Physics · Textbook solutions

Oscillations

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 44 questions

5. Oscillations — worked examples

13 q

Solved Examples

Worked · 13
  1. Solved Ex.5.1
    A body of mass 0.2 kg performs linear S.H.M. It experiences a restoring force of 0.2 N when its displacement from the mean position is 4 cm. Determine (i) force constant (ii) period of S.H.M. and (iii) acceleration of the body when its displacement from the mean position is 1 cm.
  2. Solved Ex.5.2
    A particle performs linear S.H.M. of period 4 seconds and amplitude 4 cm. Find the time taken by it to travel a distance of 1 cm from the positive extreme position.
  3. Solved Ex.5.3
    A particle performing linear S.H.M. with period 6 second is at the positive extreme position at t=0t = 0. The particle is found to be at a distance of 3 cm from this position at time t=7t = 7 s, before reaching the mean position. Find the amplitude of S.H.M.
  4. Solved Ex.5.4
    The speeds of a particle performing linear S.H.M. are 8 cm/s and 6 cm/s at respective displacements of 6 cm and 8 cm. Find its period and amplitude.
  5. Solved Ex.5.5
    The maximum velocity of a particle performing S.H.M. is 6.28 cm/s. If the length of its path is 8 cm, calculate its period.
  6. Solved Ex.5.6
    The maximum speed of a particle performing linear S.H.M is 0.08 m/s. If its maximum acceleration is 0.32 m/s2\text{m/s}^2, calculate its (i) period and (ii) amplitude.
  7. Solved Ex.5.7
    Describe the state of oscillation if the phase angle is 1110∘1110^\circ.
  8. Solved Ex.5.8
    While completing its third oscillation during linear S.H.M., a particle is at −32A\frac{-\sqrt{3}}{2} A, heading to the mean position. Determine the phase angle.
  9. Solved Ex.5.9
    The total energy of a particle of mass 200 g, performing S.H.M. is 10−210^{-2} J. Find its maximum velocity and period if the amplitude is 7 cm.
  10. Solved Ex.5.10
    The period of oscillations of a simple pendulum increases by 10%, when its length is increased by 21 cm. Find its initial length and initial period.
  11. Solved Ex.5.11
    In summer season, a pendulum clock is regulated as a second's pendulum and it keeps correct time. During winter, the length of the pendulum decreases by 1%. How much will the clock gain or lose in one day. (g=9.8 m/s2)(g = 9.8 \text{ m/s}^{2})
  12. Solved Ex.5.12
    A bar magnet of mass 120 g, in the form of a rectangular parallelepiped, has dimensions l=40l = 40 mm, b=10b = 10 mm and h=80h = 80 mm. With the dimension hh vertical, the magnet performs angular oscillations in the plane of a magnetic field with period π\pi s. If its magnetic moment is 3.4 A m2^{2}, determine the influencing magnetic field.
  13. Solved Ex.5.13
    Two magnets with the same dimensions and mass, but of magnetic moments μ1=100\mu_{1} = 100 A m2^{2} and μ2=50\mu_{2} = 50 A m2^{2} are jointly suspended in the earth's magnetic field so as to perform angular oscillations in a horizontal plane. When their like poles are joined together, the period of their angular S.H.M. is 5 s. Find the period of angular S.H.M. when their unlike poles are joined together.

Exercises

31 q

Choose the correct option

Practice · 5
  1. Choose the correct option.
    Ex Q.1 (i)
    A particle performs linear S.H.M. starting from the mean position. Its amplitude is AA and time period is TT. At the instance when its speed is half the maximum speed, its displacement xx is
    1. A.
      32A\dfrac{\sqrt{3}}{2} A
    2. B.
      23A\dfrac{2}{\sqrt{3}} A
    3. C.
      A/2A/2
    4. D.
      12A\dfrac{1}{\sqrt{2}} A
  2. Ex Q.1 (ii)
    A body of mass 1 kg is performing linear S.H.M. Its displacement xx (cm) at tt (second) is given by x=6sin⁡(100t+π/4)x = 6 \sin (100t + \pi/4). Maximum kinetic energy of the body is
    1. A.
      3636 J
    2. B.
      99 J
    3. C.
      2727 J
    4. D.
      1818 J
  3. Ex Q.1 (iii)
    The length of second's pendulum on the surface of earth is nearly 1 m. Its length on the surface of moon should be [Given: acceleration due to gravity (g)(g) on moon is 1/6th1/6^{\text{th}} of that on the earth's surface]
    1. A.
      1/61/6 m
    2. B.
      66 m
    3. C.
      1/361/36 m
    4. D.
      16\dfrac{1}{\sqrt{6}} m
  4. Ex Q.1 (iv)
    Two identical springs of constant kk are connected, first in series and then in parallel. A metal block of mass mm is suspended from their combination. The ratio of their frequencies of vertical oscillations will be in a ratio
    1. A.
      1:41:4
    2. B.
      1:21:2
    3. C.
      2:12:1
    4. D.
      4:14:1
  5. Ex Q.1 (v)
    The graph shows variation of displacement of a particle performing S.H.M. with time tt. Which of the following statements is correct from the graph?
    1. A.
      The acceleration is maximum at time TT.
    2. B.
      The force is maximum at time 3T/43T/4.
    3. C.
      The velocity is zero at time T/2T/2.
    4. D.
      The kinetic energy is equal to total energy at time T/4T/4.

