# Simple Harmonic Motion-02-Objective Unsol. Page( 26-38)

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### Simple Harmonic Motion-02-Objective Unsol. Page( 26-38)

• 1. U UN NS SO OL LV VE ED D O OB BJ JE EC CT TI IV VE E L LE EV VE EL L – – I I 1. In a simple harmonic motion (a) the potential energy is always equal to the kinetic energy (b) the potential energy is never equal to the kinetic energy (c) the average potential energy in any time interval is equal to the average kinetic energy in that time interval (d) the average potential energy in one time period is equal to the average kinetic energy in this period. 2. For a simple pendulum the graph between L and T will be : (a) hyperbola (b) parabola (c) a curved line (d) a straight line. 3. The time period of a particle in simple harmonic motion is equal to the time between consecutive appearances of the particle at a particular point in its motion. This point is (a) the mean position (b) an extreme position (c) between the mean position and the positive extreme (d) between the mean position and the negative extreme. 4. A pendulum clock keeping correct time is taken to high altitudes, (a) it will keep correct time (b) its length should be increased to keep correct time (c) its length should be decreased to keep correct time (d) it cannot keep correct time even if the length is changed. 5. The time period of a simple pendulum in a freely falling lift is : (a) zero (b) infinite (c) finite (d) none of these. 6. A body performs S.H.M. Its kinetic energy, K , varies with time t , as indicated in the graph : (a) K t (b) K t (c) (d) . 7. Which of the following graphs describes the variation of acceleration a of a particle executing SHM with its displacement x. (a) a x (b) a x
• 2. (c) a x (d) a x . 8. A particle moves on the X -axis according to the equation 2 0 sin   x x t . The motion is simple harmonic (a) with amplitude 0 x (b) with amplitude 0 2x (c) with time period 2  (d) with time period   . 9. The motion of a particle is given by sin cos     x A t B t . The motion of the particle is (a) not simple harmonic (b) simple harmonic with amplitude  A B (c) simple harmonic with amplitude ( )/ 2  A B (d) simple harmonic with amplitude 2 2  A B . 10. Two SHMs are respectively represented by sin( )    y a t kx and cos( )    y b t kx . The phase difference between the two is : (a) / 2  (b) / 4  (c) / 6  (d) 3 / 4  . 11. The displacement of a particle executing SHM is given by 0.01 sin 100 ( 0.05)    x t . The time period is : (a) 0.01 sec (b) 0.02 sec (c) 0.1 sec (d) 0.2 sec. 12. A particle executes simple harmonic motion under the restoring force provided by a spring. The time period is T. If the spring is divided into two equal parts and one part is used to continue the simple harmonic motion, the time period will (a) remain T (b) become 2T (c) become / 2 T (d) become / 2 T . 13. Which of the following expressions does not represent SHM : (a) cos A t (b) sin 2 A t (c) sin cos    A t B t (d) sin . t Ae  14. The graph plotted between the velocity and displacement from mean position of a particle executing S.H.M. is (a) circle (b) ellipse (c) parabola (d) straight line. 15. The total energy of the body executing S.H.M. is E . Then the kinetic energy, when the displacement is half of the amplitude, is : (a) / 2 E (b) / 4 E (c) 3 / 4 E (d) 3/ 4.E .
