5.

OSCILLATIONS
Theory :
(2) Two S.H.M. are represented by x = a, sin (ot +a) and x, = a, sin (ot - ay. Obtain the expressions for the
displacerment, amplitude and initial phase of the resultant motion. (March 200S (3)
(2) Represent graphically the displacement, velocity and acceleration against ine ior a parie pertorming lnear
S.H.M., when it starts from the mean position. (March 20os, 2013, Oct. 201o (3)
(3) Represent graphically the variations of K.E., P.E. and T.E. of aparticle pertorning near S H M. with respect to
displacement. (0ctober 2008, March 2009) (2)
(4) Give graphical representation of S.H.M. when particle starts from the posit\r e e NitIOn.
(Octoebr 2009) (2)
(5) Define angular S.H.M. State its differential equation. (March 201l0) (2)
(6) Define an ideal simple pendulum. Show that the motion of a simple penduhn uder certain onditions ts
sirnple harmonic. Obtain an expression for its period. (March 2010, 2013, yN18 (4)
(7) Show that for a small amplitude, the motion of a simple pendulunn is linar sH A Hene ind its period.
Marc 2011) (3)
(8) Show that linear S.H.M. is the projection of uniform circular motion on anv diameter, March 201I) (3)
(9) Show graphically the variation of velocity and acceleration in S.HM. mth hase, t itNt starts
position. (0ct. 2011) tron extrete
(2)
(10) A particle pertorming S.H.M. starts from extreme position, Plot the graph ot rh tINd displaement against
time. (March 2012) (2)
(11) Derive an expresSsion for the period of motion of a suple penduum. On h fkiNfrs does it depend
(4)
(12) State an expression for K.E. (kinetic energy) und P. E. (potenttal energv) at displaement ' i a rtle NTOrmng
linear S.H.M. Represent them graphically. Find the displaccmentat which K x ual to} E eb 2014) (3)
(13) Define phase of S.H.M. Show variation of displacenent, velocity and awleratin with pase tor a particle
performing linear S.H.M. graphically, wlhen it starts fronl exiente position. Xt 014) (4)
(14) Obtain an expression for potential energy of apartice petoning snple harmot totm Hene evaluate the
Summary of Board Questions 189

potential energy. (a) at mean position and (b) at extreme position. (Feb. 2015, March 2019) (4)
(15) Discuss the compOsition of two S.H.M.s along the same path having same period. Find the resultant amp
and initial phasc. (Oct. 2015) (4)
(16) Define linear S.H.M. Show that S.H.M. is a projcction of U.C.M. on any (2)
diameter. (Feb. 2016)
(17) Define practical simple pendulum.
Show that motion of bob of pendulum with small amplitude is linear S.H.M. Hence obtain an expression for its
period. What are the factors on which its period depends? (July 2016) (5)
(18) Obtain the differential equation of linear simple harmonic motion. (March 2017) (2)
(19) Prove the law of conservation of energy for a particle performing simple harmonic motion. Hence graphically
show the variation of kinetic energy and potential energy w. r. t. instantancous displacement. (March 201 7)
(4)
(20) Define linear simple harmonic motion. Assuming the expression for displacement of a particle starting from
extreme position, explain graphically the variation of velocity and acceleration w.r.t. time. (July 2016) (4)
(21) State the differential equation of linear simple harmonic motion. Hence obtain the expression for acceleration,
velocity and displacement of a particle performing linear S.H.M. (March 2018) (4)
(22) From ditterential equation of linear S.H.M., obtain an expression for acceleration, velocity and displacement of
a particle performing S.H.M. (March 2019) (3)
(23) At which position, the total energy of a particle executing linear s. H. M. is purely potential? (March 2020)
(1)
(24) Define linear S. H. M. Obtain differential equation of linear S. H. M. (March 2020) (3)
(25) At which position. The restoring force acting on a particle executing linear SHM is maximum?(Oct. 2021]2)
(26) Derive an expression for period of a simple pendulum. (Oct. 2021) (3)
(27) Write the differential equation for angular S. H. M. (March 2022) (1)
(28) Define second's pendulum. Derive a formula for the length of second's pendulum. (March 2022) (2)
Problems :

