Platform on Springs: Find Effective Spring Constant & Oscillation Amplitude

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The discussion focuses on calculating the effective spring constant of a platform supported by four springs after a clay mass is dropped onto it. The platform, initially at rest, oscillates after the clay sticks to it, settling 6 cm below its original position. To find the effective spring constant, the force required to compress the springs is determined using the formula F/(delta x), where delta x is the net displacement. The initial velocity of the clay/platform combination post-collision is derived from conservation of momentum principles. The calculations and reasoning behind these steps are central to solving the problem effectively.
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A platform of mass 0.8 kg is supported on four springs (only two springs are shown in the picture, but there are four altogether). A chunk of modeling clay of mass 0.6 kg is held above the table and dropped so that it hits the table with a speed of v = 0.9 m/s.

The clay sticks to the table so that the the table and clay oscillate up and down together. Finally, the table comes to rest 6 cm below its original position.


a) What is the effective spring constant of all four springs taken together?

b) With what amplitude does the platform oscillate immediately after the clay hits the platform?
 
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Can you show us what work and/or ideas you have put forth so far?
 
a) What is the net displacement Delta x of the spring from its former equilibrium position? What force F was required to compress the springs? The effective spring constant is the ratio F/(delta x).
 
The first thing in this problem is a conservtion of momentum. Assume the collision is instantaneous, so there are no net forces to worry about. What will be the velocity of the clay/platform combination immediately after collision?
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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