Which of the following equation does not represent a SHM?

• cosωt + sinωt

• sinωt $-$ cosωt

• 1 $-$ sin2ωt

• sinωt + cos (ωt + $\mathrm{\alpha }$ )

In simple harmonic motion, loss of kinetic energy is proportional to

• e^x

• x³

• logx

• x²

Five sinusoidal waves have the same frequency 500 Hz but their amplitudes are in the ratio 2 : $\frac{1}{2} : \frac{1}{2} : 1 : 1$ and their phase angles 0, $\frac{\mathrm{\pi }}{6}, \frac{\mathrm{\pi }}{3}, \frac{\mathrm{\pi }}{2} \mathrm{and} \mathrm{\pi }$ respectively. The phase angle of resultant wave obtained by the superposition of these five waves is

• 30^o

• 45^o

• 60^o

• 90^o

A  particle  executes  simple  harmonic  motion  of  period T  and amplitude  l along  a  rod  AB  of  length  2l. The  rod  AB  itself executes  simple  harmonic  motion of the same period and  amplitude  in  a  direction  perpendicular  to  its  length.  Initially, both the  particle  and the  rod  are  in  their  mean  positions. The  path  traced  out  by  the  particle  will  be 

• a circle of radius l

• a straight line inclined at $\frac{\mathrm{\pi }}{4}$ to the rod

• an ellipse

• a figure of eight

A ball is suspended  by  a  thread  of  length  L  at  the  point  O  on  a  wall  which  is  inclined  to  the  vertical  $\mathrm{\alpha }$. The  thread with  the ball is  displaced  by  a  small  angle  β  away  from  the  wall. If  the  ball  is  released, the  period  of  oscillation  of  the  pendulum  when   β > $\mathrm{\alpha }$ will be

• $\sqrt{\frac{\mathrm{L}}{\mathrm{g}}} \left[\mathrm{\pi } + 2 {\sin }^{-1} \frac{\mathrm{\alpha }}{\mathrm{\beta }}\right]$

• $\sqrt{\frac{\mathrm{L}}{\mathrm{g}}} \left[\mathrm{\pi } - 2 {\sin }^{-1} \frac{\mathrm{\beta }}{\mathrm{\alpha }}\right]$

• $\sqrt{\frac{\mathrm{L}}{\mathrm{g}}} \left[2 {\sin }^{-1} \frac{\mathrm{\beta }}{\mathrm{\alpha }} - \mathrm{\pi }\right]$

• $\sqrt{\frac{\mathrm{L}}{\mathrm{g}}} \left[ {\sin }^{-1} \frac{\mathrm{\alpha }}{\mathrm{\beta }} + \mathrm{\pi } \right]$

186.

Assertion:  The bob of a simple pendulum is a ball full of water, if a fine hole is made in the bottom of the ball, the time period first increases and then decreases.  

Reason: As water flows out of the bob the weight of bob decreases.

• If both assertion and reason are true and reason is the correct explanation of assertion.

• If both assertion and reason are true but reason is not the correct explanation of assertion.

• If assertion is true but reason is false.

• If both assertion and reason are false.

Assertion:  In the relation f = $\frac{1}{2\mathrm{l}} \sqrt{\frac{\mathrm{T}}{\mathrm{\mu }}}$, where symbols have standard meaning, μ represents linear mass density.

Reason: The frequency has the dimensions of inverse of time.

• If both assertion and reason are true and reason is the correct explanation of assertion

• If both assertion and reason are true but reason is not the correct explanation of assertion

• If assertion is true but reason is false

• If both assertion and reason are false

Assertion: In an SHM, kinetic and potential energies equal when the displacement is 1/$\sqrt{2}$ times the amplitude.

Reason:  In SHM, kinetic energy is zero when potential energy is maximum.

• If both assertion and reason are true and reason is the correct explanation of assertion

• If both assertion and reason are true but reason is not the correct explanation of assertion

• If assertion is true but reason is false

• If both assertion and reason are false

A large horizontal surface moves up and down in S.H.M with an amplitude of 1 cm. If  a mass of 10 kg ( which is placed on the surface) is to remain continuously in contact 

• 5 Hz

• 0.5 Hz

• 1.5 Hz

• 10 Hz

A boat at anchor is rocked by waves whose crests are 100 m apart and velocity is 25 m/sec. The boat bounces up once  in  every

• 2500 s

• 75 s

• 4 s

• 0.25 s