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NEET-XII-Physics

12: Simple Harmonic Motion

with Solutions - page 3

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  • Qstn #12
    The average energy in one time period in simple harmonic motion is
    (a)
    12m ω2A2
    (b)
    14m ω2A2
    (c) m ω2A2
    (d) zero
    digAnsr:   a
    Ans : (a)`` \frac{1}{2}m{\omega }^{2}{A}^{2}``
    It is the total energy in simple harmonic motion in one time period.
    Page No 251:
  • Qstn #13
    A particle executes simple harmonic motion with a frequency v. The frequency with which the kinetic energy oscillates is
    (a) v/2
    (b) v
    (c) 2 v
    (d) zero
    digAnsr:   c
    Ans : (c) 2v
    Because in one complete oscillation, the kinetic energy changes its value from zero to maximum, twice.
    Page No 251:
  • Qstn #14
    A particle executes simple harmonic motion under the restoring force provided by a spring. The time period is T. If the spring is divided in 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 T/2
    (d) become
    T/2
    digAnsr:   d
    Ans : (d) become T/`` \sqrt{2}``
    Time period `` \left(T\right)`` is given by,
    `` T=2\,\mathrm{\,\pi \,}\sqrt{\frac{m}{k}}``
    where m is the mass, and
    k is spring constant.
    When the spring is divided into two parts, the new spring constant k1 is given as,
    k1 = `` 2k``
    New time period T1:
    T1 = `` 2\,\mathrm{\,\pi \,}\sqrt{\frac{m}{2k}}=\frac{1}{\sqrt{2}}2\,\mathrm{\,\pi \,}\sqrt{\frac{m}{k}}=\frac{1}{\sqrt{2}}\,\mathrm{\,T\,}``
    Page No 251:
  • Qstn #15
    Two bodies A and B of equal mass are suspended from two separate massless springs of spring constant k1 and k2 respectively. If the bodies oscillate vertically such that their maximum velocities are equal, the ratio of the amplitude of A to that of B is
    (a) k1/k2
    (b)
    k1/k2
    (c) k2/k1
    (d)
    k2/k1
    digAnsr:   d
    Ans : (d)`` \sqrt{\frac{{k}_{2}}{{k}_{1}}}``
    Maximum velocity, v = Aω
    where A is amplitude and ω is the angular frequency.
    Further, ω = `` \sqrt{\frac{k}{m}}``
    Let A and B be the amplitudes of particles A and B respectively. As the maximum velocity of particles are equal,
    `` i.e.{v}_{A}={v}_{B}``
    `` \,\mathrm{\,or\,},``
    `` A{\omega }_{A}=B{\omega }_{B}``
    `` \Rightarrow A\sqrt{\frac{{k}_{1}}{{m}_{A}}}=B\sqrt{\frac{{k}_{2}}{{m}_{B}}}``
    `` \Rightarrow A\sqrt{\frac{{k}_{1}}{m}}=B\sqrt{\frac{{k}_{2}}{m}}({m}_{A}={m}_{B}=m)``
    `` ``
    `` \Rightarrow \frac{A}{B}=\sqrt{\frac{{k}_{2}}{{k}_{1}}}``
    Page No 251:
  • Qstn #16
    A spring-mass system oscillates with a frequency v. If it is taken in an elevator slowly accelerating upward, the frequency will
    (a) increase
    (b) decrease
    (c) remain same
    (d) become zero
    digAnsr:   c
    Ans : (c) remain same
    Because the frequency (`` \nu =\frac{1}{2\pi }\sqrt{\frac{k}{m}}``) of the system is independent of the acceleration of the system.
    Page No 251:
  • Qstn #17
    A spring-mass system oscillates with a car. If the car accelerates on a horizontal road, the frequency of oscillation will
    (a) increase
    (b) decrease
    (c) remain same
    (d) become zero
    digAnsr:   c
    Ans : (c) remain same
    As the frequency of the system is independent of the acceleration of the system.
    Page No 251:
  • Qstn #18
    A pendulum clock that keeps correct time on the earth is taken to the moon. It will run
    (a) at correct rate
    (b) 6 times faster
    (c)
    6times faster
    (d)
    6times slower
    digAnsr:   d
    Ans : (d) `` \sqrt{6}`` times slower
    The acceleration due to gravity at moon is g/6.
    Time period of pendulum is given by, `` T=2\pi \sqrt{\frac{l}{g}}``
