The position of a 100 g oscillating mass is given by x ( t ) = (2.0 cm) cos(20 t

Question

The position of a 100 g oscillating mass is given by x(t) = (2.0 cm) cos(20t), where t is in seconds, and the 20 is in rad/s. Determine the following.

(a) The amplitude (in cm).
cm

(b) The period.
s

(c) The spring constant, k. Remember that the spring constant is related to the mass and the frequency (or period).
N/m

(d) The maximum speed. Think about the expression you can write for v(t). Where is the maximum velocity in that expression?
cm/s

(e) The total energy. HINT: Remember that Etot = KE + SPE. You might calculate the KE and SPE energies at some special location, such as at a turnaround point, or at the equilibrium point.
J

(f) The velocity at t = 0.40 s. HINT: Use the information you have found already to write out the expression forv(t). Then evaluate your expression at t = 0.40 s.
cm/s

Explanation / Answer

A) Amplitude is A = 2 cm

B) period T = 2*pi/w = 2*3.142/20 = 0.3142 S

C) spring constant k = m*w^2 = 0.1*20^2 = 40 N/m

D) Vmax = A*w = 0.02*20 = 0.4 m/s = 40 cm/s

V(t) =(-A*W)*sin(20*t)

E) Total energy TE = 0.5*k*A^2 = 0.5*40*0.02^2 = 0.008 J =

F) v(t) = (-A*w)*sin(20*t)

at t = 0.4 S

v(t) = (-0.02*20)*sin(20*0.4) = -0.395 m/s = -39.5 cm/s

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