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Multiple choice physics oscillatory motion a few applications of linear shm simple pendulum example of simple harmonic motion

If a body mass $36 gm$ moves with S,H,M of amplitude $A=13$ and period  $T=12 sec$. At a time $t=0$ the displacement is $x=+13 cm$. The shortest time of passage from $x=+6.5$ cm to $x=-6.5$ is

  1. 4 sec

  2. 2 sec

  3. 6 sec

  4. 3 sec
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\begin{array}{l} m=3bg,\, A=13,T=125 \ displacementx\left( t \right) =13\sin  \left( { \frac { { 2\pi t } }{ T }  } \right)  \end{array}$

Shortest time is at maximum slope which crosses zero. It will be from $ - 6.5\,\,to\,\,6.5\,\,$ or 2 times from $0\,to\,\,6.5$
$\begin{array}{l} 6.5=13\sin  \left( { \frac { { 2\pi t } }{ { 12 } }  } \right)  \ 0.5=\sin  \left[ { \left( { \frac { \pi  }{ 6 }  } \right) t } \right]  \ t=1\, \sec   \ total\, \, time=\, 2\times 1=2 \end{array}$

Multiple choice physics oscillatory motion a few applications of linear shm simple pendulum example of simple harmonic motion

A function of time given by $\left(\sin{\omega t}-\cos{\omega t}\right)$ represents

  1. simple harmonic motion

  2. non-periodic motion

  3. periodic but not simple harmonic motion

  4. oscillatory but not simple harmonic motion
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\begin{array}{l} \sin  \omega t-\cos  \omega t \ =\sqrt { 2 } \left[ { \frac { 1 }{ { \sqrt { 2 }  } } \sin  \omega t-\frac { 1 }{ { \sqrt { 2 }  } } \cos  \omega t } \right]  \ =\sqrt { 2 } \left[ { \sin  \omega t\times \cos  \frac { \pi  }{ 4 } -\cos  \omega t\times \sin  \frac { \pi  }{ 4 }  } \right]  \ =\sqrt { 2 } \sin  \left( { \omega t-\frac { \pi  }{ 4 }  } \right)  \ this\, \, function\, \, represents\, \, SHM\, \, as\, \, it\, \, can\, \, be\, \, written\, \, in\, \, the\, \, form: \ a\sin  \left( { \omega t+\phi  } \right)  \ its\, \, period\, \, is,\, \, \frac { { 2\pi  } }{ \omega  }  \end{array}$

Hence,
option $(A)$ is correct answer.

Multiple choice physics oscillatory motion a few applications of linear shm simple pendulum example of simple harmonic motion

A particle is subjected to two simple harmonic motions along $x$ and $y$ directions according to $x=3\sin\ 100\pi t$ $y=4\sin\ 100\pi t$

  1. Motion of particle will be on ellipse travelling in clockwise direction.
  2. Motion of particle will be on a straight line with slope $4/3$
  3. Motion will be simple harmonic motion with amplitude $5$.
  4. Phase difference between two motions is $\pi/2$.
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice physics oscillatory motion a few applications of linear shm simple pendulum example of simple harmonic motion

A ring whose diameter is 1 meter, oscillates simple harmonically in a vertical plane about a nail fixed at its circumference and perpendicular to plane of ring. The time period will be

  1. 1/4 sec

  2. 1/2 sec

  3. 2sec

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For a physical pendulum (ring oscillating about a nail on its circumference), the time period T = 2*pi * sqrt(I / mgd). Here, I = I_cm + md^2 = (1/2)mr^2 + mr^2 = (3/2)mr^2. With d = r, T = 2*pi * sqrt((3/2)mr^2 / mgr) = 2*pi * sqrt(3r / 2g). With diameter 1m, r = 0.5m. T = 2*pi * sqrt(1.5 / 9.8) approx 2 seconds.

