Question

The amplitude of an oscillator is 8 cm and it completes 100 oscillations in 80s.
i) Calculate its time period and angular frequency. ii) If the initial phase is p/4, write
expressions for its displacement and velocity. iii) Calculate the values of maximum
velocity and acceleration

Answer 1

Answer on Question #73029 Physics / Mechanics | Relativity

The amplitude of an oscillator is $x_{\mathrm{max}} = 8$ cm and it completes $N = 100$ oscillations in $\tau = 80$ s. i) Calculate its time period $T$ and angular frequency $\omega$. ii) If the initial phase is $\varphi_0 = \pi /4$, write expressions for its displacement $x(t)$ and velocity $v(t)$. iii) Calculate the values of maximum velocity $v_{\mathrm{max}}$ and acceleration $a_{\mathrm{max}}$.

Solution:

The period of oscillation

$$T = \frac {\tau}{N} = \frac {80}{100} = 0.8 \mathrm{~s}$$

The angular frequency

$$\omega = \frac {2 \pi}{T} = \frac {2 \pi}{0.8} = 2.5 \pi \mathrm{~rad/s}$$

The oscillator displacement

$$x (t) = x _ {\max } \cos (\omega t + \varphi_ {0}) = 8 \cos \left(2.5 \pi t + \frac {\pi}{4}\right) \mathrm{~cm}$$

The velocity

$$v (t) = x ^ {\prime} (t) = 20 \pi \sin \left(2.5 \pi t + \frac {\pi}{4}\right) \frac {\mathrm{~cm}}{\mathrm{~s}}$$

The acceleration

$$a (t) = v ^ {\prime} (t) = 50 \pi^ {2} \cos \left(2.5 \pi t + \frac {\pi}{4}\right) \mathrm{~cm/s^2}$$

The maximum velocity

$$v _ {\max } = 20 \pi \frac {\mathrm{~cm}}{\mathrm{~s}} = 62.8 \frac {\mathrm{~cm}}{\mathrm{~s}}$$

The maximum acceleration

$$a _ {\max } = 50 \pi^ {2} \frac {\mathrm{~cm}}{\mathrm{~s^2}} = 493 \frac {\mathrm{~cm}}{\mathrm{~s^2}}$$

Answers:

i) 0.8 s, 2.5π rad/s

ii) $8\cos \left(2.5\pi t + \frac{\pi}{4}\right)\mathrm{cm}, 20\pi \sin \left(2.5\pi t + \frac{\pi}{4}\right)\frac{\mathrm{cm}}{\mathrm{s}}$

iii) $62.8 \frac{\mathrm{cm}}{\mathrm{s}}, 493 \frac{\mathrm{cm}}{\mathrm{s}^2}$

Answer provided by https://www.AssignmentExpert.com