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A particle is oscillating according to the equation $X = 7\cos 0.5\pi t$, where $t$ is in second. The point moves from the position of equilibrium to maximum displacement in time ..... $\sec$
A particle has simple harmonic motion. The equation of its motion is $x = 5\sin \left( {4t - \frac{\pi }{6}} \right)$, where $x$ is its displacement. If the displacement of the particle is $3$ units, then it velocity is
A block is resting on a piston which executes simple harmonic motion with a period $2.0 \,s$. The maximum velocity of the piston, at an amplitude just sufficient for the block to separate from the piston is .......... $ms ^{-1}$
On the superposition of two harmonic oscillations represented by ${x_1} = a\,\sin \,\left( {\omega t + {\phi _1}} \right)$ and ${x_2} = a\,\sin \,\left( {\omega t + {\phi _2}} \right)$ a resulting oscillation with the same time period and amplitude is obtained. The value of ${\phi _1} - {\phi _2}$ is .... $^o$
Two masses $m_1$ and $m_2$ connected by a spring of spring constant $k$ rest on a frictionless surface. If the masses are pulled apart and let go, the time period of oscillation is
A $3\ kg$ sphere dropped through air has a terminal speed of $25\ m/s$. (Assume that the drag force is $-bv$.) Now suppose the sphere is attached to a spring of force constant $k = 300\ N/m$, and that it oscillates with an initial amplitude of $20\ cm$. What is the angular frequencu of its damped $SHM$? ..... $rad/s$
If $< E >$ and $< U >$ denote the average kinetic and the average potential energies respectively of mass describing a simple harmonic motion, over one period, then the correct relation is