Multiple choice physics oscillations a few applications of linear shm simple pendulum example of simple harmonic motion

A person weighing $60\ kg$ stands on a platform which oscillates up and down at a frequency of $2\ Hz$ and amplitude $5\ cm$. The maximum and minimum apparent weights are nearly: ($g$ = 10$\ m/s^2$)

  1. $108$ kg-wt, $12$ kg-wt
  2. $108$ kg-wt, $24$ kg-wt
  3. $54$ kg-wt, $12$ kg-wt
  4. $54$ kg-wt, $24$ kg-wt
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$a=\omega^{2}x$
So, $a _{max}=$ $\omega^{2}A$
We know that $\omega=2\pi  f$
So, $a _{max}=\dfrac{(2\pi\times 2)^{2}\times 5}{100}$
Case I:
$N-mg=ma _{max}$
$N=m(a _{max}+g)$
$=60(10+\dfrac{16\times \pi^{2}\times 5}{100})$
$=1080 $ 

$ N=108$ kg-wt

Case II:
$mg-N=ma _{max}$
or, $N=mg-ma _{max}$
$=60(10-8)$
$=120\ N=12$ kg-wt

Multiple choice physics oscillations a few applications of linear shm simple pendulum example of simple harmonic motion

A body of mass $0.5$ kg is performing S.H.M. with a time period $\pi /2$ seconds. If its velocity at mean position is $1$ m/s, the restoring force acts on the body at a phase angle $60^o$ from extreme position is

  1. 0.5 N

  2. 1 N

  3. 2 N

  4. 4 N
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$T=\dfrac{ \pi}{2}$
$V _{max}=1 m/sec$
$\omega =\dfrac{2\pi}{T}$
$V=4  rad/sec$
$V _{max=}A\omega$
$A\times 4=1$
$A=\dfrac {1}{4}$
$a=\omega^{2}x$
$F= m\omega^{2}x$
$x=A   cos   60^o$
$\therefore x=\dfrac {A}{2}$
$F=0.5\times (4)^{2}\times \dfrac {1}{4}\times \dfrac{1}{2}$
$F=1N$

Multiple choice physics oscillations a few applications of linear shm simple pendulum example of simple harmonic motion

Assertion : If a block is in SHM, and a new constant force acts in the direction of change, the mean position may change.
Reason :In SHM only variable forces should act on the body, for example spring force.

  1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

  2. Both Assertion and Reason are correct and Reason is not the correct explanation for Assertion

  3. Assertion is correct and Reason is incorrect

  4. Assertion is incorrect and Reason is correct

Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In SHM a constant force brings no effective change in the motion but the mean position accelerates.

Multiple choice physics oscillatory motion a few applications of linear shm simple pendulum example of simple harmonic motion

Sitar maestro Ravi Shankar is playing sitar on its strings, and you, as a physicist (unfortunately without musical ears!), observed the following oddities.
I. The greater the length of a vibrating string, the smaller its frequency.
II. The greater the tension in the string, the greater is the frequency.
III. The heavier the mass of the string, the smaller the frequency.
IV. The thinner the wire, the higher its frequency.
The maestro signalled the following combination as correct one.

  1. II, III and IV

  2. I, II and IV

  3. I, II and III

  4. I, II, III and IV
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Guitar string is a standing wave as both the ends of guitar string are fixed.

$f \propto \cfrac{1}{length}$
$f \propto \sqrt{tension}$
$f \propto \cfrac{1}{\sqrt{\mu}}$, where $\mu$ is mass per unit length.
$f \propto \cfrac{1}{\text{thickness of wire}}$

Multiple choice physics oscillatory motion a few applications of linear shm simple pendulum example of simple harmonic motion

Three similar oscillators, A, B, C have the same small damping constant $r$, but different natural frequencies $\omega _0 = (k/m)^{\frac{1}{2}} : 1200 Hz, 1800 Hz, 2400 Hz$. If all three are driven by the same source at $1800 Hz$, which statement is correct for the phases of the velocities of the three?

  1. $\phi _A = \phi _B = \phi _c$
  2. $\phi _A < \phi _B = 0 < \phi _c$
  3. $\phi _A > \phi _B = 0 > \phi _c$
  4. $\phi _A > \phi _B > 0 > \phi _c$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$tan \phi = \dfrac{\omega L- \dfrac{1}{\omega C}}{R}$

If $\omega < \omega _0 \Longrightarrow$ circuit is capacitive

$\Longrightarrow$ it leads voltage
$\Longrightarrow$ velocity leads force.
if $\omega = \omega _0 \phi = 0$
if $\omega > {\omega} _0$ velocity lags behind force.