Simple Harmonic Motion PYQs Set 9 (PDF)

Set 9 of 13 · Questions 241-270 of 372

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Mere Teacher · Previous Year Questions JEE Main Physics · Simple Harmonic Motion · Set 9 (Q241-270)
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241. $If the time period of a two meter long simple pendulum is 2 s, the acceleration due to gravity at the placewhere pendulum is executing S.H.M. is:$ (1) $2 π^{2} ms^{- 2}$ (2) $16 m / s^{2}$ (3) $9.8 ms^{- 2}$ (4) $π^{2} ms^{- 2}$ 242. $A 20 g particle is subjected to two simple harmonic motions x_{1} = 2 sin 10 t x_{2} = 4 sin (10 t + \frac{π}{3}) . where x_{1} and x_{2} are in metre and t is in sec. Choose the incorrect option.$ (1) $The displacement of the particle at t = 0 will be 23 m$ (2) $Maximum speed of the particle will be 207 m / s$ (3) $Magnitude of maximum acceleration of the particle will be 2007 m / s^{2}$ (4) $Energy of the resultant motion will be 26 J$ 243. $When an external force with angular frequency ω_{d} acts on a system of natural angular frequency ω, the system oscillates with angular frequency ω_{d} . The condition for the amplitude of oscillations to be maximum is$ (1) $ω_{d} = 2 ω$ (2) $ω_{d} = ω$ (3) $ω_{d} = \frac{ω}{2}$ (4) $ω_{d} = 3 ω$ 244. $A particle starts executing simple harmonic motion (SHM) of amplitude 'a' and total energy E. At any instant, its kinetic energy is \frac{3 E}{4} then its displacement 'y' is given by:$ (1) $y = a$ (2) $y = \frac{a}{2}$ (3) $y = \frac{a 3}{2}$ (4) $y = \frac{a}{2}$ 245. $A particle executes SHM of amplitude A. The distance from the mean position when its kinetic energy becomes equal to its potential energy is:$ (1) $\frac{1}{2} A$ (2) $2A$ (3) $2 A$ (4) $\frac{1}{2} A$ 246. $Two SHM’s with same amplitude and time period, when acting together in perpendicular directions with a phase difference of π /2, give rise to$ (1) $straight motion$ (2) $elliptical motion$ (3) $circular motion$ (4) $none of these.$ 247. $A heavy brass sphere is hung from a spring and it executes vertical vibrations with period T . The sphere is now immersed in a non-viscous liquid with a density (1/10) th that of brass. When set into vertical vibrations with the sphere remaining inside liquid all the time, the time period will be$ (1) $(\frac{9}{10}) T$ (2) $(\frac{10}{9}) T$ (3) $(\frac{9}{10}) T$ (4) $Unchanged$ 248. $The amplitude of a simple pendulum is 10 cm . When the pendulum is at a displacement of 4 cm from the mean position, the ratio of kinetic and potential energies at that point is$ (1) $5.25$ (2) $2.5$ (3) $4.5$ (4) $7.5$ 249. $A loaded vertical spring executes S.H.M. with a time period of 4 sec. The difference between the kinetic energy and potential energy of this system varies with a period of$ (1) $2 sec$ (2) $1 sec$ (3) $8 sec$ (4) $4sec.$ 250. $Time period of a simple pendulum of length ' L' is T_{1} . Time period of a uniform rod of same length ' L' suspended from one end and oscillating in a vertical plane is T_{2} . Amplitude of oscillation is small in both the cases. Then \frac{T _{1}}{T _{2}} is$ (1) $\frac{2}{3}$ (2) $\frac{3}{2}$ (3) $\frac{4}{3}$ (4) $1$ 251. $The time period of a simple pendulum on the surface of the earth is T. The height above the surface of the earth at which the time period of the pendulum becomes 2 T is (Radius of the earth = 6400 km)$ (1) $3200 km$ (2) $6400 km$ (3) $1600 km$ (4) $800 km$ 252. $Which graph represents the difference between total energy and potential energy of a particle executing SHM vs its distance from mean position?