Question

The function x = (3.0 m) cos[(4π rad/s)t + π/5 rad] gives the simple harmonic motion of a body. Find the following values at t = 3.0 s.(a) the displacementm(b) the velocity (Include the sign of the value in your answer.)m/s(c) the acceleration (Include the sign of the value in your answer.)m/s2(d) the phase of the motionrad(e) the frequency of the motionHz`(f) the period of the motions

Answer 1

Step-by-step explanation:

x = 3 Cos (4πt + π/5)

(a) Substitute t = 3 s

x = 3 Cos (12π + π/5)

x = 3 (0.82) = 2.46 m

(b) Velocity, v = dx/dt

v = 3 (-1)(4π) Sin(4πt + π/5)

Substitute t = 3 s

v = - 37.68 Sin(12π + π/5)

v = - 37.68 (0.572)

v = - 21.55 m/s

(c) Acceleration, a = dv/dt

a = 3 (-1)(4π)(4π) Cos(4πt + π/5)

Substitute t = 3 s

a = - 473.26 (0.82)

a = - 388.21 m/s²

(d) Phase = 4πt + π/5

Substitute t = 3 s

Phase = 12 π + π/5 = 38.308 rad

(e) ω = 4π

so, 2πf = 4π

f = 2 Hz

(f) T = 1 / f = 1 / 2 = 0.5 second