simple harmonic motion

Q.1 The angular frequency of motion whose equation is  dty d 4 2 + 9y = 0 is (y = displacement and t = time) (A) 49 (B) 94 (C) 23 (D) 32 Q.2 A man is swinging on a swing made of 2 ropes of equal length L and in direction perpendicular to the plane of paper. The time period of the small oscillations about the mean position is (A) 2 g 2L (B) 2 g 2 L 3 (C) 2 g 3 2 L (D)  gL Q.3 Vertical displacement of a plank with a body of mass 'm' on it is varying according to law y = sin t + 3 cos t. The minimum value of  for which the mass just breaks off the plank and the moment it occurs first after t = 0 are given by: ( y is positive vertically upwards) (A) g 62 , 2g  (B) g 32 , 2 g  (C) g2 3 , 2g  (D) g 32 , g 2  Q.4 A particle of massm moves in a one-dimensional potential energy U(x) = –ax2 + bx4, where 'a' and 'b' are positive constants. The angular frequency of small oscillations about the minima of the potential energy is equal to (A) b 2a  (B) ma 2 (C) ma 2 (D) m 2a Q.5 A system of two identical rods (L-shaped) of massmand length l are resting on a peg P as shown in the figure. If the system is displaced in its plane by a small angle , find the period of oscillations: (A) 2 2 3 l g (B) 2 2 2 3 l g (C) 2 23l g (D) 3 l 3g Q.6 A particle of massm moves in the potential energy U shown above. The period of the motion when the particle has total energy E is (A) 2 mg /E 2 4 k /m 2   (B) k /m 2 (C) 2 mg /E 2 2 k /m   (D) 2 mg /E 2 2Q.7 A 2 Kg block moving with 10 m/s strikes a spring of constant 2 N/m attached to 2 Kg block at rest kept on a smooth floor. The time for which rear moving block remain in contact with spring will be (A) 2 sec (B) 2 1 sec (C) 1 sec (D) 21 sec Q.8 A particle is subjected to two mutually perpendicular simple harmonic motions such that its x and y coordinates are given by x = 2 sin t ; y = 2 sin        4 t The path of the particle will be : (A) an ellipse (B) a straight line (C) a parabola (D) a circle Q.9 Two simple harmonic motions y1 = A sin t and y2 = A cos t are superimposed on a particle of mass m. The total mechanical energy of the particle is: (A) 21 m2A2 (B) m2A2 (C) 41 m2A2 (D) zero Q.10 The potential energy of a particle of mass 0.1kg, moving along x-axis, is given by U = 5x(x-4)J where x is in metres. It can be concluded that (A) the particle is acted upon by a constant force. (B) the speed of the particle is maximum at x = 2 m (C) the particle executes simple harmonic motion (D) the period of oscillation of the particle is /5 s. Q.11 A mass of 0.2kg is attached to the lower end of a massless spring of force-constant 200 N/m, the upper end of which is fixed to a rigid support. Which of the following statements is/are true? (A) In equilibrium, the spring will be stretched by 1cm. (B) If the mass is raised till the spring is unstretched state and then released, it will go down by 2cm before moving upwards. (C) The frequency of oscillation will be nearly 5 Hz. (D) If the system is taken to the moon, the frequency of oscillation will be the same as on the earth. Q.12 A cylindrical block of density is partially immersed in a liquid of density 3. The plane surface of the block remains parallel to the surface of the liquid. The height of the block is 60 cm. The block performs SHM when displaced from its mean position. [Use g = 9.8 m/s2] (A) the maximum amplitude is 20 cm. (B) the maximum amplitude is 40 cm (C) the time period will be 2/7 seconds. (D) none Q.13 A system is oscillating with undamped simple harmonic motion. Then the (A) average total energy per cycle of the motion is its maximum kinetic energy.(B) average total energy per cycle of the motion is 2 1 times its maximum kinetic energy. (C) root mean square velocity is 2 1 times its maximum velocity (D) mean velocity is 1/2 of maximum velocity. Q.14 A particle of mass m performs SHM along a straight line with frequency f and amplitude A. (A) The average kinetic energy of the particle is zero. (B) The average potential energy is m 2f2A2. (C) The frequency of ocillation of kinetic energy is 2f. (D) Velocity function leads acceleration by /2. Q.15 The graph plotted between phase angle () and displacement of a particle from equilibrium position (y) is a sinusoidal curve as shown below. Then the best matching is Column A Column B (a) K.E. versus phase angle curve (i) (b) P.E. versus phase angle curve (ii) (c) T.E. versus phase angle curve (iii) (d) Velocity versus phase angle curve (iv) (A) (a)-(i), (b)-(ii), (c)-(iii) & (d)-(iv) (B) (a)-(ii), (b)-(i), (c)-(iii) & (d)-(iv) (C) (a)-(ii), (b)-(i), (c)-(iv) & (d)-(iii) (D) (a)-(ii), (b)-(iii), (c)-(iv) & (d)-(i) Q.16 Two springs with negligible masses and force constant of K1 = 200 Nm–1 and K2 = 160 Nm–1 are attached to the block of mass m = 10 kg as shown in the figure. Initially the block is at rest, at the equilibrium position in which both springs are neither stretched nor compressed. At time t = 0, a sharp impulse of 50 Ns is given to the block with a hammer. (A) Period of oscillations for the mass m is 3 s. (B) Maximum velocity of the mass m during its oscillation is 5 ms–1. (C) Data are insufficient to determine maximum velocity. (D) Amplitude of oscillation is 0.42 m. Q.17 A block of mass 100 gm attached to a spring of stiffness 100 N/m is lying on a frictionless floor as shown. The block is moved to compress the spring by 10 cm and released. If the collision with the wall is elastic the time period of motion is(A) 0.2 sec (B) 0.1 sec (C) 0.155 sec (D) 0.133 sec Q.18 A wire frame in the shape of an equilateral triangle is hinged at one vertex so that it can swing freely in a vertical plane, with the plane of the  always remaining vertical. The side of the frame is 3 /1 m. The time period in seconds of small oscillations of the frame will be (A) 2  (B) 2  (C) 6  (D) 5  Q.19 A small bob attached to a light inextensible thread of length l has a periodic time T when allowed to vibrate as a simple pendulum. The thread is now suspended from a fixed end O of a vertical rigid rod of length 4 3l (as in figure). If now the pendulum performs periodic oscillations in this arrangement, the periodic time will be (A) 4T 3 (B) 2T (C) T (D) 2T Q.20 A particle starts from a point P at a distance of A/2 from the mean position O & travels towards left as shown in the figure. If the time period of SHM, executed about O is T and amplitude A then the equation of motion of particle is : (A) x = A sin        6 t T 2 (B) x = A sin        6 5 t T 2 (C) x = A cos        6 t T 2 (D) x = A cos        3 t T 2Answer Key Q.1 C Q.2 B Q.3 A Q.4 B Q.5 B Q.6 C Q.7 C Q.8 A Q.9 B Q.10B,C,D Q.11 A,B,C,D Q.12 A,C Q.13 A,C Q.14 B,C Q.15 B Q.16 A,B Q.17 D Q.18 D Q.19 D Q.20 A