NCERT Exemplar

A ray of light incident at an angle θ on a refracting face of a prism emerges from the other face normally. If the angle of the prism is 5o and the prism is made of a material of refractive index 1.5, the angle of incidence is
a) 7.5o
b) 5o
c) 15o
d) 2.5o

Answer: a) 7.5o The distance between the refracting surfaces is negligible with thin prisms, thus the prism angle (A) is very small. Because A = r1 + r2, if A is tiny, both r1 and r2 will be little...

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A man runs across the roof-top of a tall building and jumps horizontally with the hope of landing on the roof of the next building which is of a lower height than the first. If his speed is 9 m/s, the distance between the two buildings is 10 m and the height difference is 9 m, will he be able to land on the next building?

For a free fall at 9m, the horizontal distance covered by the man should be at least 10 m. u = 0 a = 10 m/s2 s = 9 m t = t s = ut + 1/2 at2 Substituting the values, we get t = √9/3 = 3/√5 sec The...

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It is a common observation that rain clouds can be at about a kilometre altitude above the ground. a) If a rain drop falls from such a height freely under gravity, what will be its speed? Also, calculate in km/h b) A typical rain drop is about 4 mm diameter. Momentum is mass x speed in magnitude. Estimate its momentum when it hits ground. c) Estimate the time required to flatten the drop. d) Rate of change of momentum is force. Estimate how much force such a drop would exert on you. e) Estimate the order of magnitude force on umbrella. Typical lateral separation between two rain drops is 5 cm.

a) Velocity attained by the rain drop which is falling freely through the height h is: v2 = u2 – 2g(-h) As u = 0 v = √2gh = 100√2 m/s = 510 km/h b) Diameter of the drop, d = 2r = 4 mm Radius of the...

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A motor car moving at a speed of 72 km/h cannot come to a stop in less than 3 s while for a truck this time interval is 5 s. On a highway the car is behind the truck both moving at 72 km/h. The truck gives a signal that it is going to stop at emergency. At what distance the car should be from the truck so that it does not bump onto the truck. Human response time is 0.5 s.

For truck, u = 20 m/s v = 0 a = ? t = 5s v = u + at a = 4 m/s2 For car, t = 3 s u = 20 m/s v = 0 a = ac v = u + at ac = -20/3 m/s2 Let s be the distance between the car and the truck when the truck...

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A man is standing on top of a building 100 m high. He throws two balls vertically, one at t = 0 and other after a time interval. The later ball is thrown at a velocity of half the first. The vertical gap between first and second ball is +15m at t = 2s. The gap is found to remain constant. Calculate the velocity with which the balls were thrown and the exact time interval between their throw.

Let the speed of ball 1 = u1 = 2u m/s Then the speed of ball 2 = u2 = u m/s The height covered by ball 1 before coming to rest = h1 The height covered by ball 2 before coming to rest = h2 We know...

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A bird is tossing between two cars moving towards each other on a straight road. One car has a speed of 18 m/h while the other has the speed of 27 km/h. The bird starts moving from first car towards the other and is moving with the speed of 36 km/h and when the two cars were separated by 36 km. What is the total distance covered by the bird? What is the total displacement of the bird?

The relative speed of the cars = 27 + 18 = 45 km/h When the two cars meet together, time t is given as t = distance between cars/relative speed of cars = 36/(27+18) t = 4/5 h Therefore, distance...

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A particle executes the motion described by x(t) = x0 (1 – e-γt) where t ≥ 0, x0 > 0 a) Where does the particles start and with what velocity? b) Find maximum and minimum values of x(t), v(t), a(t). Show that x(t) and a(t) increase with time and v(t) decreases with time.

a) x(t) = x0 (1 – e-γt) v(t) = dx(t)/dt = +x0 γ e-γt a(t) = dv/dt = x0 γ2 e-γt v(0) = x0 γ b) x(t) is minimum at t = 0 since t = 0 and [x(t)]min = 0 x(t) is maximum at t = ∞ since t = ∞ and...

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An object falling through a fluid is observed to have acceleration given by a = g – bv where g = gravitational acceleration and b is constant. After a long time of release, it is observed to fall with constant speed. What must be the value of constant speed?

The concept used in this question will be based on the behaviour of a spherical object when it is dropped through a viscous fluid. When a spherical body of radius r is dropped, it is first...

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Give examples of a one-dimensional motion where a) the particle moving along positive x-direction comes to rest periodically and moves forward b) the particle moving along positive x-direction comes to rest periodically and moves backwardπ

When an equation has sine and cosine functions, the nature is periodic. a) When the particle is moving in positive x-direction, it is given as t > sin t When the displacement is as a function of...

