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Time period of a simple pendulum is $T$. The time taken to complete $5 / 8$ oscillations starting from mean position is $\frac{\alpha}{\beta} T$. The value of $\alpha$ is ..... .
Two bodies of masses $1\, kg$ and $4\, kg$ are connected to a vertical spring, as shown in the figure. The smaller mass executes simple harmonic motion of angular frequency $25\, rad/s$, and amplitude $1.6\, cm$ while the bigger mass remains stationary on the ground. The maximum force exerted by the system on the floor is ..... $N$ ( take $g = 10\, ms^{-2}$)
When a mass $m$ is attached to a spring, it normally extends by $0.2\, m$. The mass $m$ is given a slight addition extension and released, then its time period will be
A particle is executing $S.H.M.$ and its velocity $v$ is related to its position $(x)$ as $v^2+a x^2=b$, where $a$ and $b$ are positive constants. The frequency of oscillation of particle is ..........
A mass $m$ is suspended from a spring of force constant $k$ and just touches another identical spring fixed to the floor as shown in the figure. The time period of small oscillations is
A solid cylinder of density $\rho_0$, cross-section area $A$ and length $l$ floats in a liquid $\rho(> \rho_0)$ with its axis vertical, as shown. If it is slightly displaced downward and released, the time period will be
A body is performing simple harmonic with an amplitude of $10 \,cm$. The velocity of the body was tripled by air Jet when it is at $5 \,cm$ from its mean position. The new amplitude of vibration is $\sqrt{ x } \,cm$. The value of $x$ is________