Title 8-advanced-problem-package-.pdf ✅

Simple Harmonic Motion 1. 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) 2π [√ l g + √ m k ] (C) 2π√ l g (B) π [√ l g + √ m k ] (D) 2π√ m k 2. A simple pendulum with length L and mass m of the bob is vibrating with an amplitude ' a '. The tension in the string at the lowest point is (assuming a ≪ L ) (A) mg (B) mg [1 + ( a L ) 2 ] (C) mg [1 + a 2L ] 2 (D) mg [1 + ( a L )] 2 3. Two simple pendulums of length 1 m and 16 m respectively are both given small displacement in the same direction at the same instant. They will again be in the same phase after shorter pendulum has completed η vibrations. The value of η is (A) 5 (B) 4 (C) 1/4 (D) 4/3 4. A mass m is suspended from a spring of force constant k and just touches another identical spring fixed to the floor as shown in the figure. The time period of small oscillations is : (A) 2π√ m k (B) π√ m k + π√ m k/2 (C) π√ m 3k/2 (D) π√ m k + π√ m 2k 5. A particle of mass m is performing SHM along line PQ with amplitude 2a with mean position at O. At t = 0 particle is at point R (OR = a) and is moving towards Q with velocity v = a√3 m/sec. The equation can be expressed by : (A) x = a(√3sin t + cos t) (B) x = 2a(√3sin t + cos t) (C) x = 2a(sin t + √3cos t) (d) x = a(sin t + √3cos t) 6. Vertical displacement of a plank with a body of mass ' m ' on it is varying according to law y = sin ωt + √3cos ωt. The value of ω for which the mass just breaks off the plank and the moment it occurs first after t = 0 are given by (A) √ g 2 , √2 6 π √g
(B) g √2 , 2 3 √ π g (C) √ g 2 , π 3 √ 2 g (d) √2g, √ 2 3 π g 7. If the potential energy of a harmonic oscillator of mass 2 kg in its equilibrium position is 5 joules and the total energy is 9 joules when the amplitude is one meter then the period of the oscillator (in sec) is : (A) 1.5 (B) 3.14 (C) 6.28 (D) 4.67 (⬚ Hint : Total energy = U(0) + 1/2kA 2 ) 8. A particle executes a simple harmonic motion between x = −A and x = +A, the time taken for it to move from x = 0 to x = A 2 is T1 and to move from A 2 to A √2 is T2, then (A) T1 < T2 (B) T1 = T2 (C) T1 = 2 T2 (D) T2 = 2 T1 9. The equation for a particle in SHM with amplitude A and angular frequency ω considering all distances from one extreme position is: (A) x = A + Acos ωt (B) x = A 2 − Acos ωt (C) x = 1 + Acos ωt x = 1 − Acos ωt 10. A simple pendulum has a time period T1, when on earth's surface and T2 when taken to a height R above the earth's surface. R is the radius of the earth. The value of T2 T1 is (A) 1 (B) √2 (C) 4 (D) 2 MULTIPLE CORRECT ANSWERS TYPEEach of the following Question has 4 choices A, B, C & D, out of which ONE or MORE Choices may be Correct: 11. Which of the following quantities are always negative in a simple harmonic motion? (A) F⃗ ⋅ a⃗ (B) v⃗ ⋅ r⃗ (C) a⃗ ⋅ r⃗ (D) F⃗ ⋅ r⃗ 12. Acceleration-time graph of a particle executing SHM is as shown in the figure. Select the correct alternative(s). (A) Displacement of particle at 1 is negative. (B) Velocity of particle at 2 is positive. (C) Potential energy of particle at 3 is maximum (D) Speed of particle at 4 is decreasing 13. Density of liquid varies with depth as ρ = αh. A small ball of density ρ0 is released from the free surface of the liquid. Then (A) the ball will execute SHM of amplitude ρ0/α (B) the mean position of the ball will be at a depth ρ0/2α from the free surface. (C) the ball will sink to a maximum depth of 2ρ0/α (D) all of the above 14. A particle starts SHM at time t = 0. Its amplitude is A and angular frequency is ω. At time t = 0 its kinetic energy is E/4. Assuming potential energy to be zero at mean position, then displacement-time equation of the particle can be written as (A) x = Acos [ωt + (π/6)] (B) x = Asin [ωt + (π/3)] (C) x = Acos [ωt − (2π/3)] (D) x = Acos [ωt − (π/6)] 15. A particle moves along the x-axis according to the equation x = 4 + 3sin (2πt), here x is in cm and t in second. Select the correct alternative(s). (A) The motion of the particle is simple harmonic with mean position at x = 0 (B) The motion of the particle is simple harmonic with mean position at x = 4 cm (C) The motion of the particle is simple harmonic with mean position at x = −4 cm (D) Amplitude of oscillation is 3 cm 16. A block of mass m is attached to a massless spring of force constant k, the other end of which is fixed from the wall of a truck as shown in the figure. The