• 3. U UN NS SO OL LV VE ED D O OB BJ JE EC CT TI IV VE E L LE EV VE EL L – – I II I 1. Two simple harmonic motions are represented by : 1 10sin(4 / 4)     y t 2 5(sin 4 3cos 4 )     y t t the ratio of the amplitudes of the two SHM’s is : (a) 1 : 1 (b) 1 : 2 (c) 2 : 1 (d) 1 : 3 . 2. A particle is subjected to two simple harmonic motions given by : 1 10sin ,   y t and 2 5sin( )     y t The maximum speed of the particle is : (a) 2 2 10 5   (b) 2 2 10 5   (c) 5 (d) 15. 3. A particle executes simple harmonic motion with amplitude A and time period T. The average speed of the particle over n complete oscillations is (a) 0 (b) A nT (c) 4A nT (d) 4A T 4. A particle of mass m and charge  q moves along a diameter of uniformly charge sphere of radius R and carrying a total charge Q . The frequency of the SHM of the particle, if the amplitude does not exceed R , is : O -q Q (a) 2 0 1 2 4   qQ R m (b) 3 0 1 2 4   qQ mR (c) 0 1 2 4   qQ mR (d) 2 0 1 2 4   qQm R . 5. A simple pendulum with a solid metal has a period T . The metal bob is now immersed in a liquid with a density one-tenth that of the metal of the bob. The liquid is non-viscous. Now the period of the same pendulum with its bob remaining all the time in the liquid will be : (a) (9/10)T (b) (10 /9) T (c) unchanged (d) (9/10) T . 6. Figure (a) and (b) shows a mass m connected to two identical springs as shown. The ratio of frequency of vibration in case (a) and (b) is (a) 1 : 1 (b) 1 : 2 (c) 1 : 4 (d) 3 : 1. K (a) (b) K K K
• 4. 7. Figure shows a mass m suspended with a mass less inextensible string passing over a frictionless pulley. The spring constant is K . The time period of oscillation of mass m is : (a) 2 2  m k (b) 2 m k (c) 2 2 m k (d) none of above. T K mg m T 8. The time period of oscillation of a block of mass m attached to a light spring of spring constant k is 2 m k  in the absence of all other forces. Suppose that a block of mass m is attached to a fixed spring on one side and rests (i.e. is not attached to) against an identical spring on the other side. It is now allowed to oscillate. The frequency of small oscillations is k m k (a) 1 k m  (b) 1 (2 2) k m   (c) (4 2 2) k m   (d) (2 2) k m   . 9. If a simple pendulum of length l has maximum angular displacement  , then the maximum K.E. of the bob of mass m is : (a) (1/ 2) ( / ) m l g (b) / 2 mg l (c) (1 cos )   mgl (d) ( sin )/ 2  mgl . 10. A uniform spring whose unstressed length is l, has a force constant K. The spring is cut into two pieces of unstressed length 1 l and 2 l , where 2 1, l nl n  being an integer. Now a mass m is made to oscillate with first spring. The time period of its oscillation would be (a) 2 ( 1) mn T K n    (b) 2 m T nK   (c) 2 ( 1) m T K n    (d) ( 1) 2 m n T nK    11. A particle is vibrating in simple harmonic motion with an amplitude of 4 cm. At what displacement from the equilibrium position is its energy half potential and half kinetic ? (a) 1 cm (b) 2 cm (c) 2 cm (d) 2 2 cm . 12. One end of a long metallic wire of length L tied to the ceiling. The other end is tied with a massless spring of spring constant K. A mass hangs freely from the free end of the spring. The area of cross section and the young’s modulus of the wire are A and Y respectively. If the mass slightly pulled down and released, it will oscillate with a time period T equal to : (a) 2 ( / )  m K (b) 2 ( )/( )   m YA KL YAK (c) 2 ( / )  mYA KL (d) 2 ( / )  mL YA . 13. An ideal spring with spring-constant k is hung from the ceiling and a block of mass M is attached to its lower end. The mass is released with the spring initially unstretched. Then the maximum extension in the spring is
• 5. (a) 4Mg k (b) 2Mg k (c) Mg k (d) 2 Mg k . 14. Three simple harmonic motions in the same direction having the same amplitude A and same period are superposed. If each differs in phase from the next by 45º then (a) The resultant amplitude is (1+ 3 ) A (b) The phase of the resultant motion relative to the first is 90º (c) The energy associated with the resulting motion is (3+2 2 ) times the energy associated with any single motion. (d) The resulting motion is not simple harmonic. 15. For a particle executing SHM the displacement x is given by x = A cos t. Identify the graph which represents the variation of potential energy (PE) as a function of time t and displacement x. II I t –A A E III IV x (a) I, III (b) II, IV (c) II, III (d) I, IV.
