5) A 0.5 kg mass on the end of a spring oscillates at 5.0 Hz, with amplitude 10c

Question

5) A 0.5 kg mass on the end of a spring oscillates at 5.0 Hz, with amplitude 10cm, and maximum displacement at t=0.
A) Write the equation for its position in the form: Acos(?t + ?) where A is in meters, ? in rad/s, and ? in radians, with suitable values for each.

B) Write the corresponding equations for its velocity and acceleration as a function of time. C) How long does it take to go from one extreme position to the opposite one?
D) What is its maximum potential energy?
E) What is its maximum speed?

Explanation / Answer

A) from the given data, Amplitude A = 0.1 m

angular frequency, w = 2*pi*f

= 2*pi*5

= 31.4 rad/s

Initial phase angle, phi = 0

x = A*cos(w*t + phi)

= 0.1*cos(31.4*t + 0)

B) v = dx/dt

= -0.1*sin(31.4*t)*31.4

= -3.14*sin(31.4*t)

C) a = dv/dt

= -3.14*cos(31.4*t)*31.4

= -98.6*cos(31.4*t)

d) we know, w = sqrt(k/m)

==> k = w^2*m

= 31.4^2*0.5

= 493 N/m

so, maximum potential energy = 0.5*k*A^2

= 0.5*493*0.1^2

= 2.46 J

e) Vmax = A*w

= 0.1*31.4

= 3.14 m/s

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