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Multiple choice angular simple harmonic motion example of simple harmonic motion oscillatory motion oscillations physics

The bob cf a simple pendulum is a spherical hollowe bal filled with water A pluyged hole near the bouthmol th oscilloting bob gets suddenly unplugged. During observation, till water is coming out, the time period of would 

  1. First increase and then decrease to the original value

  2. first decrease and then increase to the original value

  3. remain unchanged

  4. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

As water leaks out, the center of mass of the bob initially moves downwards, increasing the effective length of the pendulum, which increases the time period. Once the water is empty, the center of mass returns to the center of the sphere, decreasing the period back to the original value.

Multiple choice angular simple harmonic motion example of simple harmonic motion oscillatory motion oscillations physics

A metallic disc oscillates about an axis through its edge in it's own plane. The equivalent length of the disc as a pendulum is

  1. $r$
  2. $\dfrac r3$
  3. $\dfrac { r } { 2 }$
  4. $\dfrac { 3r } { 2 }$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For a disc pivoted at its edge, I = I_cm + mR^2 = (1/2)mR^2 + mR^2 = (3/2)mR^2. The equivalent length L_eq = I/(mR) = (3/2)R.

Multiple choice angular simple harmonic motion example of simple harmonic motion oscillatory motion oscillations physics

The bob of a simple pendulum executes  $S H M$  in water with a period  $t,$  while the period of oscillation of the bob is  $t _{ 0 }$  in air. Neglecting the frictional force of water and given that the density of the bob is  $( 4 / 3 ) \times 1000 kg / { m } ^ { 3 }.$  What relationship between  $t$  and  $t _ { 0 }$  is true ?

  1. $t = t _ { 0 }$
  2. $t = 4 t _ { 0 }$
  3. $t = 2 t _ { 0 }$
  4. $t = t _ { 0 } / 2$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The effective gravity in water is g' = g(1 - rho_water/rho_bob). Given rho_bob = 4/3 * 1000 and rho_water = 1000, g' = g(1 - 3/4) = g/4. Since T is proportional to 1/sqrt(g), T_water = T_air / sqrt(1/4) = 2 * T_air.

Multiple choice angular simple harmonic motion example of simple harmonic motion oscillatory motion oscillations physics

A simple pendulum is released when $\theta = \pi/6$. The time period of oscillation is

  1. $\displaystyle 2\pi\sqrt{\frac{l}{g}}$
  2. $\displaystyle 2\pi\sqrt{\frac{l}{g}}\left(\frac{293}{288}\right)$
  3. $\displaystyle 2\pi\sqrt{\frac{l}{g}}\left(\frac{288}{293}\right)$

  4. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For large amplitudes, the time period is given by 
$T={2\pi }{\sqrt{\dfrac{L}{g}}}(1+\dfrac{\theta ^{2}}{16})$
Substitute $\theta =\dfrac{\pi }{6}$, we get answer as 
$T={2\pi }{\sqrt{\dfrac{L}{g}}}(\dfrac{293}{288})$
Option B is correct.

Multiple choice angular simple harmonic motion example of simple harmonic motion oscillatory motion oscillations physics

A pendulum suspended from the ceiling of an elevator at rest has time period ${ T } _{ 1 }$. When the elevator moves up with an acceleration 'a' its time period of oscillation becomes ${ T } _{ 2 }$ when the elevator moves down with an acceleration 'a', its period of oscillation become ${ T } _{ 3 }$ then

  1. ${ T } _{ 1 }=\sqrt { { T } _{ 2 }{ T } _{ 3 } } $
  2. ${ T } _{ 1 }=\sqrt { T _{ 2 }{ ^{ 2 }T _{ 3 } }^{ 2 } } $
  3. ${ T } _{ 1 }=\dfrac { \sqrt { 2 } { T } _{ 2 }{ T } _{ 3 } }{ \sqrt { {T _{ 2 }}^{ 2 }+{T _{ 3 } }^{ 2 } } } $
  4. ${ T } _{ 1 }=\dfrac { { T } _{ 2 }{ T } _{ 3 } }{ \sqrt { {T _{ 2 }}^{ 2 }+{T _{ 3 } }^{ 2 } } } $
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

T1 = 2*pi*sqrt(l/g). T2 = 2*pi*sqrt(l/(g+a)). T3 = 2*pi*sqrt(l/(g-a)). Thus, 1/T2^2 = (g+a)/(4*pi^2*l) and 1/T3^2 = (g-a)/(4*pi^2*l). Adding these gives 1/T2^2 + 1/T3^2 = 2g/(4*pi^2*l) = 2/T1^2. Solving for T1 gives T1 = sqrt(2)*T2*T3 / sqrt(T2^2 + T3^2).

Multiple choice angular simple harmonic motion example of simple harmonic motion oscillatory motion oscillations physics

In a conical pendulum, when the bob moves in a horizontal circle of radius r, with uniform speed V, the string of length L describe a cone of semi-vertical angle $\theta$. The tension  in the string is given by 

  1. $T = \dfrac{mgl}{(L^2 - r^2)}$
  2. $ T = \dfrac{\sqrt {L^2 - r^2}}{mgl}$
  3. $ T = \dfrac{mgL}{\sqrt {L^2 - r^2}}$
  4. $ T = \dfrac{mgL}{(L^2 - r^2)^2}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In a conical pendulum, T*cos(theta) = mg. From the geometry, cos(theta) = h/L = sqrt(L^2 - r^2)/L. Therefore, T = mg/cos(theta) = mgL / sqrt(L^2 - r^2).