Question

In-Class Assignment 2. The figure shows a position-versus-time graph for an oscillating mass m = 0.5 kg. x (cm) 20 10 0 -10 -20 I(s) 4 a. Determine the period of the motion. b. Determnine the angular frecquemcy of the motion c. Determine the amplitude of the motion. d. Determine the phase constant of the motion. e. Determine the maximum speed of the mass. f. Determine the maximum acceleration of the mass. g. Determine the total energy of the system. h. Draw a velocity-versus-time graph for the oscillating mass. i. Draw an acceleration-versus-time graph for the oscillating mass. j. Write an equation for r(t), which describes the position of the oscillating mass k. Write an equation for v(t), which describes the velocity of the oscillating mass. 1. Write an equation for a(t), which describes the acceleration of the oscillating mass.

Answer 1

(a) You can observe the three minimas to see that period is of 4 sec.

(b) $\omega=\frac{2\pi}{T}=\frac{2\pi}{4}=\frac{\pi}{2}rad/sec$

(c) Amplitude of motion is 20-(-20)=40 cm

(d) At time, t=0, the displacement was 10 cm, so,
$x=Asin(\omega t+\phi)$ , substituing the values,
$10=20sin(\frac{\pi}{2} \times 0+\phi)$
or, $\frac{1}{2}=sin(\phi)$
or, $\phi=\frac{\pi}{6}$ , this is the phase constant of the motion.

(e) $v_{max}=A\omega=20\times \frac{\pi}{2}=10\pi=31.4cm/s$

(f) $a_{max}=A\omega^2=20\times \left(\frac{\pi}{2}\right)^2=5\pi^2=49.3cm/s^2$

(g) Total energy of system is $\frac{1}{2}mv_{max}^2=\frac{1}{2}\times 0.5 \times (31.4/100)^2=0.1J$

(h) Plot, $v=31.4cos(\frac{\pi}{2}t+\frac{\pi}{6})$

30 10 4 7 20 -30

(i) Plot $a=-49.3sin(\frac{\pi}{2}t+\frac{\pi}{6})$

40 20 2 4 5 9 20

(j,k,l) I have written all the equations in between the questions.