• 6. S SU UB BJ JE EC CT TI IV VE E U UN NS SO OL LV VE ED D ( (C C. .B B. .S S. .E E. .) ) L LE EV VE EL L – – I I (REVIEW YOUR CONCEPTS) 1. A particle having mass 10 g oscillates according to the equation 1 (2.0cm)sin[(100 ) / 6].     x s t Find (a) the amplitude, the time period and the spring constant (b) the position, the velocity and the acceleration at 0  t . 2. Consider a particle moving in simple harmonic motion according to the equation 1 2.0cos(50 tan 0.75)     x t where x is in centimeter and t in second. The motion is started at 0  t . (a) when does the particle come to rest for the first time ? (b) when does the acceleration have its maximum magnitude for the first time ? (c) when does the particle come to rest for the second time ? 3. A block of mass 0.5 kg hanging from a vertical spring executes simple harmonic motion of amplitude 0.1 m and time period 0.314 s. Find the maximum force exerted by the spring on the block. 4. A spring balance has a scale that reads from 0 to 50 kg. The length of the scale is 20 cm. A body suspended from this balance, when displaced and released, oscillates with a period of 0.6 s. What is the weight of the body. 5. A simple pendulum of length l and having a bob of mass M is suspended in a car. The car is moving on a circular track of radius R with a uniform speed v. If the pendulum makes small oscillations in a radial direction about its equilibrium position, what will be its time period ? 6. A trolley of mass 3.0 kg, as shown in fig. is connected to two springs, each of spring constant 600 Nm-1 . If the trolley is displaced from its equilibrium position by 5.0 cm and released, what is (a) the period of ensuing oscillations, and 3.0 kg 600 Nm –1 600 Nm –1 (b) the maximum speed of the trolley ? 7. The acceleration due to gravity on the surface of moon is 1.7 ms-2 . What is the time period of a simple pendulum on the surface of moon if its time period on the surface of earth is 3.5 s ? (g on the surface of earth is 9.8 ms-2+) 8. Suppose that the two springs in figure have different force constant 1 k and 2 k . Show that the frequency f of oscillation of the block is then given by : 2 2 1 2   f f f where 1 f and 2 f are the frequencies at which the block would oscillate if connected only to spring 1 or spring 2. m k1 k2 9. Find the time period of the oscillation of mass m in figures (a, b, c). What is the equivalent spring constant of the pair of spring in each case ? m k1 k2 (a) m k1 k2 (b) m k1 k2 (c)
• 7. 10. A particle which is attached to a spring oscillates horizontally with simple harmonic motion with a frequency of (1/ )  Hz and total energy of 10 joule. If the maximum speed of the particle is 0.4 metre per second, what is the force constant of the spring ? What will be the maximum potential energy of the spring during this motion ?
• 8. S SU UB BJ JE EC CT TI IV VE E U UN NS SO OL LV VE ED D L LE EV VE EL L – – I II I (BRUSH UP YOUR CONCEPTS) 1. Two equal charges q are kept at –a and a along x-axis. A particle of mass m and charge ( / 2) q is brought to the origin and given a small displacement along (a) x-axis (b) along y-axis. Describe the motion in the two cases. 2. The block of mass 1 m shown in figure is fastened to the spring and the block of mass 2 m is placed against it. (a) Find the compression of the spring in the equilibrium position. (b) The block are pushed a further distance 1 2 (2/ ) ( ) sin   k m m g against the spring and released. Find the position where the two blocks separate (c) What is the common speed of blocks at the time of separation ? Neglect friction. k  m1 m2 3. In figure shown 100  k N/m, M = 1 kg and F = 10 N. (a) Find the compression of the spring in the equilibrium position (b) A sharp blow by some external agent imparts a speed of 2 m/s to the block towards left. Find the sum of the potential energy of the spring and the kinetic energy of the block at this instant. (c) Find the time period of the resulting simple harmonic motion k M F (d) Find the amplitude (e) Write the potential energy of the spring when the block is at the left extreme (f) Write the potential energy of the spring when the block is at the right extreme The answers of (b), (e) and (f) are different. Explain why this does not violate the principle of conservation of energy. 4. (a) What will be the time period of a simple pendulum if its length is equal to radius of earth (=6400 km)? (b) What is the maximum time period which an oscillating simple pendulum can have ? 5. The spring shown in the figure are all unstretched in the beginning when a man starts pulling the block. The man exerts a constant force F on the block. Find the amplitude and the frequency of the motion of the block. k1 k2 M k3 6. A particle is subjected to two simple harmonic motions, one along the X-axis and the other on a line making an angle of 45º with the X-axis. The two motions are given by 0 sin   x x t and 0 sin   s s t . Find the amplitude of the resultant motion. 7. A particle of mass m is attached to three springs A, B and C of equal force constants k as shown in figure. If the particle is pushed slightly against the spring C and released, find the time period of oscillation. A B C 45º 90º 8. Consider the situation shown in figure. Show that if the blocks are displaced slightly in opposite directions and released, they will execute simple harmonic motion. Calculate the time period. k m m
• 9. 9. A simple pendulum of length 40 cm is taken inside a deep mine. Assume for the time being that the mine is 1600 km deep. Calculate the time period of the pendulum there. Radius of the earth = 6400 km. 10. A smooth horizontal disc rotates about the vertical axis O with a constant angular velocity 0  . A thin uniform rod AB of length l performs small oscillations about the vertical axis A fixed to the disc at a distance a from the axis of the disc. Find the period of these oscillations.
