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# Kinda a math question, physics mechanical energy with spring system

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A block of mass 4 kg is attached to a spring whose spring constant is k = 1.10 103 N/m. The block is free to slide on a flat frictionless surface. The block is pushed to compress the spring by 18 cm, and then released. Assume the potential energy of the spring is zero at x = 0 where the spring is neither stretched nor compressed.

(a) Find the value or values of x where the kinetic energy of the block is momentarily 15% of the total mechanical energy of the system.

(b) Find the velocity of the block when this occurs.

Mar 1, 2019

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Total mechanical energy:  E = (1/2)Mv2 + (1/2)kx2  where M is mass, v is velocity

At 18cm, assuming we start with zero velocity,  E = (1/2)*1.1*103*(0.18)^2  Joules

(a) Let xa be the value of x when the PE is 15% of E:   (1/2)k*xa2 = 0.15*(1/2)k*(0.18)^2  or xa = sqrt(0.15)*0.18 m

(b) Let va be the velocity at this point.   (1/2)*4*va^2 + (1/2)*1.1*103*xa2 = (1/2)*1.1*103*(0.18)^2   solve for va (in m/s)

I'll leave you to do the number crunching.

Mar 1, 2019