard Examination in Physics(NSEP) 2015-2016

Option-(c) $-0.5,20$

Try with Hints

We know that displacement in $t$ sec is given by,

$$ s = ut +\frac{1}{2} a t^2$$

where $u$ is the initial velocity and $a$ is the acceleration. This will be negative for deceleration.

Using this idea we can just find the expression for the displacement in the $n^{th}$ second, for which the information is given to us.

Also, the car come to rest and at the second second it travels a distance of $0.25m$. WE can think it in opposite manner that with an acceleration of $a$ and initial velocity $0m/s$, the car moves a distance of $0.25m$ , hence,

$$0.5 =\frac{1}{2} a (1^2)$$

So, the displacement in $n^{th}$ second is,

$$ s_1 = ut+\frac{1}{2} a n^2 $$

and for the $(n-1)^{th}$ second the displacement is,

$$ s_2= ut+\frac{1}{2} a (n-1)^2 $$

Hence, we have

$$ s_n = s_1-s_2 = u + a(n-\frac{1}{2}) $$

we have,

$$ \frac{15}{13} = \frac{s_3}{s_8} = \frac{u-\frac{5a}{2}}{u - \frac{15a}{2}} $$

Also we have $0.5 =\frac{1}{2} a (1^2)$

Solving this two gives us $a = 0.25m/s^2 $. Putting this in first equation gives us $u = 20 m/s $.

NSEP 2015 Problem 7 | Simple Harmonic Motion

Try out this problem on Simple Harmonic Motion (SHM) from National Standard Examination in Physics 2015-2016.

NSEP 2015-16 ~ Problem 7

A particle execute a periodic motion according to the relation $x^2 = 4 \cos^2(50t) \sin^2(500t)$. Therefore, the motion can be considered to be the superposition of $n$ independent simple harmonic motions, where $n$ is

(a) $2$ (b) $3$ (c) $4$ (d) $5$

Key Concepts

Basic Trigonometry

Equations of SHM

Suggested Book | Source | Answer

Concept of Physics H.C. Verma

University Physics by H. D. Young and R.A. Freedman

Fundamental of Physics D. Halliday, J. Walker and R. Resnick

National Standard Examination in Physics(NSEP) 2015-2016

Option-(b) $3$

Try with Hints

For a simple harmonic motion the equation is $X$ = $A \cos(wt + \phi )$ or $X = A \sin(wt + \phi )$, where X is the position of the particle, A is the amplitude and w is the angular frequency and $\phi $ is the initial phase. We can also consider it to be zero.

If there are $n$ harmonic oscillator then the resultant can be written as ,

$X$=$A_{1} \sin \left(w_{1} t+\phi_{1}\right)$+$A_{2} \sin \left(w_{2} t+\phi_{2}\right)$+$\cdots$+$A_{n} \sin \left(w_{n} t+\phi_{n}\right)$

For this problem, we just have to use suitable trigonometric identities to convert them into sum.

$X$ =$ x^2$ = $4 \cos (50t) \sin (500t) \cos (50t)$, Now, we will use $2\cos(\theta)\sin(\phi)$= $\sin(\theta +\phi)$ +$ \sin(\theta -\phi)$, hence,

$X$=$2(\sin (550 t)$+$\sin (450 t)) \cos (50 t)$=$2 \sin (550 t) \cos (50 t)$+$2 \sin (450 t) \cos (50 t)$

Again applying the same identity,

$X$=$\sin (600 t)$+$2 \sin (500 t)$+$\sin (400 t)$

Hence, there are $3$ SHM.

NSEP 2015 Problem 6 | Surface Tension and Pressure

Try out this problem on the Surface Tension and Pressure from National Standard Examination in Physics 2015-2016.