• 10. S SU UB BJ JE EC CT TI IV VE E U UN NS SO OL LV VE ED D L LE EV VE EL L – – I II II I (CHECK YOUR SKILLS) 1. Two blocks ( 1.0  m kg and 9  M kg) and a spring of force constant ( 200  k N/m) are arranged on a horizontal frictionless surface as shown in figure. The coefficient of friction between the two blocks is 0.40. What is the maximum possible amplitude of SHM if no slipping is to occur between the blocks? [ [ 10  g m/s2 ]. k M m 2. A uniform rod of length L and mass M is pivoted at its centre. It is held in position by a system of springs as shown in figure. Show that when turned through a small angle  and released the rod undergoes SHM. Find the frequency of oscillations. k k 3. All the surfaces shown in figure are frictionless. The mass of the car is M , that of the block is m and the spring has spring constant k . Initially, the car and the block are at rest and the spring is stretched through a length 0 x when the system is released. (a) Find the amplitudes of the simple harmonic motion of the block and of the car as seen from the rod. (b) Find the time period(s) of the two simple harmonic motions. k M m 4. (a) Consider a small block of mass m , resting against a massless spring of spring constant 1 k on one side and another spring of spring constant 2 k , both the springs being unextended. If the block moves such that only one of the springs is compressed at any time, find the total time period of oscillation (b) Consider a situation similar to (a), except that the block does not rest on both the springs simultaneously, but it can move between the two springs freely for a distance, s. Find the time period of the oscillation if the total energy of the block is 0 E . m C A B k1 k2 m k1 k2 s 5. Find the frequency of small oscillations of a thin uniform vertical rod of mass m and length l hinged at the point O . The combined stiffness of the springs is k . l O 6. A thin uniform board of length L and mass M is balanced on a fixed semicircular cylinder of radius R as shown in the figure. If the plank is tilted slightly from its equilibrium position, find the frequency of small oscillations. R L 7. A pulley in the form of a circular disc of mass m and radius r has the groove cut all along its perimeter. A string whose one end is attached to the ceiling passes over this disc pulley and its other end is attached to a spring of spring constant k . The other end of the spring is attached to ceiling as shown in figure. Find the time period of vertical oscillations of the centre of mass assuming that the string does not slip over the pulley.
• 11. 8. A block B of mass m is attached to a fixed spring of spring constant k and is free to oscillate on a smooth horizontal plane as shown in the figure. A second identical small block of equal mass starts from A and moves towards B, which is at rest, as shown in the figure. Find the time taken by the free block to return to the point A, if the collision between the two blocks is (a) perfectly elastic (b) perfectly inelastic. m C B A k v L m 9. Two small masses 1 m and 2 m are attached to the ends of a rod of negligible mass as shown in figure. The rod is pivoted so that the masses and rod form a physical pendulum. Find an expression for the period of oscillation. Express your answer in terms of the parameters given in figure. m1 m2 l1 l2 Pivot 10. Find the angular frequency of oscillation of motion of block m for small angular motion of rot BD . Consider the rod to be massless. k2 D b c B k1 m
• 12. P PR RO OB BL LE EM MS S 1. A thin rod of length L and area of cross-section S is pivoted at its lowest point P inside a stationary, homogeneous and non-viscous liquid (figure). The rod is free to rotate in a vertical plane about a horizontal axis passing rough P. The density d1 of the material of the rod is smaller than the density d2 of the liquid. The rod is displaced by a small angle  from its equilibrium position and then released. Show that the motion of the rod is simple harmonic and determine its angular frequency in terms of the given parameters. d1 d2 P 2. Two masses m1 and m2 are suspended together by a massless spring of spring constant k (figure). When the masses are in equilibrium, m1 is removed without disturbing the system. Find the angular frequency and amplitude of oscillation of m2. m2 m1 3. Two light springs of force constant k1 and k2 and a block of mass m are in one line AB on a smooth horizontal table such that one end of each spring is fixed on rigid supports and the other end is free as shown in the figure. The distance CD between the free ends of the springs is 60 cms. If the block moves along AB with a velocity 120 cm/sec in between the springs, calculate the period of oscillation of the block. 