1. A block of mass m is falling from the height h on to the plate of the spring
ID: 2123955 • Letter: 1
Question
1. A block of mass m is falling from the height h on to the plate of the spring weight balance (plate%u2019s mass M, spring constant k) as shown. The block sticks to the balance and the system starts to oscillate in the vertical direction. Find the amplitude A of the oscillations.
The diagram is above, listed as Problem 4.
A block of mass m is falling from the height h on to the plate of the spring weight balance (plate%u2019s mass M, spring constant k) as shown. The block sticks to the balance and the system starts to oscillate in the vertical direction. Find the amplitude A of the oscillations.Explanation / Answer
Find the PE of the falling block
PE = mgy
PE = mgh
As the block is about to strike the plate, the PE is converted to KE. Use this KE to find the speed of the block just before impact.
KE = mgh
0.5mv^2 = mgh
mv^2 = 2mgh
v^2 = 2gh
v = sqrt(2gh)
Use conservation of momentum to determine the speed of the spring/plate/block system after the block sticks to plate
mv + Mv = (m+M)v
m(sqrt(2gh) + M(0) = (m+M)v
v = [m*sqrt(2gh)]/(m+M)
This velocity provides KE to the system
KE = 0.5mv^2
KE = 0.5(m+M)[(m*sqrt(2gh))/(m+M)]^2
KE = 0.5[(m*sqrt(2gh))]^2 /(m+M)
This KE is converted to PE of the spring
PE = 0.5kA^2
0.5[2ghm^2]/(m+M) = 0.5kA^2
2ghm^2/(m+M) = kA^2
2ghm^2/[k(m+M)] = A^2
A = sqrt[2ghm^2/[k(m+M)]]
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