$A$ particle is executing simple harmonic motion with an amplitude $A$ and time period $T$. The displacement of the particle after $2T$ time from its initial position is

Why is the periodic time $T = \frac{2\pi}{\omega}$ taken for the function $f(t) = A \cos \omega t$?

$A$ particle is executing $SHM$ about $y=0$ along the $y$-axis. Its position at an instant is given by $y = (7 \, m) \sin(\pi t)$. Its average velocity for the time interval $0$ to $0.5 \, s$ is ........... $m/s$.

$A$ particle is moving with constant angular velocity along the circumference of a circle. Which of the following statements is true?

$A$ particle of mass $m$ is under the influence of a force $F$ which varies with the displacement $x$ according to the relation $F = -kx + F_0$,where $k$ and $F_0$ are constants. The particle,when disturbed,will oscillate:

$A$ point particle is acted upon by a restoring force $F = -k x^3$. The time period of oscillation is $T$ when the amplitude is $A$. The time period for an amplitude $2A$ will be