LEVEL – I

Q:1. A particle moves in x-y plane according to the equation $ \displaystyle \vec{r} = (\hat{i}+2\hat{j})A cos\omega t$ the motion of the particle is

(A) on a straight line

(B) on an ellipse

(C) periodic

(D) simple harmonic

Q:2. Which of the following quantities are always positive in a simple harmonic motion ?

(A) $ \displaystyle \vec{F}.\vec{a} $

(B) $ \displaystyle \vec{v}.\vec{r} $

(C) $ \displaystyle \vec{a}.\vec{r} $

(D) $ \displaystyle \vec{F}.\vec{r} $

Q:3. The magnitude of average acceleration in half time period in a simple harmonic motion is

(A) 2 ω2A /π

(B) ω2A /2 π

(C) ω2A /√2 π

(D) Zero

Q:4. A small block oscillates back and forth on a smooth concave surface of radius R. The time period of small oscillation is

(A) $ \displaystyle T = 2\pi \sqrt{\frac{R}{g}} $

(B) $ \displaystyle T = 2\pi \sqrt{\frac{2 R}{g}} $

(C) $ \displaystyle T = 2\pi \sqrt{\frac{R}{2 g}} $

(D) None of these

Q:5. A particle of mass 10 gm lies in a potential field v = 50 x2 + 100. The value of frequency of oscillations in Hz is

(A) 5 Hz

(B) 5/π Hz

(C) 10π/3 Hz

(D) none of these.

Q:6. When two mutually perpendicular simple harmonic motions of same frequency , amplitude and phase are superimposed

(A) the resulting motion is uniform circular motion.

(B) the resulting motion is a linear simple harmonic motion along a straight line inclined equally to the straight lines of motion of component ones.

(C) the resulting motion is an elliptical motion, symmetrical about the lines of motion of the components.
(D) the two S.H.M. will cancel each other.

Q:7. The angular frequency of small oscillations of the system shown in the figure is

(A) $ \displaystyle \sqrt{\frac{K}{2 m}}$

(B) $ \displaystyle \sqrt{\frac{2 K}{ m}}$

(C) $ \displaystyle \sqrt{\frac{K}{4 m}}$

(D) $ \displaystyle \sqrt{\frac{4 K}{ m}}$

Q:8. A particle executes SHM with a frequency f. The frequency with which it’s KE oscillates is

(A) f/2

(B) f

(C) 2f

(D) 4f

Q:9. A simple pendulum has some time period T. What will be the percentage change in its time period if its amplitudes is decreased by 5 % ?

(A) 6 %

(B) 3 %

(C) 1.5 %

(D) 0 %

Q:10. The work done by the string of a simple pendulum during one complete oscillation is equal to
(A) total energy of the pendulum

(B) KE of the pendulum

(C) PE of the pendulum

(D) Zero

Answers:

1. (A)  2. (A)  3. (A)  4. (A)  5. (B)  6. (B)  7. (C)  8. (C)  9. (D)  10. (D)

LEVEL – I

Q:11. Two uniform rods are welded together to form a letter T as shown in the figure.

Each rod is of mass M and length ‘ l ‘ . If this combination is hinged at ‘ A ‘ and kept in vertical plane then time period of small oscillations about A is equal to

(A) $ \displaystyle 2 \pi \sqrt{\frac{l}{6 \sqrt{3} g}}$

(B) $ \displaystyle 2 \pi \sqrt{\frac{3 l}{2 \sqrt{2} g}}$

(C) $ \displaystyle 2 \pi \sqrt{\frac{l}{2 g}}$

(D) $ \displaystyle 2 \pi \sqrt{\frac{11 l}{6 \sqrt{5} g}}$

Q:12. A cylindrical piston of mass M slides smoothly inside a long cylinder closed at one end, enclosing a certain mass of a gas.

