Question

A spring of stiffness 2o KN/m supports a mass of 4 Kg. the mass is pulled down 8 mm and released to produce linear oscillations. calculate the frequency and periodic time. sketch the graphs of displacement, velocity and acceleration. calculate the displacement, velocity and acceleration 0.05 s after released.

note X= A sin(w t + phase angle)

X : displacement
A : amplitude
w : omega
t : periodic time
ceta : phase angle

Answer 1

Answer on Question #45776, Physics, Mechanics | Kinematics | Dynamics

A spring of stiffness 20 KN/m supports a mass of 4 Kg. the mass is pulled down 8 mm and released to produce linear oscillations. calculate the frequency and periodic time. sketch the graphs of displacement, velocity and acceleration. calculate the displacement, velocity and acceleration 0.05 s after released.

note X = A sin(w t + phase angle)

X : displacement

A : amplitude

w : omega

t : periodic time

ceta : phase angle

$$\omega = \sqrt {\frac {k}{m}} = \sqrt {\frac {20 * 10^{3} N / m}{4 k g}} = 10 \sqrt {50} \, s^{-1} \approx 70.7 \, s^{-1}$$ $$A = 8 \, \text{mm} = 8 \cdot 10^{-3} \, \text{m}$$ $$\varphi (\text{phase angle}) = - \frac {\pi}{2}$$

Phase angle take in such way, that at the moment $(t=0)$ we have displacement equal to $-A$.

$$X = 8 \cdot 10^{-3} \, \text{m} \cdot \sin \left(70.7 \, s^{-1} \cdot t - \frac {\pi}{2}\right)$$ $$\begin{array}{l} \frac {d X}{d t} = V = 70.7 \, s^{-1} \cdot 8 \cdot 10^{-3} \, \text{m} \cdot \cos \left(70.7 \, s^{-1} \cdot t - \frac {\pi}{2}\right) \\ = 70.7 \, s^{-1} \cdot 8 \cdot 10^{-3} \, \text{m} \cdot \cos \left(70.7 \, s^{-1} \cdot t - \frac {\pi}{2}\right) \\ = 0.5656 \cdot \cos \left(70.7 \, s^{-1} \cdot t - \frac {\pi}{2}\right) \, \text{m/s} \end{array}$$ $$\frac {d V}{d t} = a = - 0.5656 \cdot 70.7 \, s^{-1} \cdot \sin \left(70.7 \, s^{-1} \cdot t - \frac {\pi}{2}\right) \, \text{m/s} = - 40 \cdot \sin \left(70.7 \, s^{-1} \cdot t - \frac {\pi}{2}\right) \frac {\text{m}}{s^{2}}$$

X, m

V, m|s

a, m|s^2

And

$$X(t = 0.05\,s) = 0.00738886607\,m$$

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