Did I Solve the Tray Oscillation Problem Correctly?

[kky1638]

I think I solved the problem but tell me if I made any errors

Homework Statement

A tray of mass m, when suspended from a spring attached to the ceiling, stretches the spring by a distance Δx. A glob of peanut butter with mass M is dropped from a height h onto the tray, which is at rest, and sticks there.

a) What is the spring constant k of the spring in terms of the variables given above?
b) Find the maximum extension of the spring below its initial equilibrium position before the tray had been attached.

Homework Equations

x = Acos(wt + @)
v = -Awsin(wt + @)
w = sqrt (k/mass) etc.

The Attempt at a Solution

(a) Since the spring stretches Δx when tray is attached,
kΔx = mg, k = Δx/g

(b) The butter of mass M drops height h, so the velocity just before it sticks to the tray is
sqrt(2gh). And using conservation of momentum during the very short time period it sticks to the tray, M * sqrt(2gh) = (M + m)Vi, Vi = (M*sqrt(2gh))/(M + m)
Vi can be found from this and it is the v at time 0.
v(0) = -Awsin(@)
0 = x(0) = Acos(@)

A = sqrt((v(0)/-w)^2 + 0^2) = v(0)/w = Vi/w

So the maximum extension is (M*sqrt(2gh))/(M + m) / (sqrt(k/(M+m))) + Δx

Is this correct?
Is there a faster way to do this?

 

[tiny-tim]

Is this correct?
Is there a faster way to do this?

Hi kky1638!

Your v_i is right, but I'm not following what you've done after that.

(why are you using angles?)

Hint: use conservation of energy …

that is, KE + gravitational PE + spring energy.

 

[thomate1]

Your solution appears to be correct. The spring constant k can be expressed as k = (M+m)g/Δx, which is equivalent to your solution of k = Δx/g. As for a faster way to solve this problem, it may depend on your specific approach and what you consider to be "faster." However, one possible approach could be to use the principle of conservation of energy. The initial potential energy of the system is mgh, and after the peanut butter is dropped, the potential energy is converted to kinetic energy of the combined system (tray + peanut butter). This kinetic energy can then be used to find the maximum extension of the spring.