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Right choice is (a) Velocity

The explanation is: We know from Stoke’s LAW that damping FORCE is proportional to velocity of a BODY. It also depends on the surface area in contact with the source of particles causing damping, like air.

Right option is (a) Increase

Easiest EXPLANATION: The time PERIOD of a simple pendulum is GIVEN by: T = 2π √(l/g).If the TEMPERATURE increases,length of STRING will increase and thus,the time period will increase.

The CORRECT option is (b) False

Explanation: In a SHM, the speed is maximum at the mean position and goes on DECREASING towards the EXTREME position.So the speed at every position from 0 to A/2 is greater than that from A/2 to A.Thus, the first half is covered FASTER.

Right ANSWER is (a) π/60 s

The best explanation: LET the equation of motion be: x = 0.1sin(wt) where w is the angular frequency.

On derivating this equation w.r.t time we get: v = 0.1wcos(wt).

Given that MAXIMUM speed is 1m/s, we get 0.1w = 1.

∴ w = 10s^-1.

Now our equation of motion has become: x = 0.1sin(10t).

Assume we start from x=0 at t=0.Then by PUTTING x=0.05 in the equation we can find the time.0.05 = 0.1 sin(10t)∴ sin(10t) = 0.5∴ 10t = π/6∴ t = π/60 s.

Correct option is (C) Decreases EXPONENTIALLY

The explanation is: Energy in PRESENCE of damping forces is given by: 1/2kA^2e^-bt/m.This shows that kinetic energy decreases exponentially with TIME.

Right CHOICE is (b) A/√2

To EXPLAIN: Let x = Asin(wt).

∴ Kinetic ENERGY = 1/2 mA^2w^2cos^2(wt)

potential energy = 1/2kA^2sin^2(wt).

According to given condition:

1/2mA^2w^2cos^2(wt) = 1/2A^2w^2sin^2(wt)USING, k = mw^2, we get:tan^2(wt) = 1.

∴ wt = π/4.∴ x = Asin(π/4) = A/√2.

The correct answer is (b) T/2

The explanation is: Amplitude = Ae^-bt/2m& ENERGY = 1/2kA^2e^-bt/m.For amplitude to BECOME HALF, e^-bt/2m = 1/2

Or bT/2m = LN(2)

Or T = 2m ln(2)/bFor energy to become half, e^-bt/m = 1/2

Or t = m ln(2)/b = T/2.

The correct option is (c) T1 = T2 = 2π √(m/k)

Explanation: For the spring MASS system on the horizontal SURFACE, F = -kx.

So, T1 = 2π/w = 2π √(m/k).When the spring mass system is HUNG from the ceiling, at mean position MG = kx0.Let spring be pulled further by x, extension = x+x0,F = k(x+x0)-mg

= kx + kx0– mg = kx.The force is proportional to x.Thus, w = √(k/m) & T2 = 2π √(m/k).Therefore, T1 = T2 = 2π √(m/k).

Right option is (a) Frequency of forced oscillator

For EXPLANATION I WOULD SAY: In the case of forced oscillations, the oscillations MADE by the body do not depend on their NATURAL frequency, instead they oscillate with the frequency of the forced oscillator.

The CORRECT choice is (b) 0.94s

Explanation: W = √2/2 = 1. Time period of SHM = 2π/w = 2π s.Speed of the block will be maximum at the mean position.Time to reach mean position from extreme position = T/4

= 2π/4 = π/2 =1.57s.

Time for the SMALL BALL to fall down = √(2h/g) = √(4/9.8) = 0.63s.∴ Time when it should be released= 1.57 – 0.63 s= 0.94s.

Correct answer is (b) They both will reach the mean position at the same TIME

Easiest explanation: Their time period will be the same as LENGTH is the same. So time to reach the mean position from their RESPECTIVE extremes will be the same. Therefore they will reach the mean position at the same time. They both have different potential energies at the START, so their maximum speeds will be different. Another way to look at this is that the speed of the 2nd pendulum should be more as it has to cover greater arc length in the same AMOUNT oftime.

Correct option is (a) Periodic

The best I can EXPLAIN: The motion is periodic as it repeats its PATH at regular INTERVALS. It is not OSCILLATORY because there is no specific equilibrium position. Rectilinear motion means LINEAR motion.

Correct answer is (a) True

Explanation: Resonance is the phenomenon of oscillation with increased amplitude when frequency of forced oscillator is CLOSE to NATURAL frequency. The REASON for this increase in amplitude is that the vibrations of the external oscillator match with the vibrations of PARTICLES being oscillated and THUS there is a net increase in vibrations.