(1) The displacement of particle performing S.H.M. is given by

x=5sin nt +12 sin nt +em
2 Determine the amplitude, period and initial phase of the motion.
(Oct. 2008) (4)
(2) The period of a simple pendulum increases by 10% when its length is increased by 21 cm. Find the original
length and period of the pendulum. (g 9.8 m/s) (Oct. 2009) (4)
(3) S.H.M. is given by the equation x= 8 sin (4nt) + 6 cos (4nt) cm.
Find its (a) amplitude (b) initial phase (c) period (d) frequency (Oct. 2010) (4)
(4) A particle of mass 10 gm executes linear S.H.M. of amplitude 5 cm. With a period of 2 second. Find its potential
th

energy, kinetic energy second after it has passed through the mean position. (Oct. 2011) (4)
(5) The period of simple pendulum is found to increase by 50% when the length of the pendulum is increased by
0.6 m. Calculate the initial length and initial period of oscillation at a place where g =9.8 m/s.
(March 2012) (4)
(6) Abody of mass 1 kg is made to oscillate on a spring of force constant 16 N/m. Calculate:
(a) angular frequency, (b) frequency of vibration. (Oct 2013) (2)
(7) When the length of a simple pendulum is decreased by 20 cm, the period changes by 10%. Find the original
length of the pendulum. (Feb. 2014) (3)
(8) The maximum velocity of a particle performing linear S.H.M. is 0.16 m/s. If its maximum acceleration is
0.64 m/s', calculate its period. (Oct. 2014) (2)
(9) Aparticle in S.H.M. has aperiod of 2 seconds and amplitude of 10 cm. Calculate the acceleration when it is at
4cm from its positive extreme position. (Feb. 2015) (2)
(10) The periodic time of alinear harmonic oscillator is 2r second, with maximum displacement of 1 cm. If the
particle starts from extreme position, find the displacement of the particle alter 3 seconds. (Feb. 2015) (2)
(11) Aparticle performing linear S.H.M. has a period of 6.28 seconds and a pathlength of 20 cm. What is the velocity
when its displacement is 6 cm from mean position? (Feb. 2016) (2)
(12) A particle executes S.H.M. with a period of 10 seconds. Find the time in which its potential energy will be half
of its total energy. (July 2016) (2)
(13) Aclock regulated by seconds pendulum, keeps correct time. During summer, length of pendulum increases to
1.005 m. How much will the clock gain of loose in one day? (g - 9.8 m/s and z =3.142) (July 2017) (3)
(14) Aparticle performing linear S.H.M. has maximum velocity of 25 cm/s and maximum acceleration of 100 cm/s
Find the amplitude and period of oscillation (7 3.14) (March 2018) (2)
 Solution
Uttam's XIl Physics Papers
190
potential energy
of leDgth 1m and mass 10 g oscillates freely with amplitude 2 cm. Find its (2)
(15) Asimple pendulum m/s) (July 2018) the number
(P.E) at the extreme position. (g- 9.8 times its initial length. Calculate
lengh of the sccond's pendulum in aclock is increased to 4 (2)
(16 The new pendulum in one minute. (March 2019)
of osc1llations completed by the N/m. Calculate
oscillate on a spring of force constant 16 (2)
(1) A boiy of mass 1kg is made to vibrations. (March 2019)
(a) Angular frequency, (b) Frequency of amplitude of 5 cm. Calculate its
pendulum of length 1 m has mass 10 g. and oscillates freely with (3)
(1S) A Smple 202O)
potential energy at extreme position. (March maximum velocity 25 cm/s and maximum acceleration 100
cm/s2.
H. M. has
(19) Aparticle pertorming linear S. (2)
Find period of oscillation. (March 2022)
mIOT OR WAVES