Answer to Question #245710 in Calculus for luka

Question

A body oscillates with simple harmonic motion according to the equation x(t)=5sin(4pie.t+pie/4

(x in meters). At time t = 5 seconds what are : /(6 x

2.5mrks)

(a) the displacement (b) the velocity

(c) the acceleration (d) the phase of the motion

(e) the frequency (f) the period of the motion.

Answer 1

(a)

$$x(5)=5\sin(4\pi(5)+\dfrac{\pi}{4})=2.5\sqrt{2}\ m$$

(b)

$$v(t)=20\pi\cos(4\pi t+\dfrac{\pi}{4})$$

$$v(5)=20\pi\cos(4\pi (5)+\dfrac{\pi}{4})=10\pi\sqrt{2}\ m/s$$

(c)

$$a(t)=-80\pi^2\sin(4\pi t+\dfrac{\pi}{4})$$

$$a(5)=-80\pi^2\sin(4\pi (5)+\dfrac{\pi}{4})=-40\pi^2\sqrt{2}\ m/s^2$$

(d)

$$\varphi(t)=4\pi t+\dfrac{\pi}{4}$$

$$\varphi(5)=4\pi (5)+\dfrac{\pi}{4}=\dfrac{81\pi}{4} rad$$

(e)

$$f=\dfrac{4\pi}{2\pi}=2\ s^{-1}$$

(f)

$$T=\dfrac{1}{f}=\dfrac{1}{2}\ s$$