The correct option is (a) A/√7

Explanation: Let X = Asin(wt).

U = kA^2sin^2(wt)K = 1/2mA^2w^2cos^2(wt).Given: U = 1/3K.kA^2sin^2(wt) = (1/3)*(1/2)mA^2w^2cos^2(wt),

and k = mw^2.

∴ TAN^2(wt) = 1/6. Or tan(wt) = 1/√6.

∴ SIN(wt) = 1/√7.x = Asin(wt) = A/√7.

The correct choice is (b) 0.06

The explanation: When the block is released, the spring FORCE will act towards the mean position and friction force will act against velocity. The work done by these forces should be EQUAL to potential energy of the SYSTEM in the compressed state.∴ Work done in dx DISPLACEMENT = (μmg – kx)dx.

On integrating this, with dx varying from 0 to 0.1, we get

Work = (μmgx – kx√/2)0^0.1= 0.5μ – 0.015.

This should be equal to initial potential energy of the system.

0.5μ – 0.015 = 1/2kA√

∴ 0.5μ– 0.015 = 1/2 * 3 * 0.01

∴ 0.5μ– 0.015 = 0.015

∴ 0.5μ = 0.03 OR μ = 0.06.

Correct option is (a) 1m

For EXPLANATION I would say: Time period of SECONDS PENDULUM = 2s.∴ 2 = 2π √(L/9.8)OR L = 1m.

Correct ANSWER is (a) 50J

To explain I would SAY: 1/2ka^2 = 8 & 1/2kb^2 = 18.

We have to find value of 1/2k(a + b) ^2.

∴ 1/2k(a + b)^2 = 8 + 18 + kab

= 26+(K*4*6/k)

= 26 + 24 = 50J.

Right option is (a) 2S

The explanation is: The KINETIC energy of a particle in SHM is given by:

0.5mA^2w^2cos^2(WT + a), where a is the phase constant.

cos^2(wt + a) = (1 + cos2(wt+a))/2.

Therefore the time period of oscillation of kinetic energy is 4/2 = 2s.

Right CHOICE is (d) MOTION is RECTILINEAR

For explanation: Force always acts in the negative direction, so motion is not PERIODIC. Motion is rectilinear since the particle will always travel in a straight line. Force is zero at x=1.

The correct option is (b) Speed

For explanation I WOULD say: At the extreme position in SHM, the body comes to instantaneous rest. The FORCE, and therefore ACCELERATION, is maximum at this point according to the force eqn: F = -kx.

Right ANSWER is (b) 2HZ

Best explanation: The motion will repeat itself after the ball HITS the ground every time. The time taken for the ball to reach the topmost POSITION is t = u/g.

This will ALSO be the time to reach the ground.

Thus, total time = 2u/g.

This will be the time period.

Frequency = 1/T = g/2u

= 10/5 =2Hz.

Right option is (b) Velocity is towards AMPLITUDE & acceleration is towards MEAN position

Best explanation: The force on a PARTICLE in SHM is always towards the mean position. The particle is currently going towards an extreme, thus velocity will be towards the amplitude.

Correct OPTION is (a) 0.0866m

The explanation: Let the first particle go towards NEGATIVE displacement.Its equation will be:X1 = 0.1sin(w1t + π) & maximum speed = 0.1w1=10

W1 = 100s^-1.

Let the second particle go towards positive displacement.Its equation will be:X2 = 0.1sin(w2t) & maximum speed = 0.1w2 = 5

W2 = 50s^-1.

Now, in their phasor DIAGRAM: w1t + w2t =2π.

∴ t = 2π/150.

At t = π/150

x1 = 0.1sin(50*2π/150) = 0.0866m.

Right ANSWER is (B) It is a SHM with T = π/w

The best EXPLANATION: X = cos^2wt – sin^2wt = cos 2wt.Thus, TIME period T = 2π/2w = π/w.Also the amplitude of motion is 1m.

The correct option is (b) π√2 s

For explanation: Using k = mw^2,

we get: 2 = 1w^2

or w = √2.

Thus, TIME PERIOD = 2π/w = π√2 s.

Correct CHOICE is (a) 4√2m/s

The explanation is: F = -dU/dx = 8 – 2x.

Particle STARTS from x=0, so force will increase its speed till force becomes zero at x=4.

Let acceleration be ‘a’.

a = dv/dt = v dv/dx.

And a = F/m = F/1 = 8-2x.

adx = vdv. Now, integrate both sides.The LIMITS of dx will be 0 to 4 and the limits of dv will be 0 to v.