1 2 ( 1.8 / , 3.2 / , 200 )    k N m k N m m gm . 60cm A B C D m k1 k2  4. An object of mass 0.2 Kg executes simple harmonic motion oscillate along the x-axis with a frequency of 25        Hz. At the position x = 0.04 m, the object has kinetic energy of 0.5 J and potential energy of 0.4 J. Find the amplitude of oscillation 5. Two identical balls A and B of mass 0.1 kg are attached to two identical massless springs. The spring-mass system is constrained to move inside a rigid smooth pipe bent in the form of a circle as shown in the figure. The pipe is fixed in a horizontal plane. The centers of the balls can move in a circle, of radius 0.06 metre. Each spring has a natural length 0.06  meter and spring constant 0.1 N/m. Initially, both the balls are displaced by an angle 6    radian with respect to the diameter PQ of the circle (as shown in the figure) and released from rest. P Q O 6  6  A B (a) Calculate the frequency of oscillation of ball B. (b) Find the speed of ball A when A and B are at the two ends of the diameter PQ. (c) What is the total energy of the system ? 6. A point particle of mass M is attached to one end of a massless rigid non-conducting rod of length L . Another point particle of the same mass is attached to the other end of the rod. The two particles carry charges q and q respectively. This arrangement is held in a region of a uniform electric field E such that the rod makes a small angle  (say of about 5º) with the field direction. Find an expression for the minimum time needed for the rod to become parallel to the field after it is set free. +q M  -q E M d2
• 13. A AN NS SW WE ER RS S O OB BJ JE EC CT TI IV VE E U UN NS SO OL LV VE ED D L LE EV VE EL L – – I I 1. (d) 2. (b) 3. (b) 4. (c) 5. (b) 6. (a) 7. (c) 8. (d) 9. (d) 10. (a) 11. (b) 12. (d) 13. (d) 14. (b) 15. (c) O OB BJ JE EC CT TI IV VE E U UN NS SO OL LV VE ED D L LE EV VE EL L – – I II I 1. (a) 2. (c) 3. (d) 4. (b) 5. (b) 6. (b) 7. (b) 8. (b) 9. (c) 10. (c) 11. (d) 12. (b) 13. (b) 14. (c) 15. (a) S SU UB BJ JE EC CT TI IV VE E U UN NS SO OL LV VE ED D ( (C C. .B B. .S S. .E E. .) ) L LE EV VE EL L – – I I 1. (a) 2.0 cm, 0.063 s, 100 N/m (b) 1.0 cm, 1.73 m/s, 100 m/s2 2. (a) 2 1.6 10  s (b) 2 1.6 10  s (c) 2 3.6 10  s. 3. 25 N 4. 219 N 5. 2 4 2 1 2 / T g v R   
• 14. 6. (a) 0.31 s (b) 1.0 ms-1 7. 8.4 s 9. (a) 1 2 2  m k k (b) 1 2 2  m k k (c) 1 2 1 2 ( ) 2   m k k k k 10. 500 N/m and 10 Joule. S SU UB BJ JE EC CT TI IV VE E U UN NS SO OL LV VE ED D L LE EV VE EL L – – I II I 1. (a) SHM with the time period 3 0 (2 / ) 2   q ma (b) Accelerated translatory 2. (a) 1 2 ( ) sin   m m g k (b) when the spring acquires its natural length (c) 1 2 3 ( ) sin   m m g k 3. (a) 10 cm (b) 2.5 J (c) /5  s (d) 20 cm (e) 4.5 J (f) 0.5 J. 4. (a) 2 ( / 2 )   T R g 1 hour (b) 2 ( / ) 84.6   T R g min. 5. 2 3 1 2 2 3 3 1 1 2 2 3 3 1 2 3 ( ) 1 , 2 ( )        F k k k k k k k k k k k k k k M k k 6. 2 2 0 0 0 0 2 x s x s   7. 2 2  m k 8. 2 2  m k 9. 1.47 s 10. 2 0 2 2 3 T a    
• 15. S SU UB BJ JE EC CT TI IV VE E U UN NS SO OL LV VE ED D L LE EV VE EL L – – I II II I 1. 20 cm 2. 1 6 2   k f M 3. (a) 0 0 ,   Mx mx M m M m (b) 2 ( )   mM k M m 4. (a) 1 2 m m k k           (b) 2 1 2 0 2 m m S m k k E     5. 3 3 2    g k l M rad/s. 6. 3gR f L   7. 3 2   m T k sec. 8. (a) 2L m t v k    (b) 3 2 L m v k   9. 2 2 1 1 2 2 2 2 1 1 2 ( )     m l m l T m gl m gl s. 10. 2 1 2 2 2 1 2 rad /s. [ ( ) k k c m k c k b c     S SU UB BJ JE EC CT TI IV VE E U UN NS SO OL LV VE ED D L LE EV VE EL L 1. 2 1 1 3 2          d d g d L 2. 1 m g A k  Angular frequency = 2 k m 3. T = 2.83 sec 4. 2 6 10 m   . 5. (a) 1   f (b) v = .02  m/s (c) 2 5 4 10    E J 6. 1 2 min s. 2 2 ML T q E        