Question

The equation of SHM is given by y= 10 sin (2pi t /45 + alpha) if the displacement is 5 cm at t=0 ,then the total phase at t=7.5 s will be

Answer 1

Simple Harmonic Motion

A Simple Harmonic Motion is one which has some characteristics:

• A Restoring Force acts on the Oscillating Body.

• The Restoring Force is always directed towards the Equilibrium Position.

• The Magnitude of the Restoring Force is proportional to the displacement from the Equilibrium Position.

If we consider the body to be oscillating along Y direction, then we can represent its motion with time with a sinusoidal equation, as:

$\large\boxed{y = A\sin(\omega t + \phi)}$

Here,

y = Position of body

A = Amplitude

$\omega$ = Angular Frequency

t = Time

$\phi$ = Phase Constant.

The term inside the sine, i.e. $(\omega t + \phi)$ is known as Phase.

$\rule{300}{1}$

Question:

The equation of SHM is given by $y=10\sin\left(\frac{2\pi}{45}t+\alpha\right)$. If the displacement is 5 cm at t = 0, then the total phase at t = 7.5 s will be -

Answer:

The SHM Equation is given:

$y = 10\sin\left(\dfrac{2\pi}{45}t+\alpha\right)$

We can find $\alpha$ by using the piece of information given to us: The Displacement is y = 5 cm at t = 0 s.

Put y = 5 and t = 0.

$\implies 5 = 10\sin\left(\dfrac{2\pi}{45} \times 0 + \alpha \right) \\\\\\ \implies 5 = 10\sin(0+\alpha) \\\\\\ \implies \sin\alpha = \dfrac{1}{2} \\\\\\ \implies \alpha = \dfrac{\pi}{6}$

Thus, the equation for the given SHM is:

$y = 10\sin\left(\dfrac{2\pi}{45}t+\dfrac{\pi}{6}\right)$

Thus, the Phase is:

$\text{Phase} = \dfrac{2\pi}{45}t+\dfrac{\pi}{6}$

We need the Phase at t = 7.5 s. We can calculate it easily.

$\displaystyle\text{Phase} = \frac{2\pi}{45}t+\frac{\pi}{6} \\\\\\ \implies \text{Phase} = \frac{2\pi}{45} \times 7.5 + \frac{\pi}{6} \\\\\\ \implies \text{Phase} = \frac{\pi}{3}+\frac{\pi}{6} \\\\\\ \implies \text{Phase} = \frac{3\pi}{6} \\\\\\ \implies \Large\boxed{\boxed{\text{Phase} = \frac{\pi}{2}}}$

Thus, The Total Phase at t = 7.5 s is $\frac{\pi}{2}$.