(8x – x^2)0^4 = v^2/2.

∴ v = 4√2 m/s.

Correct option is (a) True

To elaborate: exp(sin2t) is an exponential function.But the value of exponent sin2t repeats after EVERY interval of π,so the GIVEN function is periodic with a time PERIOD π.

Right choice is (b) False

For explanation I would SAY: Periodic MOTION is motion that repeats itself at regular intervals of time. Oscillatory motion is periodic motion where there is an equilibrium point. When a BODY is given a small displacement from this equilibrium position, a force will act on it towards the equilibrium position. Thus, we can say that every oscillatory motion is periodic but not VICE versa.

The correct choice is (a) True

To EXPLAIN I WOULD say: The TOTAL energy of the particle is the SUM of kinetic & potential energy. Potential energy is frame dependent, so it can be negative in some cases. Thus, the value of kinetic energy can exceed total energy.

Correct choice is (a) Speed is constant, acceleration is zero

To explain: For the particle in uniform CIRCULAR motion, magnitude of velocity is constant, but direction is CONTINUOUSLY changing. THUS, speed is constant but acceleration is non-zero.

Correct OPTION is (a) 0.1sin(20t + 3π/4)

Easy explanation: Let the equation of motion be: x = A sin(WT + a).The amplitude is 0.1m,w = 2/0.1 = 20s^-1.a will be either π/4 or 3π/4 since at t=0, x = A/√2.The ball is travelling in anticlockwise direction so its VELOCITY of projection at t=0 should be towards the origin and therefore negative.v = Aw cos(wt + a).

At t=0, v is negative, so a will be 3π/4.Therefore EQN is: x = 0.1sin(20t + 3π/4)

Right OPTION is (b) It is linearly proportional to POSITION

Easy EXPLANATION: Force on a particle is always directed towards the MEAN position.F = -mAw^2sin(wt), when x = ASIN(wt).So, force is linearly proportional to position.

The CORRECT CHOICE is (B) √3 / √2

Explanation: w = √(k/m), Maximum SPEED = Aw.According to question: A1w1 = A2w2.Their extensions from mean positions will be their amplitudes.

A1/A2 = w2/w1

= √k2/√k1

= √3 / √2.

Correct OPTION is (b) False

The best I can explain: Higher the value of DAMPING LESSER will be the EFFECT of forced oscillations in increasing the amplitude close to natural frequency. Therefore, higher the value of damping constant, lesser will be the amplitude at resonance.

Right CHOICE is (C) m = k/w^2

The best explanation: The force on a PARTICLE in SHM = -mw^2x(t).

GIVEN that: F = -kx(t). We get:

k = mw^2

Or m = k/w^2.

Right choice is (a) 2√2 m

The best EXPLANATION: Using, W = √ (k/m), we GET: w = √ (1/2)s^-1.Maximum VELOCITY = Aw.2 = A/√2∴A = 2√2 m.

Right OPTION is (a) Ae^-at, where a is some constant

To EXPLAIN: The given FORCE represents damping SHM.x(t) = Ae^-atcos(bt+c).The term Ae^-at is the amplitude which DECREASES with increase in time,until eventually the particle COMES to a stop.

Correct CHOICE is (a) True

To elaborate: Let the angular speed of the BODY be w.Let it start from rest from the x-axis.At any time the angle it has COVERED is WT.The projection on x-axis is then given by: x = Asin(wt).THUS, angular speed is the same as angular frequency of SHM.

Correct option is (a) 3

To elaborate: For motion to be OSCILLATORY there has to be an equilibrium position TOWARDS which the force is acting throughout the motion.In the expression, F = -kx^n,If n is even, force will ALWAYS act in one direction.If n is odd, force will change direction when x changes direction and will always act towards the origin.Hence, motion will be oscillatory if n is odd.Thus, from the given options 3 is correct.

Right answer is (a) Number of oscillations per UNIT time

Explanation: Frequency is DEFINED as the number of oscillations per unit time. Its unit is Hz. It is related to the ANGULAR frequency W by the RELATION: f = w/2π.

The CORRECT option is (d) 250m/s^2

Explanation: vmax = AW.

5 = 0.1w.

w = 50s^-1.

amin at extreme = Aw^2

= 0.1*50*50

= 250m/s^2.

The correct option is (a) W > 1

Explanation: Let the velocity EQUATION of SHM be: v = Awcos(wt),acceleration will be given by: a = -Aw^2sin(wt).Maximum magnitude of VEL = AwMaximum magnitude of ACC = Aw^2.For, Aw^2 > Aw

w > 1.