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A ball is bouncing elastically with a speed 1 m/s between walls of a railway compartment of size 10 m in a direction perpendicular to walls. The train is moving at a constant velocity of 10 m/s parallel to the direction of motion of the ball. As seen from the ground, a) the direction of motion of the ball changes every 10 seconds b) speed of ball changes every 10 seconds c) average speed of ball over any 20 seconds intervals is fixed d) the acceleration of ball is the same as from the train

The correct option is b) speed of ball changes every 10 seconds, c) average speed of ball over any 20 seconds intervals is fixed, and d) the acceleration of the ball is the same as from the train

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A spring with one end attached to a mass and the other to a rigid support is stretched and released. a) magnitude of acceleration, when just released is maximum b) magnitude of acceleration, when at equilibrium position is maximum c) speed is maximum when mass is at equilibrium position d) magnitude of displacement is always maximum whenever speed is minimum

The correct answer is a) magnitude of acceleration, when just released is maximum and c) speed is maximum when mass is at equilibrium position

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A graph of x versus t is shown in the figure. Choose correct alternatives from below. a) the particle was released from rest at t=0 b) at B, the acceleration a>0 c) at C, the velocity and the acceleration vanish d) average velocity for the motion A and D is positive e) the speed at D exceeds that at E

The correct answer is a) the particle was released from rest at t=0, c) at C, the velocity and the acceleration vanish and e) the speed at D exceeds that at E

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The variation of quantity A with quantity B, plotted in figure describes the motion of a particle in a straight line. a) quantity B may represent time b) quantity A is velocity if motion is uniform c) quantity A is displacement if motion is uniform d) quantity A is velocity if motion is uniformly accelerated

The correct answer is a) quantity B may represent time, c) quantity A is displacement if motion is uniform, and d) quantity A is velocity if motion is uniformly accelerated

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A lift is coming from 8th floor and is just about to reach 4th floor. Taking ground floor as origin and positive direction upwards for all quantities, which one of the following is correct? a) x<0, v<0, a>0 b) x>0, v<0, a<0 c) x>0, v<0, a>0 d) x>0, v>0, a<0

The correct answer is a) x<0, v<0, a<0 The value of x and v becomes negative as the lift is moving from the 8th floor to the 4th floor whereas acceleration is acting upwards and stays...

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What are the coordinates of the vertices of a cube whose edge is 2 units, one of whose vertices coincides with the origin and the three edges passing through the origin, coincides with the positive direction of the axes through the origin?

Solution: It is given that a cube with 2 units edge, one of whose vertices coincides with the origin and the 3 edges passing through the origin, coincides with the positive direction of the axes...

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In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.
(a) 7x + 5y + 6z + 30 = 0 and 3x – y – 10z + 4 = 0
(b) 2x + y + 3z – 2 = 0 and x – 2y + 5 = 0

Solution: (a) $7 x+5 y+6 z+30=0$ and $3 x-y-10 z+4=0$ It is given that The eq. of the given planes are $7 x+5 y+6 z+30=0$ and $3 x-y-10 z+4=0$ Two planes are $\perp$ if the direction ratio of the...

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Lifetimes of the molecules in the excited states are often measured by using pulsed radiation source of duration nearly in the nanosecond range. If the radiation source has the duration of 2 \mathrm{~ns} and the number of photons emitted during the pulse source is 2.5 \times 10^{15} J, calculate the energy of the source.

Frequency of radiation $(\nu)$, $\nu=\frac{1}{2.0 \times 10^{-9} s}$ $\nu=5.0 \times 10^{8} s^{-1}$ Energy $(E)$ of source $=$ Nhv Where, $N$ is the no. photons emitted $\mathrm{h}$ is Planck's...

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What is the energy in joules, required to shift the electron of the hydrogen atom from the first Bohr orbit to the fifth Bohr orbit and what is the wavelength of the light emitted when the electron returns to the ground state? The ground state electron energy is -2.18\times10^{-11} ergs. The ground-state electron energy is -2.18\times10^{-11}ergs.

$ E_{5}=\frac{-\left(2.18 \times 10^{-18}\right) Z^{2}}{(n)^{2}} $ Where, $Z$ denotes the atom's atomic number Ground state energy $=-2.18 \times 10^{-11}$ ergs $=-2.18 \times 10^{-11} \times...

