Simple Harmonic Motion PYQs Set 12 (PDF)

Set 12 of 13 · Questions 331-360 of 372

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Mere Teacher · Previous Year Questions JEE Main Physics · Simple Harmonic Motion · Set 12 (Q331-360)
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331. $Two particles are oscillating along two close parallel straight lines side by side, with the same frequency and amplitudes. They pass each other, moving in opposite directions when their displacement is half of the amplitude. The mean positions of the two particles lie on a straight line perpendicular to the paths of the two particles. The phase difference is$ (1) $\frac{π}{6}$ (2) $0$ (3) $\frac{2 π}{3}$ (4) $π$ 332. $Two blocks with masses 100 g and 200 g are attached to the ends of springs A and B as shown in figure. The energy stored in A is E . The energy stored in B, when spring constants k_{A}, k_{B} of A and B, respectively satisfy the relation 4 k_{A} = 3 k_{B^{’}}, is :$ (1) $\frac{4}{3} E$ (2) $4 E$ (3) $3E$ (4) $2 E$ 333. $Y = Asin (ωt + φ_{0}) is the time - displacement equation of a SHM, At t = 0 the displacement of the particleis Y = \frac{A}{2} and it is moving along negative x - direction. Then the initial phase angle φ_{0} will be$ (1) $\frac{π}{6}$ (2) $\frac{π}{3}$ (3) $\frac{2 π}{3}$ (4) $\frac{5 π}{6}$ 334. $A particle executes simple harmonic oscillation with an amplitude a . The period of oscillation is T . The minimum time taken by the particle to travel half of the amplitude from the equilibrium position is$ (1) $T /8$ (2) $T/12$ (3) $T /2$ (4) $T /4.$ 335. $When a particle executes SHM, the nature of graphical representation of velocity as a function of displacement is:$ (1) $elliptical$ (2) $parabolic$ (3) $straight line$ (4) $circular$ 336. $In simple harmonic motion, the total mechanical energy of given system is E . If mass of oscillating particle P is doubled then the new energy of the system for same amplitude is:$ (1) $E$ (2) $E / 2$ (3) $2 E$ (4) $E 2$ 337. $A rod of mass ’ M ’ and length ’ 2 L ’ is suspended at its middle by a wire. It exhibits torsional oscillations; If two masses each of ' m' are attached at distance ’ L /2 ’ from its centre on both sides, it reduces the oscillation frequency by 20 % . The value of ratio m / M is close to$ (1) $0.77$ (2) $0.57$ (3) $0.37$ (4) $0.17$ 338. $A block of mass m is connected to a light spring of force constant k . The system is placed inside a damping medium of damping constant b . The instantaneous values of displacement, acceleration and energy of the block are x, a and E respectively. The initial amplitude of oscillation is A and ω^{'} is the angular frequency of oscillations. The incorrect expression related to the damped oscillations is$ (1) $x = A e^{\frac{- b}{m}} cos (ω^{'} t + ϕ)$ (2) $ω^{'} = \frac{k}{m} - \frac{b ^{2}}{4 m ^{2}}$ (3) $E = \frac{1}{2} k A^{2} e^{- \frac{b t}{m}}$ (4) $m \frac{d ^{2} x}{d t ^{2}} + b \frac{d x}{d t} + k x = 0$ 339. $An electromagnetic wave of intensity enters in a medium of refractive index '' without any loss. The ratio of the magnitudes of electric fields, and the ratio of the magnitudes of magnetic fields of the wave before and after entering into the medium are respectively, given by:$ (1) $(\frac{1}{n}, \frac{1}{n})$ (2) $\frac{F}{3 mR}$ (3) $(n, \frac{1}{n})$ (4) $(\frac{1}{n}, n)$ 340. $The time period of a simple harmonic oscillator is T = 2 π \frac{m}{k} . Measured value of mass (m) of the object is 10 g with an accuracy of 10 mg and time for 50 oscillations of the spring is found to be 60 s using a watch of 2 s resolution. Percentage error in determination of spring constant (k) is____1% .