(10 marks) A 15 kg mass attached to a spring undergoes simple harmonic motion wi

Question

(10 marks) A 15 kg mass attached to a spring undergoes simple harmonic motion with an amplitude of A = 0.05 m. The spring constant is 20 N/m, and the system is at the equilibrium position x = 0 m at t = 0 s but moving in the positive x-direction (to the right). Calculate (a) the displacement, velocity, and acceleration as a function of time, the maximum value of its speed and maximum acceleration, (c) the speed and magnitude of acceleration when the mass is at x = 0.04 m, and the time it takes for the mass to move from x = 0 to x = 0.04 m. (e) What is the period of oscillation?

Explanation / Answer

k = m*w^2 = 20

w = sqrt(k/m) = sqrt(20/15) = 1.15 rad/s

a. apply

x = A*sin(wt)

v = dx/dt = w*A*cos(wt) = w*sqrt(A^2-x^2)

a = d/dt = -w^2*A*sinwt = -w^2*x

---------------

b)

speed is maximum when cos(wt) is maximum

maximum value of coswt = 1

maximum speed = vmax = w*A    <<---answer

acceleration is maximum when sin(wt) is maximum

maximum value of sinwt = 1

maximum acceleration = amax = w^2*A <<---answer

++++++++++++++

c)

v = 1.15*Sqrt(0.05^2-0.04^2) = 0.0345 m/s    <<---asnwer

a = -w^2*x = 1.15^2*0.04 = 0.0529 m/s^2    <<---asnwer

d)

from x = A*sinwt)

0.04 = 0.05*sin(1.15*t)

t = 0.806 s <<---asnwer

+++++++++++++++

e)

T = 2*pi/w = (2*3.14)/1.15) = 5.46 s

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