Simple Harmonic Motion 5 Question 14
19. Column I describes some situations in which a small object moves. Column II describes some characteristics of these motions. Match the situations in Column I with the characteristics in Column II.
$(2007,6$ M)
Column I
Column II
(A) The object moves on the $x$-axis under a conservative force in such a way that its speed and position
(p) The object executes a simple harmonic motion. satisfy $v=c _1 \sqrt{c _2-x^{2}}$, where $c _1$ and $c _2$ are positive constants.
(B) The object moves on the $x$-axis in such a way that its velocity and its displacement from the origin satisfy $v=-k x$, where $k$ is a positive constant.
(C) The object is attached to one end of a mass-less spring of a given spring constant. The other end of the spring is attached to the ceiling of an elevator. Initially everything is at rest. The elevator starts going upwards with a constant acceleration $a$. The motion of the object is observed from the elevator during the period it maintains this acceleration.
(D) The object is projected from the earth’s surface vertically upwards with a speed $2 \sqrt{\frac{G M _e}{R _e}}$, where $M _e$ is the mass of the earth and $R _e$ is the radius of the earth. Neglect forces from objects other than the earth.
Objective Questions II (One or more correct option)
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Answer:
Correct Answer: 19. $A \rightarrow p \quad B \rightarrow q, r \quad C \rightarrow p \quad D \rightarrow q, r$
Solution:
- (A) Compare with the standard equation of SHM
$$ v=\omega \sqrt{A^{2}-x^{2}} $$
we see that the given motion is SHM with,
$$ \omega=C _1 \text { and } A^{2}=C _2 $$
(B) The equation shows that the object does not change its direction and kinetic energy of the object keeps on decreasing.
(C) A pseudo force (with respect to elevator) will start acting on the object. Its means position is now changed and it starts SHM.
(D) The given velocity is greater than the escapevelocity
$$ =\sqrt{\frac{2 G M _e}{R _e}} . \text { Therefore it keeps on moving } $$
towards infinity with decreasing speed.
