You are holding a shallow circular container of radius $R$,filled with water to a height $h$ $(h \ll R)$. When you walk with speed $v$,it is seen that water starts spilling over. This happens due to the resonance of the periodic impulse given to the container (due to walking) with the oscillation of the water in the container. If the time period of water oscillating in the container is inversely proportional to $\sqrt{h}$,then $v$ is proportional to
- A$R$
- B$\sqrt{R}$
- C$1 / \sqrt{R}$
- D$1 / R$
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Column $I$ describes some situations in which a small object moves. Column $II$ describes some characteristics of these motions. Match the situation in Column $I$ with the characteristics in Column $II$.
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.
| 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. |
Two particles $P$ and $Q$ describe simple harmonic motions of same period,same amplitude,along the same line about the same equilibrium position $O.$ When $P$ and $Q$ are on opposite sides of $O$ at the same distance from $O,$ they have the same speed of $1.2 \, m/s$ in the same direction. When their displacements are the same,they have the same speed of $1.6 \, m/s$ in opposite directions. The maximum velocity in $m/s$ of either particle is
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 $s$)
Two pendulums with identical bobs and lengths are suspended from a common support such that in the rest position the two bobs are in contact. After being displaced by $5^o$,the bob $A$ is released from rest at $t = 0$. Subsequently,it collides elastically head-on with the other bob $B$. Identify the graph showing the variation in energy of pendulum $A$ with time for $0 \leqslant t \leqslant T$ (where $T$ is the period of either pendulum).
Medium
View SolutionObtain the ratio of maximum acceleration and maximum velocity of a $SHM$ particle.
Medium
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