$ (1) $7.60$ (2) $3.35$ (3) $3.43$ (4) $6.76$ 341. $If a simple harmonic oscillator has got a displacement of 0.02 m and acceleration equal to 0.02 m / s^{2} at any time, the angular frequency of the oscillator is equal to Oscillations 123$ (1) $10 rad / s$ (2) $0.1 rad / s$ (3) $100 rad / s$ (4) $1 rad / s$ 342. $A particle of mass m executes simple harmonic motion with amplitude a and frequency v. The average kinetic energy during its motion from the position of equilibrium to the end is:$ (1) $π^{2} m a^{2} v^{2}$ (2) $\frac{1}{4} m a^{2} v^{2}$ (3) $4 π^{2} m a^{2} v^{2}$ (4) $2 π^{2} m a^{2} v^{2}$ 343. $Amplitude of a wave is represented by A = \frac{c}{a + b - c} Then resonance will occur when$ (1) $b = - c /2$ (2) $b = 0 and a = c$ (3) $b = - a /2$ (4) $None of these$ 344. $A mass is suspended from a vertical spring which is executing. S.H.M. of frequency 5 Hz . The spring is unstretched at the highest point of oscillation. Maximum speed of the mass is [acceleration due to gravity g = 10 m / s^{2}]$ (1) $2 πm / s$ (2) $πm / s$ (3) $\frac{1}{2 π} m / s$ (4) $\frac{1}{π} m/s$ 345. $A body of mass 5 kg hangs from a spring and oscillates with a time period of 2 π seconds. If the ball is removed, the length of the spring will decrease by$ (1) $\frac{g}{k} metres$ (2) $\frac{k}{g} metres$ (3) $2 π metres$ (4) $g metres.$ 346. $Consider two SHMs along the same straight line x_{1} = A_{1} sin (ωt + φ_{1}), x_{2} = A_{2} sin (ωt + φ_{2}), where A_{1} and A_{2} are their amplitudes and φ_{1} and φ_{2} are their initial phase angle. If the two SHMs meet simultaneously and ' R' is the resultant amplitude, match column I with column II. Column-I (A). The two SHMs are in phase, A1=A2=A (B). The two SHMs are in phase, A1 \neq = A2 (C). The two SHMs are 90 out of phase, A1=A2=A (D). The two SHMs are 180^{\circ} out of phase, A1=A2 Column-II (I). R = A_{1} + A_{2} (II). R = 0 (III). R = 2 A (IV). R = 2 A$ (1) $A-III, B-I, C-IV, D-II$ (2) $A-IV, B-III, C-II, D-I$ (3) $A-I, B-III, C-II, D-IV$ (4) $A-III, B-IV, C-I, D-II$ 347. $Resonance is an example of$ (1) $Tuning fork$ (2) $Forced vibration$ (3) $Free vibration$ (4) $Damped vibration$ 348. $A simple pendulum of length 2 m is given a horizontal push through angular displacement of 60^{\circ} . If the mass of bob is 200 gram, the angular velocity of the bob will be (Take acceleration due to gravity = \frac{10 m}{s ^{2}}) (si n 30^{\circ} = co s 60^{\circ} = 0.5, co s 30^{\circ} = si n 60^{\circ} = \frac{3}{2})$ (1) $22 rad / s$ (2) $32 rad / s$ (3) $2 2.5 rad / s$ (4) $3 2.5 rad / s$ 349. $When two displacements represented by y_{1} = a sin (ω t) and y_{2} = b cos (ω t) are superimposed the motion is$ (1) $simple harmonic with amplitude a^{2} + b^{2}$ (2) $simple harmonic with amplitude \frac{( a + b )}{2}$ (3) $not a simple harmonic$ (4) $simple harmonic with amplitude \frac{a}{b}$ 350. $The displacement of a particle varies according to the relation x = 4(cosπt + sinπt). The amplitude of the particle is$ (1) $8$ (2) $- 4$ (3) $4$ (4) $42$ 351. $Which of the following statements regarding damping force of a damped oscillator is NOT correct?$ (1) $Damping force depends on the nature of the surrounding medium.$ (2) $Damping force is generally proportional to the velocity of the body making oscillations.$ (3) $Damping force acts in the direction of the velocity of the body.$ (4) $Ratio of the damping force and velocity of the body depends on the size and shape of the body.