A box containing a pebble is attached to an ideal horizontal spring and is oscillating on a friction-free air table. When the box has reached its maximum distance from the equilibrium point, the pebble is suddenly lifted out vertically without disturbing the box. Will the following characteristics of the motion increase, decrease, or remain the same in the subsequent motion of the box? Justify your answer.
a. frequency; b. period; c. amplitude; d. maximum kinetic energy of the box; e. maximum speed of the box.

I correctly reasoned that the frequency will increase, and the period will decrease using the relationships about SHM we’ve learned, but I’m having issues understanding the other three.

I know the equation for amplitude has angular frequency in it, and since I know that decreasing the mass will increase the angular frequency, I would think that the amplitude after removing the pebble would be less than the amplitude before, even though intuitively I would think the opposite. The equation I have for A is A = sqrt (x^2 +(v/w)^2) but I don’t really understand this equation – what is v? Just linear velocity? What about on a pendulum – how would that work?

For d., I would of course assume that decreasing the mass in the system would decrease the kinetic energy, but linear conservation of momentum says that it doesn’t, so if the mass decreases, the speed must increase to make the total mechanical energy remain the same. But I don’t understand how removing mass, and therefore weight, from a system doesn’t change its total mechanical energy. Especially if you’ve already determined that your amplitude is going to increase, and you have E = 1/2*k*A^2 : if your amplitude increased, shouldn’t your total mechanical energy increase?

I’m used to figuring if kinetic energy is conserved for collisions, but this isn’t really a collision, so I have no idea what conservation of linear motion applies here.

Thanks!

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