Oscillations help in physics please... best answers will be awarded?

Question

The velocity of a simple harmonic oscillator is given by

v = -8.45sin(26.6t)

a) What is its angular frequency?

b) What is the amplitude of the motion in meters to two decimal places?

c) To the nearest hundredth of a meter where is the mass at the time t=27.84 seconds?

d) If the mass is 0.16 kg, what is the spring s potential energy to the nearest tenth of a joule

e) What is its kinetic energy to the nearest tenth of a joule?

Answer 1

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Needing to assume some units, I'll assume m/s for velocity.

a) the coefficient of t in the trig argument -- here 26.6 rad/s

b) Because velocity is a sin function, displacement was a cos function.

x = Acos(26.6t) and

v = dx/dt = (-A * 26.6rad/s)sin(26.6t)

Therefore -A * 26.6rad/s = -8.45 m/s

A = 0.32 m

c) x = 0.32m*cos(26.6*27.84) → make sure you're in radians mode

x = 0.20 m

d) k = ω²m = (26.6rad/s)² * 0.16kg = 113.21 kg/s²

Presuming we're interested in the point where x = 144.62 m, then

SPE = ½ * 113.21kg/s² * (0.20m)² = 2.4 J

e) v = -8.45m/s*sin(26.6*27.84) = 6.47 m/s

KE = ½ * 0.16kg * (6.47m/s)² = 3.4 J

Bonus: so the total energy is 5.7 J.

Max KE = ½mVmax² = ½ * 0.16kg * (8.45m/s)² = 5.7 J √√√

Max SPE = ½kA² = ½ * 113.21kg/s² * (0.32m)² = 5.7 J √√√

Hope this helps!