Exercises

Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial ( $\mathrm{t}=0$) position of the particle, the radius of the circle, and the angular speed of the rotating particle. For simplicity, the sense of rotation may be fixed to be anticlockwise in every case: ($\mathrm{x}$ is in $\mathrm{cm}$ and $\mathrm{t}$ is in $\mathrm{s}$).
$\mathrm{x}=3\sin \left(2\mathrm{\pi t}+\mathrm{\pi }/4\right)$

Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial ( $\mathrm{t}=0$) position of the particle, the radius of the circle, and the angular speed of the rotating particle. For simplicity, the sense of rotation may be fixed to be anticlockwise in every case: ($\mathrm{x}$ is in $\mathrm{cm}$ and $\mathrm{t}$ is in $\mathrm{s}$).
$\mathrm{x}=2\cos \left(\mathrm{\pi t}\right)$ 

Figure (a) shows a spring of force constant $\mathrm{k}$ clamped rigidly at one end and a mass $\mathrm{m}$ attached to its free end. A force $\mathrm{F}$ applied at the free end stretches the spring. Figure (b) shows the same spring with both ends free and attached to a mass $\mathrm{m}$ at either end. Each end of the spring in figure (b) is stretched by the same force $\mathrm{F}$.

 If the mass in Fig. (a) and the two masses in Fig. (b) are released, what is the period of oscillation in each case?

The motion of a simple pendulum is approximately simple harmonic for small angle oscillations. For larger angles of oscillation, a more involved analysis shows that T is greater than $2\pi \sqrt{\frac{l}{g}}$. Think of a qualitative argument to appreciate this result.

A man with a wristwatch on his hand falls from the top of a tower. Does the watch give correct time during the free fall? 

What is the frequency of oscillation of a simple pendulum mounted in a cabin that is freely falling under gravity? 

A body describes simple harmonic motion with an amplitude of $5 \mathrm{cm}$ and a period of $0.2 \mathrm{s}.$ Find the acceleration and velocity of the body when the displacement is $3 \mathrm{cm}$.

A body describes simple harmonic motion with an amplitude of $5 \mathrm{cm}$ and a period of $0.2 \mathrm{s}.$ Find the acceleration and velocity of the body when the displacement is  $0 \mathrm{cm}.$