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A helicopter of mass 2000 kg rises with a vertical acceleration of 15 m/s2. The total mass of the crew and passengers is 500 kg. Give the magnitude and direction of the a) force on the floor of the helicopter 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

Given, M = 2000 kg helicopter mass m = 500 kg m = 500 kg m = 500 kg m = 500 kg m = 500 kg m = 500 kg Helicopter acceleration with crew and passengers = 15 m/s2 a) Force exerted by the crew and...

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A racing car travels on a track ABCDEFA. ABC is a circular arc of radius 2 R. CD and FA are straight paths of length R and DEF is a circular arc of radius R = 100 m. The coefficient of friction on the road is μ = 0.1. The maximum speed of the car is 50 m/s. Find the minimum time for completing one round.

Time taken from $A$ to $B$ to $C$ $\mathrm{S} 1=$ length $\mathrm{pf}$ path $=3 / 42 \pi(2 \mathrm{R})=300 \pi \mathrm{m}$ $\mathrm{V} 1=$ speed(maximum) along the circular path of the car $...

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When a body slides down from rest along a smooth inclined plane making an angle of 45 degree with the horizontal, it takes time T. When the same body slides down from rest along a rough inclined plane making the same angle and through the same distance, it is seen to take time pT, where p is some number greater than 1. Calculate the coefficient of friction between the body and the rough plane.

The inclined plane angle is $45^{\circ}$ such that $ \begin{array}{l} u=0 \\ s=s \\ t=T \\ a=g \sin 45^{\circ}=g / \sqrt{2} \\ s=u t+1 / 2 a t^{2} \\ s=g T^{2} / 2 \sqrt{2} \end{array} $ $...

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A bock of mass M is held against a rough vertical wall by pressing it with a finger. If the coefficient of friction between the block and the wall is μ and the acceleration due to gravity is g, calculate the minimum force required to be applied by the ginger to hold the block against the wall?

F is the force exerted by the finger on a body of mass M that is resting on the wall. Using the balanced state as a starting point, $ \begin{array}{l} \mathrm{F}=\mathrm{N} \\ \mathrm{f}=\mathrm{Mg}...

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Consider the non-empty set consisting of children in a family and a relation R defined as aRb if a is brother of b. Then R is (A) symmetric but not transitive (B) transitive but not symmetric (C) neither symmetric nor transitive (D) both symmetric and transitive

The correct option is (B) transitive but not symmetric Given aRb ⇒ a is brother of b. This does not mean b is also a brother of a as b can be a sister of a. Therefore, R is not symmetric. aRb ⇒ a is...

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The earth has a radius of 6400 km. The inner core of the 1000 km radius is solid. Outside it, there is a region from 1000 km to a radius of 3500 km which is in a molten state. Then again from 3500 km to 6400 km the earth is solid. Only longitudinal (P) waves can travel inside a liquid. Assume that the P wave has a speed of 8 km/s in solid parts and of 5 km/s in liquid parts of the earth. An earthquake occurs at someplace close to the surface of the earth. Calculate the time after which it will be recorded in a seismometer at a diametrically opposite point on the earth if wave travels along diameter?

Answer: According to the question, r1 = 1000 km, r2 = 3500 km, r3 = 6400 km and d1 = 1000 km And we can calculate, d2 = 3500 – 1000 d2 = 2500 km d3 = 6400 – 3500 d3 = 2900 km Expression for the...

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A tuning fork vibrating with a frequency of 512 Hz is kept close to the open end of a tube filled with water. The water level in the tube is gradually lowered. When the water level is 17 cm below the open end, the maximum intensity of sound is heard. If the room temperature is 20oC, calculate

c) if the water in the tube is replaced with mercury, will there be any difference in your observations? Answer: (c) Sound is reflected into the air column by water and mercury in the tube, forming...

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A tuning fork vibrating with a frequency of 512 Hz is kept close to the open end of a tube filled with water. The water level in the tube is gradually lowered. When the water level is 17 cm below the open end, the maximum intensity of sound is heard. If the room temperature is 20oC, calculate

a) speed of sound in air at room temperature b) speed of sound in air at 0oC Answer: According to the question, the frequency of the tuning fork is f = 512 Hz a)  When the first maxima are taken...

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A train standing at the outer signal of a railway station blows a whistle of frequency 400 Hz still air. The train beings to move with a speed of 10 m/s towards the platform. What is the frequency of the sound for an observer standing on the platform?

Answer: It is given that v0 = 400 Hz and vz = 10 m/s We know that the velocity of sound in air is va = 330 m/s The frequency heard by the observer on the platform is v'. Therefore, we can write: v’...

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