Question

do it with explanation​

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 $\frac{\pi}{2}$.

Answer 2

☘️ See the attachment picture for Explanation.

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