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(B) The frequency of the spring-based clock $S$ is given by $f_S = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$. This frequency depends only on the spring const

Vedclass · Accepted Answer

$P$ will run faster than $S$

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(B) The frequency of the spring-based clock $S$ is given by $f_S = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$. This frequency depends only on the spring constant $k$ and mass $m$,and is independent of the acceleration due to gravity $g$.The frequency of the pendulum-based clock $P$ is given by $f_P = \frac{1}{2\pi} \sqrt{\frac{g}{l}}$. This frequency depends on the acceleration due to gravity $g$.The acceleration due to gravity is $g = \frac{GM}{R^2} = \frac{G}{R^2} \cdot \rho \cdot \frac{4}{3} \pi R^3 = \frac{4}{3} \pi \rho G R$.Since the density $\rho$ remains the same and the radius $R$ is doubled,the new acceleration due to gravity $g'$ becomes $2g$.Consequently,the frequency of the pendulum clock $P$ becomes $f_P' = \sqrt{2} f_P$,while the frequency of the spring clock $S$ remains unchanged.Therefore,the pendulum clock $P$ will run faster than the spring clock $S$.

Column $I$	Column $II$
$(A)$ The object moves on the $x$-axis under a conservative force such that its speed $v = c_1 \sqrt{c_2 - x^2}$,where $c_1, c_2 > 0$.	$(p)$ The object executes simple harmonic motion.
$(B)$ The object moves on the $x$-axis such that its velocity $v = -kx$,where $k > 0$.	$(q)$ The object does not change its direction.
$(C)$ An object is attached to a spring in an elevator accelerating upwards with constant acceleration $a$. The motion is observed from the elevator.	$(r)$ The kinetic energy of the object keeps on decreasing.
$(D)$ The object is projected vertically upwards with speed $2 \sqrt{GM_e / R_e}$.	$(s)$ The object can change its direction only once.

$A$ clock $S$ is based on the oscillation of a spring,and a clock $P$ is based on pendulum motion. Both clocks run at the same rate on Earth. On a planet having the same density as Earth but twice the radius,which of the following is true?

Similar Questions

$A$ man weighing $60\ kg$ stands on the horizontal platform of a spring balance. The platform starts executing simple harmonic motion of amplitude $0.1\ m$ and frequency $\frac{2}{\pi}\ Hz$. Which of the following statements is correct?

$A$ body executing simple harmonic motion has a maximum acceleration equal to $24 \, m/s^2$ and a maximum velocity equal to $16 \, m/s$. The amplitude of the simple harmonic motion is:

$Assertion :$ In simple harmonic motion,the velocity is maximum when the acceleration is minimum.
$Reason :$ Displacement and velocity of $S.H.M.$ differ in phase by $\frac{\pi }{2}$.

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