block is placed over a smooth surface and initially the spring is unstretched. Suddenly the truck starts moving towards right with a constant acceleration a0. As seen from the truck (A) the particle will execute SHM. (B) the time period of oscillations will be 2π√ m k (C) the amplitude of oscillations will be ma0 k (D) the energy of oscillations will be m2a0 2 k 17. A particle is executing SHM on a straight line. A and B are two points at which its velocity is zero. It passes through a certain point P(AP < BP) at successive intervals of 0.5sec and 1.5sec with a speed of 3 m/s. (A) The maximum speed of particle is 3√2 m/s (B) The maximum speed of particle is √2 m/s (C) The ratio AP BP is √2−1 √2+1 (D) The ratio AP BP is 1 √2 18. Two particles undergo SHM along the same line with the same time period (T) and equal amplitudes (A). At a particular instant one particle is at x = −A and the other is at x = 0. They move in the same direction. They will cross each other at x = −A x = 0 x = +A (A) t = 4 T/3 (B) t = 3 T/8 (C) x = A/2 (D) x = A/√2 19. A spring mass system as shown in figure is suspended in a constant electric field E. A charge q is given to the mass. Then the time period of oscillation of the system is (A) 2π√ m k (B) 2π√ m 2k (C) 2π√ mg−qE k (D) 2π√ qE−mg k 20. The potential energy of a particle of mass 0.1 kg moving along the x-axis is given U = 5x(x − 4)J, where x is in meters. 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 π/5 second 21. For a simple harmonic motion with given angular frequency ω, two arbitrary initial conditions are necessary and sufficient to determine the motion complete. These initial conditions may be : (A) Initial position and initial velocity (B) Amplitude and initial phase (C) Total energy of oscillation and amplitude (D) Total energy of oscillation and initial phase 22. An object of mass m is performing simple harmonic motion on a smooth horizontal surface as shown in the figure. Just as the oscillating object reaches its extreme position, another object of mass 2m is dropped on to oscillating object, which sticks to it. For this situation mark out the correct statement(s). (A) Amplitude of oscillation remains unchanged (B) Time period of oscillation remains unchanged (C) The total mechanical energy of the system does not change (D) The maximum speed of the oscillating object changes 23. The given figure (a) shows a spring of force constant k fixed at one end and carrying a mass m at the other end placed on a horizontal frictionless surface. The spring is stretched by a force F. Figure
(b) shows the same spring with both ends free and a mass m fixed at each free end. Each of the spring is stretched by the same force F. The mass in case (a) and the masses in case (b) are then released. Which of the following statements are true? (A) While oscillating, the maximum extension of the spring is more in case (a) than in case (b). (B) The maximum extension of the spring is same is both cases (C) The time period of oscillation is the same is both cases (D) The time period of oscillation in case (a) is √2 time that in case (b) 24. Two blocks connected by a spring rest on a smooth horizontal plane as shown in the given figure. A constant force F starts acting on block m2 as shown in the figure. Which of the following statements are not correct? (A) Length of the spring increases continuously if m1 > m2. (B) Blocks start performing SHM about centre of mass of the system, which moves rectilinearly with constant acceleration. (C) Blocks start performing oscillations about centre of mass of the system with increasing amplitude. (D) Acceleration of m2 is maximum at initial moment of time only. 25. A block of mass m is suspended by a rubber cord of natural length l = mg/k, where k is force constant of the cord. The block is lifted upwards so that the cord becomes just tight and then block is released suddenly. Which of the following will not be true? (A) Block performs periodic motion with amplitude greater than l. (B) Block performs SHM with amplitude equal to l. (C) Block will never return to the position from where it was released. (D) Angular frequency ω is equal to 1rad/s. 26. When the point of suspension of pendulum is moved, its period of oscillation (A) Decreases when it moves vertically upwards with an acceleration a (B) Decreases when it moves vertically downwards with acceleration greater than 2 g (C) Increases when it moves horizontally with acceleration a (D) All of the above 27. A linear harmonic oscillator of force constant 2 × 106 N/m and amplitude 0.01 m has a total mechanical energy of 160 J. Its: (A) Maximum potential energy is 100 J (B) Maximum kinetic energy is 100 J (C) Maximum potential energy is 160 J (D) Minimum potential energy zero 28. A spring of spring constant K is fixed to the ceiling of a lift. The other end of the spring is attached to a block of mass m. The mass is in equilibrium. Now the lift accelerates downwards with an acceleration. Now the lift accelerates downwards with an acceleration 2g : (A) The block will not perform SHM and it will stick to the ceiling. (B) The block will perform SHM with time period 2π√m/K. (C) The amplitude of the block will be 2mg/K if it perform SHM. (D) The min. potential energy of the spring during the motion of the block will be 0 . 29. Simple pendulum is kept suspended vertically in a stationary bus. The bus starts moving with an acceleration a towards left. As observed inside the bus : (Neglect frictional forces on pendulum and assume size of the ball to the very small.) : (A) Time period of oscillation of the pendulum will be 2π√ l √a2+g2 for any value of a (B) Time period of oscillation of the pendulum will