NSEP 2015-16 ~ Problem 6

Two air bubble with radius $r_1$ and $r_2$ $(r_2>r_1)$ formed of the same liquid stick to each other to form a common interface. Therefore, the radius of curvature of the common surface is

(a) $\sqrt{r_1 r_2}$ (b) Infinity (c) $\frac{r_2}{r_1}\sqrt{{r_2}^2-{r_2}^2}$ (d) $\frac{r_1 r_2}{r_2 - r_1}$

Key Concepts

Basic surface Tension

Relation between surface Tension and Radius of curvature

Suggested Book | Source | Answer

Concept of Physics H.C. Verma

University Physics by H. D. Young and R.A. Freedman

Fundamental of Physics D. Halliday, J. Walker and R. Resnick

National Standard Examination in Physics(NSEP) 2015-2016

Option-(d) $\frac{r_1 r_2}{r_2 - r_1}$

Try with Hints

For a bubble of radius $r$ with double surface and whose inside pressure is $p_{in}$ and outside pressure is $p_{out}$ and also the surface tension is $T$, the relation between pressure and radius is,

$$p_{in} - p_{out} = \Delta p = \frac{4T}{r}$$

When 2 bubble merge on their common interface the pressure difference is Just the difference between the pressure(inside) of both two bubbles. Also, the surface tension remains same.

We know $p_1 - p_0 = \frac{4T}{r_1}$ and $p_2 - p_0 = \frac{4T}{r_2}$, Then,

$$ p_1 - p_2 = \frac{4T}{R} $$

here $R$ is the radius of curvature of the interface.

Hence,

$p_{1}-p_{2}$=$\left(p_{1}-p_{0}\right)-\left(p_{2}-p_{0}\right)$=$4 T\left(\frac{1}{r_{1}}-\frac{1}{r_{2}}\right)$

This gives us,

$$ \frac{1}{R} = \frac{1}{r_1}-\frac{1}{r_2} \to R= \frac{r_1 r_2}{r_2 - r_1} $$

NSEP 2015 Problem 5 | Rotational Mechanics & Small Oscillation

Try out this problem on the one rotational mechanics and small oscillation from National Standard Examination in Physics 2015-2016.

NSEP 2015-16 ~ Problem 5

A thin rod of length l in the shape of a semicircle is pivoted at one of its ends such that it is free to oscillate in its own place. The frequency f of small oscillation of the semicircular rod is

(a) $\frac{1}{2\pi} \sqrt{\frac{g\pi}{2l}}$ (b) $\frac{1}{2\pi} \sqrt{\frac{g\sqrt{\pi^2+4}}{2l}}$ (c) $\frac{1}{2\pi} \sqrt{\frac{g(\pi+4)}{l}}$ (d) $\frac{1}{2\pi} \sqrt{\frac{g(\pi^2+1)}{2\pi l}}$

Key Concepts

Rotational Mechanics and Moment of Inertia

Small Oscillations

Suggested Book | Source | Answer

Concept of Physics H.C. Verma

University Physics by H. D. Young and R.A. Freedman

Fundamental of Physics D. Halliday, J. Walker and R. Resnick

National Standard Examination in Physics(NSEP) 2015-2016

Option-(b) $\frac{1}{2\pi} \sqrt{\frac{g\sqrt{\pi^2+4}}{2l}}$

Try with Hints

Suppose, we have $\Gamma = -k\theta $, where $\theta $ is the angular displacement and $\Gamma $ is the torque.

Then, we can write the angular acceleration $\alpha $ as,

$$ \alpha = \frac{\Gamma}{I}= -\frac{k}{I}\theta = w^2 \theta$$

where I is the moment of inertia, i.e., mass's counterpart in rotational mechanics.

Finally we have, $w = 2\pi f= \sqrt{\frac{\alpha }{\theta}} = \sqrt{\frac{\Gamma }{I\theta}} $, where f is the frequency of the oscillation.

Now for a force $\vec{F}$, $\Gamma = F r \sin(\theta)$. Now using small angle approximation $\sin(\theta) = \theta $. Hence,

$$ f = \frac{1}{2\pi }\sqrt{\frac{Fr}{I}} $$

Now, we can consider the semi-circle of mass m as a point of mass m concentrated into it's centre of mass. This will give us as shown in the figure.

