In class we examined an oscillating system that consisted of a massive cart atta

Question

In class we examined an oscillating system that consisted of a massive cart attached to two opposing springs. I stated that the motion of the cart was simply harmonic when it was set to oscillating. However, I did not prove that statement. Please prove that the motion of cart in that situation is simple harmonic motion. Due Monday. No late work accepted. The picture: The picture below shows the cart at its equilibrium position x' at rest. The unstretched lengths of each spring are x_1 and x_2 and the spring constant of each spring is k_1 and k_2 respectively. The picture below shows the cart away from its equilibrium position x. Using Newton's 2^nd Law F_Net = sigma_all i F_ , Hooke's Law F_ = -k middot delta x, and a fair bit of simple algebra prove that the motion of the cart must be simple harmonic motion. Remember...the criterion for simple harmonic motion is that the net force on the object must be restoring and proportional to its position away from equilibrium.

Explanation / Answer

when the cart is displaced by x to the right
force due to right spring = Fr = -k2*x ( towards left)
force due to left spring, Fl = -k1*x ( towards left)
Net force = F = Fl + Fr = -(k1 + k2)x [ towards left]
for mass of body m, acceleration at any time t, from newtons second la
ma = -(k1 + k2)x
this is just like the spring mass equaiton
ma = -kx

where k = (k1 + k2)
hence this will definitely follow SHM with angular frequency, w = sqroot((k1 + k2)/m)

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