The cylinder is kept with its axis horizontal. If the piston is slightly compressed isothermally from its equilibrium position, it oscillates simple harmonically, the period of oscillation will be

(A) $ \displaystyle 2 \pi \sqrt{\frac{M h}{P A}}$

(B) $\displaystyle 2 \pi \sqrt{\frac{M A}{P h}}$

(C) $ \displaystyle 2 \pi \sqrt{\frac{M }{P A h}}$

(D) $ \displaystyle 2 \pi \sqrt{M P h A}$

Q:13. The period of the free oscillations of the system shown here if mass M1 is pulled down a little and force constant of the spring is k and masses of the fixed pulleys are negligible, is

(A) $ \displaystyle 2 \pi \sqrt{\frac{M_1 + M_2}{k}}$

(B) $ \displaystyle 2 \pi \sqrt{\frac{M_1 + 4 M_2}{k}}$

(C) $ \displaystyle 2 \pi \sqrt{\frac{M_2 + 4 M_1}{k}}$

(D)$ \displaystyle 2 \pi \sqrt{\frac{M_2 + 3 M_1}{k}}$

Q:14. The period of small oscillations of a simple pendulum of length l if its point of suspension O moves with a constant acceleration α = α1 i^ + α2 j^ with respect to earth is

(A) $ \displaystyle 2 \pi \sqrt{\frac{l}{((g-\alpha_2)^2 + \alpha_1^2)^{1/2}}}$

(B) $ \displaystyle 2 \pi \sqrt{\frac{l}{((g +\alpha_1)^2 + \alpha_2^2)^{1/2}}}$

(C) $ \displaystyle 2 \pi \sqrt{\frac{l}{g}}$

(D) $ \displaystyle 2 \pi \sqrt{\frac{l}{(g^2 +\alpha_1^2)^{1/2}}}$

Q:15. A particle moves along the X-axis according to the equation x = 10 sin3 ( πt ). The amplitudes and frequencies of component SHMs are

(A) amplitude 30/4 , 10/4 ; frequencies 3/2 , 1/2

(B) amplitude 30/4 , 10/4 ; frequencies 1/2 , 3/2

(C) amplitude 10 , 10 ; frequencies 1/2 , 1/2

(D) amplitude 30/4 , 10 ; frequencies 3/2 , 2

Q:16. A pendulum makes perfectly elastic collision with block of m lying on a frictionless surface attached to a spring of force constant k. Pendulum is slightly displaced and released. Time period of oscillation of the system is

(A) $ \displaystyle 2 \pi [ \sqrt{\frac{l}{g}} + \sqrt{\frac{m}{k}} ] $

(B) $ \displaystyle \pi [ \sqrt{\frac{l}{g}} + \sqrt{\frac{m}{k}} ] $

(C) $ \displaystyle 2 \pi \sqrt{\frac{l}{g}} $

(D) $ \displaystyle 2 \pi \sqrt{\frac{m}{k}} $

Q:17. Three springs of each force constant k are connected as shown figure.
Point mass m is slightly displaced to compress A and released. Time period of oscillation

(A) $ \displaystyle 2 \pi \sqrt{\frac{m}{2 k}} $

(B) $ \displaystyle 2 \pi \sqrt{\frac{m}{3 k}} $

(C) $ \displaystyle 2 \pi \sqrt{\frac{m}{k}} $

(D) $ \displaystyle 2 \pi \sqrt{\frac{m}{k + \sqrt{2}(k+1)}} $

Q:18. Two blocks each of mass m are connected with springs of force constant k. Initially springs are relaxed. Mass A is displaced to left and B is displaced towards right by same amount and released then time period of oscillation of any one block (Assume collision to be perfectly elastic)

(A) $ \displaystyle 2 \pi \sqrt{\frac{m}{k}} $

(B) $ \displaystyle 2 \pi \sqrt{\frac{m}{2 k}} $

(C) $ \displaystyle \pi \sqrt{\frac{m}{k}} $

(B) $ \displaystyle \pi \sqrt{\frac{m}{2 k}} $

Q:19. If for a particle moving in SHM, there is a sudden increase of 1% in restoring force just as particle passing through mean position, percentage change in amplitude will be

(A) 1%

(B) 2%

(C) 0.5%

(D) zero.

Q:20. S1 and S2 are two identical springs. The oscillation frequency is f. If one spring is removed, frequency will be

(A) f

(B) 2f

(C) √2 f

(D) f /√2

Answers:

11. (D)  12. (A)  13. (C)  14. (A)  15. (B)  16. (B)  17. (A)  18. (C)  19. (C)  20. (C)

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