    Therefore, on moon, time period will be:
    Tmoon = `` 2\pi \sqrt{\frac{l}{{g}_{moon}}}=2\pi \sqrt{\frac{l}{\left({\displaystyle \raisebox{1ex}{$g$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\right)}}=\sqrt{6}\left(2\pi \sqrt{\frac{l}{g}}\right)=\sqrt{6}T``
    Page No 251:
  • Qstn #19
    A wall clock uses a vertical spring-mass system to measure the time. Each time the mass reaches an extreme position, the clock advances by a second. The clock gives correct time at the equator. If the clock is taken to the poles it will
    (a) run slow
    (b) run fast
    (c) stop working
    (d) give correct time
    digAnsr:   d
    Ans : (d) give correct time
    Because the time period of a spring-mass system does not depend on the acceleration due to gravity.
    Page No 251:
  • Qstn #20
    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
    digAnsr:   c
    Ans : (c) its length should be decreased to keep correct time
    Time period of pendulum,
    T = `` 2\pi \sqrt{\frac{l}{g}}``
    At higher altitudes, the value of acceleration due to gravity decreases. Therefore, the length of the pendulum should be decreased to compensate for the decrease in the value of acceleration due to gravity.
    Page No 251:
  • Qstn #21
    The free end of a simple pendulum is attached to the ceiling of a box. The box is taken to a height and the pendulum is oscillated. When the bob is at its lowest point, the box is released to fall freely. As seen from the box during this period, the bob will
    (a) continue its oscillation as before
    (b) stop
    (c) will go in a circular path
    (d) move on a straight line
    digAnsr:   c
    Ans : (c) will go in a circular path
    As the acceleration due to gravity acting on the bob of pendulum, due to free fall gives a torque to the pendulum, the bob goes in a circular path.
    Page No 251:
  • #
    Section : iii
  • Qstn #1
    Select the correct statements.
    (a) A simple harmonic motion is necessarily periodic.
    (b) A simple harmonic motion is necessarily oscillatory.
    (c) An oscillatory motion is necessarily periodic.
    (d) A periodic motion is necessarily oscillatory.
    digAnsr:   a,b
    Ans : (a) A simple harmonic motion is necessarily periodic.
    (b) A simple harmonic motion is necessarily oscillatory.
    A periodic motion need not be necessarily oscillatory. For example, the moon revolving around the earth.
    Also, an oscillatory motion need not be necessarily periodic. For example, damped harmonic motion.
    Page No 251:
  • Qstn #2
    A particle moves in a circular path with a uniform speed. Its motion is
    (a) periodic
    (b) oscillatory
    (c) simple harmonic
    (d) angular simple harmonic
    digAnsr:   a
    Ans : (a) periodic
    Because the particle covers one rotation after a fixed interval of time but does not oscillate around a mean position.
    Page No 251:
  • Qstn #3
    A particle is fastened at the end of a string and is whirled in a vertical circle with the other end of the string being fixed. The motion of the particle is
    (a) periodic
    (b) oscillatory
    (c) simple harmonic
    (d) angular simple harmonic
    digAnsr:   a
    Ans : (a) periodic
    Because the particle completes one rotation in a fixed interval of time but does not oscillate around a mean position.
    Page No 251:
  • Qstn #4
    A particle moves in a circular path with a continuously increasing speed. Its motion is
    (a) periodic
    (b) oscillatory
    (c) simple harmonic
    (d) none of them
    digAnsr:   d
    Ans : (d) none of them
    As the particle does not complete one rotation in a fixed interval of time, neither does it oscillate around a mean position.
    Page No 251:
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