$ 352. $Two masses m and \frac{m}{2} are connected at the two ends of a massless rigid rod of length l . The rod is suspended by a thin wire of torsional constant k at the centre of mass of the rod-mass system (see figure). Because of torsional constant k, the restoring toruque is τ = kθ for angular displacement θ . If the rod is rotated by θ_{0} and released, the tension in it when it passes through its mean position will be$ (1) $\frac{3 k θ _{0}^{2}}{l}$ (2) $\frac{2 k θ _{0}^{2}}{l}$ (3) $\frac{k θ _{0}^{2}}{l}$ (4) $\frac{k θ _{0}^{2}}{2 l}$ 353. $Two blocks of masses m and M, (M > m), are placed on a frictionless table as shown in figure. A massless spring with spring constant k is attached with the lower block. If the system is slightly displaced and released, then (μ = coefficient of friction between the two blocks)(A) The time period of small oscillation of the two blocks is T = 2 π \frac{( m + M )}{k} (B) The acceleration of the blocks is a = - \frac{kx}{M + m} (x = displacement of the blocks from the mean position) (C) The magnitude of the frictional force on the upper block is \frac{mμ \| x \|}{M + m} (D) The maximum amplitude of the upper block, if it does not slip, is \frac{μ ( M + m ) g}{k} (E) Maximum frictional force can be μ(M+m)g. Choose the correct answer from the options given below:$ (1) $B, C, D Only$ (2) $C, D, E Only$ (3) $A, B, D Only$ (4) $A, B, C Only$ 354. $A small block is connected to one end of a massless spring of un-stretched length 4.9 m. The other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is stretched by 0.2 m and released from rest at t = 0. It then executes simple harmonic motion with angular frequency ω = \frac{π}{3} rad / s . Simultaneously at t = 0, a small pebble is projected with speed v from point P at an angle of 45º as shown in the figure. Point P is at a horizontal distance of 10 cm from O. If the pebble hits the block at t = 1s, the value of v is (take g = 10 m/ s^{2})$ (1) $50 m / s$ (2) $51 m / s$ (3) $52 m / s$ (4) $53 m / s$ 355. $In case of a forced vibration, the resonance peak becomes very sharp when the$ (1) $damping force is small$ (2) $restoring force is small$ (3) $applied periodic force is small$ (4) $quality factor is small$ 356. $An equilateral triangle ABC is cut from a thin solid sheet of wood. (See figure) D, E and F are the mid-points of its sides as shown and G is the centre of the triangle. The moment of inertia of the triangle about an axis passing through G and perpendicular to the plane of the triangle is I_{0} . If the smaller triangle DEF is removed from ABC, the moment of inertia of the remaining figure about the same axis is I . Then:$ (1) $I = \frac{15}{16} I_{0}$ (2) $I = \frac{3}{4} I_{0}$ (3) $I = \frac{9}{16} I_{0}$ (4) $I = \frac{I _{0}}{4}$ 357. $The total energy of particle performing SHM depend on$ (1) $k, a, m$ (2) $k, a$ (3) $k, a, x$ (4) $k, x .$ 358. $The S.H.M. of a particle is given by the equation y = 3 sin ωt + 4 cos ωt. The amplitude is:$ (1) $7$ (2) $1$ (3) $5$ (4) $12$ 359. $A pendulum bob has a speed of 3m/s at its lowest position. The pendulum is 50cm long. The speed of bob, when the length makes an angle 6 0^{0} of to the vertical will be (g=10m/ s^{2}) m/s$ 360. $A spring-block system is resting on a frictionless floor as shown in the figure. The spring constant is 2.0 N m^{- 1} and the mass of the block is 2.0 kg. Ignore the mass of the spring. Initially the spring is in an unstretched condition. Another block of mass 1.0 kg moving with a speed of 2.0 m s^{-1} collides elastically with the first block. The collision is such that the 2.0 kg block does not hit the wall. The distance, in meters, between the two blocks when the spring returns to its unstretched position for the first time after the collision is ________$