For lunch, you and your friends decide to stop at the nearest deli and have a sa

Question

For lunch, you and your friends decide to stop at the nearest deli and have a sandwich made fresh for you with 0.113 kg of turkey. The slices of turkey are weighed on a plate of mass 0.420 kg and placed atop a vertical spring of negligible mass, with a force constant of k = 220 N/m. The slices of turkey are dropped on the plate all at the same time from a height of 0.170 m. They make a totally inelastic collision with the plate and set the scale into vertical simple harmonic motion (SHM). You may assume that the collision time is extremely small.
Once the turkey is on the plate, the scale oscillates around a new equilibrium position.

How far from the new equilibrium position was the plate when the slices of turkey landed on it?

Find the velocity of the turkey immediately after the collision.

Using your previous result, calculate the total mechanical energy of the system (turkey and plate) after the collision, by adding the kinetic energy to the elastic potential energy determined by the deviation from the new equilibrium position. Ignore gravitational potential energy. Remember that a vertical SHM is equivalent to a horizontal SHM with the vertical equilibrium position redefined.

Use the total mechanical energy to calculate the amplitude of the oscillation of the turkey.

Explanation / Answer

a) kx = mg

initially 210x1 = 0.350*9.81

x1 = 0.0164 m

finally , 210x2 = (0.350 + 0.103)*9.81

x2 = 0.0217 m

x2 - x1 = 5.3 x 10^-3 m   = 5.3 mm

b) using energy conservation to find speeed just before the collision,

mgh = mv^2 /2

9.81 x 0.280 = v^2 / 2

v = 2.34 m/s

using momentum conservation,

initial momentum = final momentum

0.103 x 2.34 = (0.103 + 0.350)vf

vf = 0.532 m/s

c) at this time spring is stretched by 4.76 x 10^-3 m.

total mechanical energy = KE + PE

      = m v^2 /2 + kx^2 /2

      = [(0.103+0.350)x0.532^2 / 2 ] +   [ 210 x (5.3 x 10^-3)^2 / 2)]
   
      = 0.067 J

d) total mechanical energy = k A^2 / 2

0.067 = 210 A^2 / 2

A = 0.0253 m

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