A system consists of 2 masses (6 kg and 4 kg) connected by a compressed spring (

Question

A system consists of 2 masses (6 kg and 4 kg) connected by a compressed spring (k 10,000 N/m, spring compression distance 10 cm). The system is sliding across a level surface with a speed of 8 m/s. At t 0, the spring suddenly expands a) Calculate the amount of work done upon each block in the center of mass frame. b) Calculate the amount of work done upon each block in the earth frame. c) Calculate the difference in the total kinetic energy of each frame. Why is there a difference? Support your response with calculations.

Explanation / Answer

given two masses

M = 6 kg

m = 4 kg

spring constant k = 10,000 N/m

spring compression, x = 0.1 m

speed u = 8 m/s

at t= 0, the spring expands

a. let the body M move by distance X and m move by distance x due to expansion of the psring

then x + X = 0.1 m

and as this is an internal force the position of center of mass relative to the masses will not change

MX = mx

6X = 4x

X = 4x/6

x + 4x/6 = 0.1

x = 0.06 m

X = 0.04 m

hence work done on the block m = 0.5kx^2 = 0.5*10,000*0.06^2 = 18 J

hence work done on the block M = 0.5kX^2 = 0.5*10,000*0.04^2 = 8 J

b. in earth frame

initial energy of the system = 0.5(M + m)u^2 + 0.5k*0.1^2 = 0.5(10)8^2 + 0.5*10,000*0.1^2 = 370 J

when the psring is uncompressed, let speed of both the blocks be v

then from conservation of energy

0.5(M + m)v^2 = 370

v = 8.602 m/s

so work done on M = 0.5M(v^2 - u^2) = 30 J

work done on m = 0.5m(v^2 - u^2) = 20 J

c. difference in total of each frame = (30 + 20) - (18 + 8) = 24 J

this is because in center of mass frame the speed of center of mass also changes which is not considered in the center of mass frame

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