The figure below shows two examples of SHM, labeled A and B. (The vertical axis

Question

The figure below shows two examples of SHM, labeled A and B. (The vertical axis is marked in increments of 10 m and the horizontal axis is marked in increments of 2.5 s.) . Both of these describe the motion of a mass attached to a spring, that oscillates on a horizontal frictionless surface.

a)What is the angular frequency of the oscillator labelled B ?
b)Write down the equation of motion for the oscillator labelled B and use it to calculate the position at t = 7 s
c)What is the first time after t=0 that the oscillator

Explanation / Answer

a)

frequency,f = 1/T = 1/(2*2.5) = 1/5 = 0.2 Hz

So, angular frequency ,W = 2*pi*f = 2*pi*0.2 = 0.4*pi = 1.26 rad/s <---answer

b)

general equation of motion can be writen as : x = A*cos(Wt)

where A = amplitude of oscillation = 3.5*10 = 35

So, x = 35*cos(1.26t)

So, at t=7s, x = 35*cos(1.26*7) = -28.3 m <-------answer

c)

at minima it has maximum positive acceleration , so, at t = 2.5 s <-------answer

d)

differentiating x,

we get,

v = A*W*sin(Wt) = 35*1.26*cos(Wt)

So, at equilibrium, v = 35*1.26 = 44.1 m/s

e)

conserving energy,

P.E stored in spring = K.E at equilibrium

So, 0.5*kx^2 = 0.5*mv^2

where x = amplitude = 35 m

So, 0.5*k*35^2 = 0.5*2*44.1^2

So, k = 3.18 N/m <------answer

f)

total enegy of system = 0.5k*x^2 = 0.5*3.18*35^2 = 1944.8 J <----------answer

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