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If a block B is attached to a spring and oscillates with amplitude $A_1$. Another block A is added on the block B at equilibrium phase.

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At equilibrium, the net force acting on the block B is zero. So by conserving momentum, we get

$m_1v_1=(m_1+m_2)v_2$ Using the formula $v=A\omega$ and $\omega^2=\frac{k}{m}$. We can get $A_2=\sqrt{\frac{m_1}{m_1+m_2}}A_1$. So amplitude has decreased. The net energy of spring block system $E=0.5kA^2$. So the net energy has essentially decreased. Where did the energy go?

Qmechanic
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Ziaul Hasan Hamim
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1 Answers1

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The energy difference arises because block A needs to be accelerated to match the speed of block B.

I've assembled a list of a couple dozen similar questions, of which the best match describes sand being dropped on a conveyor belt. It may be useful to review these to see how dissipative processes necessarily arise when suddenly attaching a load to a system with a restoring force.

One can't assume that both blocks are perfectly rigid and that the attachment is immediate and perfect, as an unphysical infinite acceleration would be required.

(Isn't idealization of perfect rigidity reasonable as a limiting case? Often, yes, but not in this case. Idealization works when a certain parameter is close to zero and reduces to zero when increasing a stiffness to infinity, say. But here, the dissipative aspect never reduces to zero.)

Let's consider some examples.

One could posit sliding as block A gets up to speed, for example; this would turn some bulk kinetic energy into frictional heat.

Even without sliding, both blocks must deform over a finite time, essentially composing additional internal spring-mass systems. The blocks flex and oscillate. One could assume that this internal oscillation persists, in which case one would have to add that kinetic energy to the total energy, making up the energy difference in question.

In reality, damping occurs in the flexing blocks due to mechanical hysteresis, also termed internal friction. Here again, bulk kinetic energy is converted to thermal energy.

As a result, there's no avoiding the need to consider the kinetic and thermal aspects of a sudden load or "jolt," as described in the examples linked above.

Chemomechanics
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