A mass m oscillates at the end of an ideal, massless spring k, passing through p

Question

A mass m oscillates at the end of an ideal, massless spring k, passing through points PQRSP, and repeating indefinitely. (Neglect all friction, unless told otherwise.)

Let m=0.75 kg, k= 13 N/m, and maximum amplitude of displacement A=18 cm.

A. Find the period of oscillation.
(I found T= 1.51 s. Don't know if that's correct)

B. Find the maximum speed v of the mass.
(I found Vmax= 0.749 m/s. Don't know if that's correct either)

At time t=0, the mass is released at point Q (where x=A and v=0).

C. Find the mass's displacement x at t=2.0 seconds

D. Find the mass's velocity v at t= 2.0 seconds

E. Find the mass's acceleration a at t=2.0 seconds

F. Find the horizontal component of force acting on the mass at t=2.0 seconds

G. Suppose that friction and air resistance cause the amplitude of oscillations to diminish gradually over time, from 18 cm to 9.0 cm. Find the total energy dissipated by friction over this time period.

Explanation / Answer

here w = (k/m)^0.5 = 4.163

A). T = 2*pi/w = 2*3.14/4.163 = 1.51 seconds

B). max speed = Aw = (0.18)(4.163) = 0.7494 m/s

C). here , the equation of displacement

x = Acos(wt) , at t = 2 s

x = 18cos(4.163*2) = -8.195 cm

D). v = dx/dt = -Aw*sin(wt)

at t=2 s , v = 0.667 m/s

E). a = -w^2*x

at t= 2 s , a = -0.1164 m/s^2

F). F = ma

at t = 2 s , F = (0.75)(-0.1164) = -0.0873 N

G). total energy = 0.5*k*A^2

heat dissipated = 0.5*13*(0.18^2 - 0.09^2) {from conservation of energy}

heat dissipated = 0.15795 J

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