What is the phase constant for the harmonic oscillator with the position function $\mathrm{x}\left(\mathrm{t}\right)$ given in figure, if the position function has the form $\mathrm{x}={\mathrm{x}}_{\mathrm{m}}\cos (\mathrm{\omega t} +\mathrm{\phi })$? The vertical axis scale is set by ${\mathrm{x}}_{\mathrm{s}}=9.0 \mathrm{cm}$.

The angle of the pendulum of is given by $\theta ={\theta }_{\mathrm{M}}\cos \left[(5.63 \mathrm{rad} {\mathrm{s}}^{-1})t +\phi \right]$. If at $t=0, \theta =0.040 \mathrm{rad}$ and $\frac{\mathrm{d}\theta }{\text{d}t}=-0.200 \mathrm{rad} {\mathrm{s}}^{-1},$ what are

(a) the phase constant $\phi$ and 

(b) the maximum angle ${\mathrm{\theta }}_{\mathrm{M}}$? (hint: Don't confuse the rate $\frac{\mathrm{d\theta }}{dt}$ at which $\mathrm{\theta }$ changes with the $\omega$ of the SHM).

A loudspeaker produces a musical sound by means of the oscillation of a diaphragm whose amplitude is limited to $4 \mu \mathrm{m}$.

(a) At what frequency is the magnitude of $a$ of the diaphragm's acceleration equal to $g$?

(b) For greater frequencies, is $a$ greater than or less than $g$?

A body undergoes simple harmonic motion of amplitude $4.2 \mathrm{cm}$ and period $0.200 \mathrm{s}$. The magnitude of the maximum force acting on it is $10.0 \mathrm{N}$.

(a) What is the mass?

(b) If the oscillations are produced by a spring, what is the spring constant?

In the figure shown below, a physical pendulum consists of a uniform solid disk (of radius $R=2.35\mathrm{cm}$) support in a vertical plane by a pivot located a distance $d=1.50 \mathrm{cm}$ from the centre of the disk. The disk is displaced by a small angle and released. What is the period of the resulting simple harmonic motion?

The overhead view of the arrangement of the wall, the spring and rod system is shown in the figure. A long uniform rod of mass $0.600 \mathrm{kg}$ is free to rotate in a horizontal plane about a vertical axis through its centre. A spring with force constant $k=1530 \mathrm{N} {\mathrm{m}}^{-1}$ is connected horizontally between one end of the rod and a fixed wall. When the rod is in equilibrium, it is parallel to the wall. What is the period of the small oscillations that result when the rod is rotated slightly and released?

For the damped oscillator system shown in the figure shown below with $m=250 \mathrm{g}, k=85 \mathrm{N} {\mathrm{m}}^{-1} \mathrm{and}\mathit{ }b=70 \mathrm{g} {\mathrm{s}}^{-1}$, what is the ratio of the oscillation amplitude at the end of $10$ cycles to the time initial oscillation amplitude?