$ (1) (2) (3) (4) 253. $Which of the following quantity does not change due to damping of oscillations?$ (1) $Angular frequency$ (2) $Time period$ (3) $Initial phase$ (4) $Amplitude$ 254. $If the length of a simple pendulum is increased by 2%, then the time period$ (1) $increases by 1%$ (2) $decreases by 1%$ (3) $increases by 2%$ (4) $decreases by 2%.$ 255. $A particle performs simple harmonic motion with a time period of 16 s . At a time t = 2 s, the particle passes through the origin and at t = 4 s its velocity is 4 m / s . The amplitude of the motion is$ (1) $32 π 2.$ (2) $\frac{32 2}{π}$ (3) $32 π$ (4) $32$ 256. $A block of mass 100 g moving at a speed of 2 m / s compresses a spring through a distance 2 cm before its speed is halved. Find the spring constant of the spring.$ (1) $12500 N / m$ (2) $750 N / m$ (3) $1000 N / m$ (4) $1500 N / m$ 257. $A tray of mass 12 kg is supported by two identical springs as shown in figure. When the tray is pressed down slightly and then released, it executes SHM with a time period of 1, 5 s . The spring constant of each spring is$ (1) $105 N m^{- 1}$ (2) $\infty$ (3) $50 N m^{- 1}$ (4) $0$ 258. $Two masses m and \frac{m}{2} are connected at the two ends of a massless rigid rod of length λ. The rod is suspended by a thin wire of torsional constant k at the center of mass of the rod-mass system (see figure). Because of torsional constant k, the restoring torque is τ = kθ. for angular displacement θ. If the rod is rotated by θ_{0.} and released, the tension in it when it passes through its mean position will be:$ (1) $\frac{3 k θ _{0}^{2}}{l}$ (2) $\frac{k θ _{0}^{2}}{2 l}$ (3) $\frac{2 k θ _{0}^{2}}{l}$ (4) $\frac{k θ _{0}^{2}}{l}$ 259. $A hydrometer executes simple harmonic motion when it is pushed down vertically in a liquid of density ρ . If the mass of hydrometer is m and the radius of the hydrometer tube is r, then the time period of oscillation is$ (1) $T = 2 π \frac{m}{π r ^{2} ρg}$ (2) $T = 2 π \frac{π r ^{2} ρg}{m}$ (3) $T = \frac{1}{2 π} \frac{m}{π r ^{2} ρg}$ (4) $T = \frac{1}{2 π} \frac{π r ^{2} ρg}{m}$ 260. $Two springs, of force constants k_{1} and k_{2}, are connected to a mass m as shown. The frequency of oscillation of mass is ƒ. If both k_{1} and k_{2} are made four times their original values, the frequency of oscillation becomes:$ (1) $ƒ/2$ (2) $ƒ/4$ (3) $4f$ (4) $2ƒ$ 261. $The displacement time graph of a particle executing SHM is given in figure (sketch is schematic and not to scale) Which of the following statements is/are true for this motion? (A) The force is zero at t = \frac{3 T}{4} (B) The acceleration is maximum at t=T (C) The speed is maximum at t = \frac{T}{4} (D) The P.E. is equal to K.E. of the oscillation at t = \frac{T}{2}$ (1) $(B), (C) and (D)$ (2) $(A) and (D)$ (3) $(A), (B) and (C)$ (4) $(A), (B) and (D)$ 262. $For particle P revolving round the centre O with radius of circular path r and regular velocity a, as shown