Now, for a semi-circle the centre of mass is at a height of $\frac{2R}{\pi}$ distance above the radius, i.e., $AO = \frac{2R}{\pi}$, where $AP=R=\frac{l}{\pi}$.

Now, $F r = mg l_{c} = mg \sqrt{R^2 + (\frac{2R}{\pi})^2}$

and $I = 2mR^2$

Putting these values gives us,

$f=\frac{1}{2 \pi} \sqrt{\frac{F r}{I}}$

=$\frac{1}{2 \pi} \sqrt{\frac{g \sqrt{\pi^{2}+4}}{2 \pi R}}$

=$\frac{1}{2 \pi} \sqrt{\frac{g \sqrt{\pi^{2}+4}}{2 l}}$

NSEP 2015 Problem 4 | Rotational Motion

Try out this problem on the one dimensional motion from National Standard Examination in Physics 2015-2016.

NSEP 2015-16 ~ Problem 4

A body released from a height H hits elastically an inclined plane at a point P. After the impact the body starts moving horizontally and hits the ground. The height at which point P should be situated so as to have the total time of travel maximum is

(a) H (b) 2H (c) $\frac{H}{4} (d) $\frac{H}{2}$

Key Concepts

Freely falling body

Time needed for a free body

Suggested Book | Source | Answer

Concept of Physics H.C. Verma

University Physics by H. D. Young and R.A. Freedman

Fundamental of Physics D. Halliday, J. Walker and R. Resnick

National Standard Examination in Physics(NSEP) 2015-2016

Option-(d) $\frac{H}{2}$

Try with Hints

From the height H up to the point P, the ball will have a free fall. Hence, the time is fixed.

We just have to think how to get the path of the maximum time in the next part of the motion.

Hint: It can also be considered as free fall by only considering y direction.

B is the point at height H. From P the ball goes horizontally. Then due to gravity it has some projectile motion.

The path is randomly drawn.

From P to ground, the height is h. Then, as the height is h, the time needed to fall is,

$$ t_{PA} = \sqrt{\frac{2x}{g}}$$

From B to P, the time needed is,

$$ t_{BP} = \sqrt{\frac{2(H-x)}{g}}$$

The total time is,

$$t = t_{BP}+t_{PA} = \sqrt{\frac{2(H-x)}{g}} +\sqrt{\frac{2x}{g}}$$

For maximum,

$$ \frac{dt}{dx}=0 $$

Calculating this,

$$ \frac{1}{2}\sqrt{\frac{2}{g}}\frac{-1}{\sqrt{H-x}} + \frac{1}{2}\sqrt{\frac{2}{g}}\frac{1}{\sqrt{x}} $$

solving this,

$$x=\frac{H}{2}$$

NSEP 2015-16 Problem 12

Key Concepts

Suggested Book | Source | Answer

Try with Hints

Other useful links

Related Program

Subscribe to Cheenta at Youtube

NSEP 2015-16 ~ Problem 9

Key Concepts

Suggested Book | Source | Answer

Try with Hints

Other useful links

Related Program

Subscribe to Cheenta at Youtube

NSEP 2015-16 ~ Problem 8

Key Concepts

Suggested Book | Source | Answer

Try with Hints

Other useful links

Related Program

Subscribe to Cheenta at Youtube

NSEP 2015-16 ~ Problem 7

Key Concepts

Suggested Book | Source | Answer

Try with Hints

Other useful links

Related Program

Subscribe to Cheenta at Youtube

NSEP 2015-16 ~ Problem 6

Key Concepts

Suggested Book | Source | Answer

Try with Hints

Other useful links

Related Program

Subscribe to Cheenta at Youtube

NSEP 2015-16 ~ Problem 5

Key Concepts

Suggested Book | Source | Answer

Try with Hints

Other useful links

Related Program

Subscribe to Cheenta at Youtube

NSEP 2015-16 ~ Problem 4

Key Concepts

Suggested Book | Source | Answer

Try with Hints

Other useful links

Related Program

Subscribe to Cheenta at Youtube