Answer in brief

Practice · 5
  1. Answer in brief.
    Ex Q.2 (i)
    Define linear simple harmonic motion.
  2. Ex Q.2 (ii)
    Using differential equation of linear S.H.M, obtain the expression for (a) velocity in S.H.M., (b) acceleration in S.H.M.
  3. Ex Q.2 (iii)
    Obtain the expression for the period of a simple pendulum performing S.H.M.
  4. Ex Q.2 (iv)
    State the laws of simple pendulum.
  5. Ex Q.2 (v)
    Prove that under certain conditions a magnet vibrating in uniform magnetic field performs angular S.H.M.

Solve the following

Practice · 21
  1. Ex Q.3
    Obtain the expression for the period of a magnet vibrating in a uniform magnetic field and performing S.H.M.
  2. Ex Q.4
    Show that a linear S.H.M. is the projection of a U.C.M. along any of its diameter.
  3. Ex Q.5
    Draw graphs of displacement, velocity and acceleration against phase angle, for a particle performing linear S.H.M. from (a) the mean position (b) the positive extreme position. Deduce your conclusions from the graph.
  4. Ex Q.6
    Deduce the expressions for the kinetic energy and potential energy of a particle executing S.H.M. Hence obtain the expression for total energy of a particle performing S.H.M and show that the total energy is conserved. State the factors on which total energy depends.
  5. Ex Q.7
    Deduce the expression for period of simple pendulum. Hence state the factors on which its period depends.
  6. Ex Q.8
    At what distance from the mean position is the speed of a particle performing S.H.M. half its maximum speed. Given path length of S.H.M. =10 cm= 10\ \text{cm}.
  7. Ex Q.9
    In SI units, the differential equation of an S.H.M. is d2xdt2=−36x\dfrac{d^2x}{dt^2} = -36x. Find its frequency and period.
  8. Ex Q.10
    A needle of a sewing machine moves along a path of amplitude 4 cm with frequency 5 Hz. Find its acceleration (130)\left(\dfrac{1}{30}\right) s after it has crossed the mean position.
  9. Ex Q.11
    Potential energy of a particle performing linear S.H.M is 0.1 π2x20.1\ \pi^2 x^2 joule. If mass of the particle is 20 g, find the frequency of S.H.M.
  10. Ex Q.12
    The total energy of a body of mass 2 kg performing S.H.M. is 40 J. Find its speed while crossing the centre of the path.
  11. Ex Q.13
    A simple pendulum performs S.H.M of period 4 seconds. How much time after crossing the mean position, will the displacement of the bob be one third of its amplitude.
  12. Ex Q.14
    A simple pendulum of length 100 cm performs S.H.M. Find the restoring force acting on its bob of mass 50 g when the displacement from the mean position is 3 cm.
  13. Ex Q.15
    Find the change in length of a second's pendulum, if the acceleration due to gravity at the place changes from 9.75 m/s29.75\ \text{m/s}^2 to 9.80 m/s29.80\ \text{m/s}^2.
  14. Ex Q.16
    At what distance from the mean position is the kinetic energy of a particle performing S.H.M. of amplitude 8 cm, three times its potential energy?
  15. Ex Q.17
    A particle performing linear S.H.M. of period 2π2\pi seconds about the mean position O is observed to have a speed of b3 m/sb\sqrt{3}\ \text{m/s}, when at a distance bb (metre) from O. If the particle is moving away from O at that instant, find the time required by the particle, to travel a further distance bb.
  16. Ex Q.18
    The period of oscillation of a body of mass m1m_1 suspended from a light spring is TT. When a body of mass m2m_2 is tied to the first body and the system is made to oscillate, the period is 2T2T. Compare the masses m1m_1 and m2m_2
  17. Ex Q.19
    The displacement of an oscillating particle is given by x=asin⁡ωt+bcos⁡ωtx = a\sin\omega t + b\cos\omega t where aa, bb and ω\omega are constants. Prove that the particle performs a linear S.H.M. with amplitude A=a2+b2A = \sqrt{a^2 + b^2}
  18. Ex Q.20
    Two parallel S.H.M.s represented by x1=5sin⁡(4πt+π/3)x_1 = 5\sin(4\pi t + \pi/3) cm and x2=3sin⁡(4πt+π/4)x_2 = 3\sin(4\pi t + \pi/4) cm are superposed on a particle. Determine the amplitude and epoch of the resultant S.H.M.
  19. Ex Q.21
    A 20 cm wide thin circular disc of mass 200 g is suspended to a rigid support from a thin metallic string. By holding the rim of the disc, the string is twisted through 60∘60^\circ and released. It now performs angular oscillations of period 1 second. Calculate the maximum restoring torque generated in the string under undamped conditions. (π3≈31)(\pi^3 \approx 31)
  20. Ex Q.22
    Find the number of oscillations performed per minute by a magnet is vibrating in the plane of a uniform field of 1.6×10−5 Wb/m21.6 \times 10^{-5}\ \text{Wb/m}^2. The magnet has moment of inertia 3×10−6 kg m23 \times 10^{-6}\ \text{kg m}^2 and magnetic moment 3 A m23\ \text{A m}^2.
  21. Ex Q.23
    A wooden block of mass m is kept on a piston that can perform vertical vibrations of adjustable frequency and amplitude. During vibrations, we don't want the block to leave the contact with the piston. How much maximum frequency is possible if the amplitude of vibrations is restricted to 25 cm? In this case, how much is the energy per unit mass of the block? (g≈π2≈10 m s−2)(g \approx \pi^2 \approx 10\ \text{m s}^{-2})