When a body is in S.H.M.,match the following: | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) In $S.H.M.$,the displacement is $x = A \sin \omega t$.Velocity $v = \omega \sqrt{A^2 - x^2}$ and acceleration $a = -\omega^2 x$.$A$. Velocity is m

Vedclass · Accepted Answer

$A-II, B-III, C-IV, D-I$

Vedclass · Answer

(C) In $S.H.M.$,the displacement is $x = A \sin \omega t$.Velocity $v = \omega \sqrt{A^2 - x^2}$ and acceleration $a = -\omega^2 x$.$A$. Velocity is maximum at the mean position $(x = 0)$,so $A-II$.$B$. Kinetic Energy $(K.E.)$ is $\frac{1}{2} m \omega^2 (A^2 - x^2)$. For $K.E. = \frac{3}{4} E_{total} = \frac{3}{4} (\frac{1}{2} m \omega^2 A^2)$,we get $A^2 - x^2 = \frac{3}{4} A^2$,which implies $x^2 = \frac{1}{4} A^2$ or $x = \frac{A}{2}$. So $B-III$.$C$. Potential Energy $(P.E.)$ is $\frac{1}{2} m \omega^2 x^2$. For $P.E. = \frac{3}{4} E_{total} = \frac{3}{4} (\frac{1}{2} m \omega^2 A^2)$,we get $x^2 = \frac{3}{4} A^2$,which implies $x = \frac{\sqrt{3}}{2} A$. So $C-IV$.$D$. Acceleration is maximum at the extreme position $(x = A)$,so $D-I$.Thus,the correct match is $A-II, B-III, C-IV, D-I$.

List-$I$	List-$II$
$A$. Velocity is maximum	$I$. Acceleration is maximum
$B$. $K.E.$ is $\left(\frac{3}{4}\right)^{\text{th}}$ of total energy	$II$. At mean position
$C$. $P.E.$ is $\left(\frac{3}{4}\right)^{\text{th}}$ of total energy	$III$. At half of the amplitude
$D$. Acceleration is maximum	$IV$. At $\frac{\sqrt{3}}{2}$ times the amplitude

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.

Similar Questions

The displacement-time graph of a particle executing $SHM$ is shown. Which of the following statements is wrong?

Two oscillating systems,a simple pendulum and a vertical spring-mass system,have the same time period of motion on the surface of the Earth. If both are taken to the moon,then-

The displacement-time graph of a particle executing $SHM$ is shown. Which of the following statements is/are true?

At $t=0$,a particle executing $SHM$ with a time period $3 \ s$ is in phase with another particle executing $SHM$. The time period of the second particle is $T$ (less than $3 \ s$). If they are again in the same phase for the third time after $45 \ s$,then the value of $T$ is .... . (in $s$)

Vedclass Test Series

Exam Paper Generator

Online Exam Module