A body oscillates with simple harmonic motion according to the following equation. x = (2.5 m) cos[ (6π rad/s) t + π/4 rad] (a) At t = 7.0 s, find the displacement. 1.77 Correct: Your answer is correct. m (b) At t = 7.0 s, find the velocity. - 33.32 Correct: Your answer is correct. m/s (c) At t = 7.0 s, find the acceleration. - 628.1 Correct: Your answer is correct. m/s2 (d) At t = 7.0 s, find the phase of the motion. rad (e) At t = 7.0 s, find the frequency of the motion. Hz (f) At t = 7.0 s, find the period of the motion. s

A. 1.77m

B. - 33.3m/s

C. - 628.27m/s²

D. 2π² or 26°

E. 3Hz

F. 0.33s

Explanation:

Given X = 2.5cos (6πt + π/4)

(a) at t = 7

X = 2.5cos[6π (7) + π/4)

= 2.5cos[42π+π/4] = 2.5cos (169π/4)

= 2.5cos[21*360 + 45]

= 2.5cos (π/4)

= 1.7677m

~=1.77m

Note: π = 180°

Hence 169π/4 = 7605° = [21*360 + 45] = 45°

(B) velocity = dX/dt

dX/dt = - 2.5sin (6πt + π/4) * 6π ...

= - 15πsin[6π (7) + π/4]

= - 15πsin (169π/4)

= - 15πsin (π/4)

= - 33.325

~ = - 33.3m/s

(C) acceleration; a = d²x/dt² = X"

x" = d (dx/dt) / dt

x" = - 15sin (6πt+π/4) * 6π

= - 90π²cos (6π + π/4)

= - 90π²cos (π/4)

= - 628.26m/s²

(D) phase angle = wπT

= (2πf) π * 1/f

= 2π² = 180π = 566° = 360+206

= 206 = 180° + 26°

= 26°

Note π=180°

(E) using the acceleration, a we use the formula:

a = - w²x

w = 2πf

a = - (2πf) ²x

a = - 4π²f²x

f = √ (a/4π²x) = 1 / (2π) √ (a/x)

= 0.1591√ (628.26/1.77)

= 2.998

~ = 3Hz

At t = 7.0, x = 1.77m

(F) T = 1/f = 1/2.998

T = 0.3335s

Given:

Displacement, x = (2.5 m) cos[ (6π rad/s) t + π/4 rad]

A.

At t = 7s,

x = 2.5 * cos (42π + π/4)

= 2.5 * cos (169/4 * π)

= 5/4 * sqrt2

= 1.77 m

B.

dx/dt = v = - (2.5 * 6π) * sin[ (6π rad/s) t + π/4 rad]

= - 15π * sin[ (6π rad/s) t + π/4 rad]

At t = 7s,

= - 15π * sin[ (42π rad/s) t + π/4 rad]

= - 15π * sin (169/4 * π)

= - 15/2 * π * sqrt2

= - 33.32 m/s

C.

dv/dt = a = - (2.5 * (6π) ^2) * cos[ (6π rad/s) t + π/4 rad]

= - 90 * (π) ^2) * cos[ (6π rad/s) t + π/4 rad]

At t = 7s,

= - 90 * (π) ^2) * cos[ (42π rad/s) + π/4 rad]

= - 45 * (π) ^2) * sqrt2

= - 628.1 m/s^2

D.

Comparing,

x = Acos (wt + phil)

With,

x = (2.5 m) cos[ (6π rad/s) t + π/4 rad]

Phase angle, phil = π/4 rad

Since 2π rad = 360°

π/4 rad = 360/8

= 45°

E.

angular velocity, w = 2π/t

= 2π * f

Comparing the above equations,

w = 6π rad/s

Frequency, f = 6π/2π

= 3 Hz

F.

Period, t = 1/f

= 1/3

= 0.33 s.