Q: Two identical point masses P and Q , suspended from two separate massless springs of spring constants k _1 and k _2, respectively, oscillate vertically. If their maximum speeds are the same, the ratio

aligned & Given m _ P = m _ Q \\ & Also ( V _ )_ P = ( V _ )_ Q \\ & A _ P _ P = A _ Q _ Q \\ & A _ P k _1 ~m _ P = A _ Q k _2 ~m _ Q [ = k m ] \\ & A _ Q A _ P = k _1 k _2 aligned

Q: In an oscillating spring mass system, a spring is connected to a box filled with sand. As the box oscillates, sand leaks slowly out of the box vertically so that the average frequency (t) and average

aligned & = k m \\ & m aligned E = 1 2 kA ^2 The part of sand being removed takes away a part of energy. Total energy decreases. Energy of SHM decreases As a result Amplitude also decreases.

Q: If the mass of the bob in a simple pendulum is increased to thrice its original mass and its length is made half its original length, then the new time period of oscillation is x 2 times its original

aligned & T^ =2 ^ g where ^ = 2 \\ & T=2 g \\ & T^ = x 2 T \\ & 2 2 g = x 2 2 g \\ & 1 2 = x 2 x= 2 aligned

Q: If x=5 ( t+ 3 ) m represents the motion of a particle executing simple harmonic motion, the amplitude and time period of motion, respectively, are

aligned & x=5 ( t+ 3 ) m \\ & Amplitude =5 ~m \\ & = = 2 T \\ & T= 2 =2 ~s aligned

Question 1

Savitha, a XI standard student, while conducting an experiment to determine the effective length of a simple pendulum L, notes down the data of time taken to complete 30 oscillations as 60 s and hence

Accepted Answer

Time period T = 60 30 = 2 s. The formula for the time period of a simple pendulum is T = 2 L g . Squaring both sides, we get T^2 = 4 ^2 L g . Substituting the given values T = 2 s, ^2 = 9.8, and g = 9.8 m/s^2: (2)^2 = 4 9.8 L 9.8 4 = 4L L = 1 m. Answer: 1 m

Question 2

A cylindrical cork of uniform density floats in a liquid of density _1. If the cork is depressed slightly and released, it oscillates harmonically with time period T. If the same cork floats in anothe

Accepted Answer

Let m be the mass of the cork and A be its cross-sectional area. When the cork is depressed by a small distance x in a liquid of density _L, the extra buoyant force provides the restoring force: F = -A _L g x The acceleration of the cork is: a = - A _L g m x Comparing this with the standard equation of SHM a = - ^2 x, we get: = A _L g m The time period of oscillation is: T = 2 = 2 m A _L g This shows that T 1 _L . For the first liquid of density _1: T 1 _1 For the second liquid of density _2: 2T

Question 3

For a simple pendulum, having time period T, the variation of kinetic energy (K.E.) with time (t) is represented by :

Accepted Answer

The displacement of a simple pendulum executing simple harmonic motion is given by x = A ( t + ). The velocity of the pendulum is v = dx dt = A ( t + ). The kinetic energy is given by K = 1 2 m v^2 = 1 2 m A^2 ^2 ^2( t + ). Since kinetic energy is proportional to the square of velocity, it can never be negative. This eliminates the graphs that show negative values. Also, the kinetic energy of a pendulum is not constant, which eliminates the constant graph. Using the trigonometric identity ^2 = 1

Question 4

The sum of kinetic energy and potential energy of a simple pendulum bob is 0.02 joule. The speed of the simple pendulum bob at equilibrium position is approximately : (Consider mass of the bob = 20 g)

Accepted Answer

Total energy of the simple pendulum is given as E = 0.02 J. Mass of the bob, m = 20 g = 0.02 kg. At the equilibrium position, the potential energy is zero and the total energy is purely kinetic. E = 1 2 m v^2 0.02 = 1 2 0.02 v^2 v^2 = 2 v = 2 1.41 m/s. Answer: 1.41 m/s

Question 5

Two identical point masses P and Q , suspended from two separate massless springs of spring constants k _1 and k _2, respectively, oscillate vertically. If their maximum speeds are the same, the ratio

Accepted Answer

aligned & Given m _ P = m _ Q \ & Also ( V _ )_ P = ( V _ )_ Q \ & A _ P _ P = A _ Q _ Q \ & A _ P k _1 ~m _ P = A _ Q k _2 ~m _ Q [ = k m ] \ & A _ Q A _ P = k _1 k _2 aligned

Question 6

In an oscillating spring mass system, a spring is connected to a box filled with sand. As the box oscillates, sand leaks slowly out of the box vertically so that the average frequency (t) and average

Accepted Answer

aligned & = k m \ & m aligned E = 1 2 kA ^2 The part of sand being removed takes away a part of energy. Total energy decreases. Energy of SHM decreases As a result Amplitude also decreases.

Question 7

The two-dimensional motion of a particle, described by r =( i +2 j ) A t is a/an: A. parabolic path B. elliptical path C. periodic motion D. simple harmonic motion Choose the correct answer from the o

Accepted Answer

r =( i +2 j ) A t x=A t y=2 A t y=2 x The path is straight line. The motion is SHM and periodic as d r d t =-( i +2 j ) A t d^2 r d t^2 =-( i +2 j ) ^2 A t a =- ^2 r

Question 8

Let _1, _2 and _3 be the angular speed of the second hand, minute hand and hour hand of a smoothly running analog clock, respectively. If x_1, x_2 and x_3 are their respective angular distances in 1 m

Accepted Answer

_1= 2 60 ; x_1= 2 60 60=2 _2= 2 3600 ; x_2= 2 3600 60= 2 60 _3= 2 3600 12 ; x_3= 2 3600 12 60= 2 720 _1 x_1 = _2 x_2 = _3 x_3 = 1 60 =k

Question 9

If the mass of the bob in a simple pendulum is increased to thrice its original mass and its length is made half its original length, then the new time period of oscillation is x 2 times its original

Accepted Answer

aligned & T^ =2 ^ g where ^ = 2 \ & T=2 g \ & T^ = x 2 T \ & 2 2 g = x 2 2 g \ & 1 2 = x 2 x= 2 aligned

Question 10

If x=5 ( t+ 3 ) m represents the motion of a particle executing simple harmonic motion, the amplitude and time period of motion, respectively, are

Accepted Answer

aligned & x=5 ( t+ 3 ) m \ & Amplitude =5 ~m \ & = = 2 T \ & T= 2 =2 ~s aligned

Question 11

A particle executing simple harmonic motion with amplitude A has the same potential and kinetic energies at the displacement

Accepted Answer

Potential energy = 1 2 k x^2 Kinetic energy = 1 2 k A^2- 1 2 k x^2 According to given condition 1 2 k x^2= 1 2 k A^2- 1 2 k x^2 k 2 x^2=k A^2 x^2= A^2 2 x= A 2

Oscillations — Previous Year Questions & Solutions