A 1.15-kg mass oscillates according to the equation x = 0.650 cos (8.40t) where x is in meters and t in seconds. Determine (a) the amplitude, (b) the frequency, (c) the total energy, and (d) the kinetic energy and potential energy when x = 0.360 m.

(a) A = 0.650 m

(b) f = 1.3368 Hz

(c) E = 17.1416 J

(d) K = 11.8835 J

U = 5.2581 J

Explanation:

Given

m = 1.15 kg

x = 0.650 cos (8.40t)

(a) the amplitude,

A = 0.650 m

(b) the frequency,

if we know that

ω = 2πf = 8.40 ⇒ f = 8.40 / (2π)

⇒ f = 1.3368 Hz

(c) the total energy,

we use the formula

E = m*ω²*A² / 2

⇒ E = (1.15) (8.40) ² (0.650) ² / 2

⇒ E = 17.1416 J

(d) the kinetic energy and potential energy when x = 0.360 m.

We use the formulas

K = (1/2) * m*ω² * (A² - x²) (the kinetic energy)

and

U = (1/2) * m*ω²*x² (the potential energy)

then

K = (1/2) * (1.15) * (8.40) ² * ((0.650) ² - (0.360) ²)

⇒ K = 11.8835 J

U = (1/2) * (1.15) * (8.40) ² * (0.360) ²

⇒ U = 5.2581 J