in below figure, the projection of OP on the x-axis at time t is$ (1) $x (t) = r cos (ωt - \frac{π}{6})$ (2) $x (t) = r cos (ωt + \frac{π}{6})$ (3) $x (t) = r sin (ωt + \frac{π}{6})$ (4) $x (t) = r cos (ωt)$ 263. $Two independent harmonic oscillators of equal mass are oscillating about the origin with angular frequencies ꞷ1 and ꞷ2 and have total energies E_{1} and E_{2}, respectively. The variations of their momenta p with positions x are shown in figures. If \frac{a}{b} = n^{2} and \frac{a}{R} = n, then the correct equation(s) is (are):$ (1) $E_{1} ꞷ_{1} = E_{2} ꞷ_{2}$ (2) $\frac{ω _{2}}{ω _{1}} = n^{2}$ (3) $ꞷ_{1} ꞷ_{2} = n^{2}$ (4) $\frac{E _{1}}{ω _{1}} = \frac{E _{2}}{ω _{2}}$ 264. $A simple pendulum of length l has maximum angular displacement θ . The maximum kinetic energy of the bob of mass m is (g = acceleration due to gravity)$ (1) $mgl (1 + cos θ)$ (2) $mgl (1 + cos^{2} θ)$ (3) $mgl (1 - cos θ)$ (4) $mgl (cos θ - 1)$ 265. $As shown in the figure, a spring is kept in a stretched position with some extension by holding the masses 1 kg and 0.2 kg with a separation more than spring natural length and are released. Assuming the horizontal surface to be frictionless, the angular frequency (in SI unit) of the system is : k = 150 N / m$ (1) $5$ (2) $27$ (3) $20$ (4) $30$ 266. $Two identical tennis balls each having mass 'm' and charge 'q' are suspended from a fixed point by threads of length ‘r. What is the equilibrium separation when each thread makes a small angle ' θ' with the vertical?$ (1) $x = (\frac{q ^{2} l}{2 π ϵ _{0} mg})^{\frac{1}{2}}$ (2) $x = (\frac{q ^{2} l}{2 π ϵ _{0} mg})^{\frac{1}{3}}$ (3) $x = (\frac{q ^{2} l ^{2}}{2 π ϵ _{0} m ^{2} g})^{\frac{1}{3}}$ (4) $x = (\frac{q ^{2} l ^{2}}{2 π ϵ _{0} m ^{2} g ^{2}})^{\frac{1}{3}}$ 267. $A solid ball is suspended from the ceiling of a motor car through a light string. A transverse pulse travels at the speed 60 cm^{- 1} on the string, when the car is at rest. When the car accelerates on a horizontal road, then speed of the pulse is 66 cm^{- 1} . The acceleration of the car is nearly (g = 10 ms^{- 2})$ (1) $4.3 ms^{- 2}$ (2) $2.9 ms^{- 2}$ (3) $6.8 ms^{- 2}$ (4) $5.5 ms^{- 2}$ 268. $In figure (a), mass '2m' is fixed on mass 'm' which is attached to two springs of spring constant k. In figure (b), mass ‘m' is attached to two spring of spring constant 'k' and '2k'. If mass 'm' in (a) and (b) are displaced by distance V horizontally and then released, then time period T_{1} and T_{2} corresponding to (a) and (b) respectively follow the relation.$ (1) $\frac{T _{1}}{T _{2}} = \frac{3}{2}$ (2) $\frac{T _{1}}{T _{2}} = \frac{3}{2}$ (3) $\frac{T _{1}}{T _{2}} = \frac{2}{3}$ (4) $\frac{T _{1}}{T _{2}} = \frac{2}{3}$ 269. $Two simple harmonic motions are represented by the equations x_{1} = 5 sin (2 πt + \frac{π}{4}) and x_{2} = 52 (sin 2 πt + cos 2 πt) The amplitude of second motion is times the amplitude in first motion.$ 270. $A particle of mass 1kg is hanging from a spring of force constant 100 N m^{- 1} . The mass is pulled slightly downward and released so that it executes free simple harmonic motion with time period T. The time when the kinetic energy and potential energy of the system will become equal, is \frac{T}{x} . The value of x is ...$