A mass on a vertical spring begins its motion at rest at $y = 0\ cm$. It reaches a maximum height of $y = 10\ cm$. The two forces acting on the mass are gravity and the spring force. The graph of its kinetic energy ($KE$) versus position is given below. Net force on the mass varies with $y$ as
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A block is placed on a frictionless horizontal table. The mass of the block is m and springs are attached on either side with force constants ${K_1}$ and ${K_2}$. If the block is displaced a little and left to oscillate, then the angular frequency of oscillation will be
A pendulum suspended from the ceiling of a train has a period $T$ when the train is at rest. When the train travels same distance per unit time, the period of oscillation is
On a frictionless horizontal plane, a bob of mass $m=0.1 kg$ is attached to a spring with natural length $l_0=0.1 m$. The spring constant is $k_1=0.009 Nm ^{-1}$ when the length of the spring $I > l_0$ and is $k_2=0.016 Nm ^{-1}$ when $ I < l_0$. Initially the bob is released from $l=0.15 m$. Assume that Hooke's law remains valid throughout the motion. If the time period of the full oscillation is $T=(n \pi) s$, then the integer closest to $n$ is. . . . .
For a particle executing $S.H.M.,\, x =$ displacement from equilibrium position, $v =$ velocity at any instant and $a =$ acceleration at any instant, then
Equations ${y_1} = A\sin \omega t$ and ${y_2} = \frac{A}{2}\sin \omega t + \frac{A}{2}\cos \omega t$ represent $S.H.M.$ The ratio of the amplitudes of the two motions is
At time $t = 0$, a simple harmonic oscillator is at its extreme position. If it covers half of the amplitude distance in $1\, second$, then the time period of oscillation is ..... $s$
A large horizontal surface moves up and down in $SHM$ with an amplitude of $1 \,cm$. If a mass of $10\, kg$ (which is placed on the surface) is to remain continually in contact with it, the maximum frequency of $S.H.M.$ will be ... $Hz$
Two, spring $P$ and $Q$ of force constants $k_p$ and ${k_Q}\left( {{k_Q} = \frac{{{k_p}}}{2}} \right)$ are stretched by applying forces of equal magnitude. If the energy stored in $Q$ is $E$, then the energy stored in $P$ is
If $x=5 \sin \left(\pi t+\frac{\pi}{3}\right) \mathrm{m}$ represents the motion of a particle executing simple harmonic motion, the amplitude and time period of motion, respectively, are