Question

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

Answer 1

Answer:

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

Answer 2

Answer:

Explanation:

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.