A particle executes simple hormonic motion between $x =\, -A$ and $x = +A$ . It starts from $x = 0$ moves in $+x-$ direction. The time taken for it to move from $x = 0$ to $x = \frac {A}{2}$ is $T_1$ and to move from $\frac {A}{2}$ to $\frac {A}{\sqrt 2}$ is $T_2$ , then
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Two simple harmonic motions are represented by the equations ${y_1} = 0.1\sin \left( {100\pi t + \frac{\pi }{3}} \right)$ and ${y_2} = 0.1\cos \pi t.$ The phase difference of the velocity of particle $1$ with respect to the velocity of particle $2$ is
Three simple harmonic motions in the same direction having the same amplitude a and same period are superposed. If each differs in phase from the next by ${45^o}$, then
When the displacement of a simple harmonic oscillator is one third of its amplitude, the ratio of total energy to the kinetic energy is $\frac{x}{8}$, where $x=$_____________.
The displacement time graph of a particle executing $S.H.M.$ is given in figure: (sketch is schematic and not to scale) Which of the following statements is are true for this motion?
$(A)$ The force is zero $t=\frac{3 T}{4}$
$(B)$ The acceleration is maximum at $t=T$
$(C)$ The speed is maximum at $t =\frac{ T }{4}$
$(D)$ The $P.E.$ is equal to $K.E.$ of the oscillation at $t=\frac{T}{2}$
A simple harmonic motion is represented by $y\, = 5\,(\sin \,3\pi t\, + \,\sqrt 3 \,\cos \,3\pi t)\,cm$ The amplitude and time period of the motion are
The displacement of a body executing $SHM$ is given by $x = A \sin (2\pi t + \pi /3).$ The first time from $t = 0$ when the velocity is maximum is .... $\sec$
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