An object of mass 5 kg slides with an initial velocity vnot in a hollow, frictio

Question

An object of mass 5 kg slides with an initial velocity vnot in a hollow, frictionless tube. The end of the tube contains two nested springs, as shown. The object hits the inner spring at point A and compresses it to point B before it also hits the outer spring. The object eventually comes to rest at point C. The graph shows the magnitude of the force exerted on the object by the springs as a function of distance.

What is the spring constant of the inner spring?

What is the change in kinetic energy of the object as it moves from points A to B?

What is the change in kinetic energy of the object as it moves from points B to C?

What is the initial speed of the object?

What is the spring constant of the outer spring?

Explanation / Answer

1)slope of the grapch from 0 to 0.1 m will give the spring constant of inner spring that is K1= 20/0.1 = 200N/m

2) The change in kynetic energy would be equals to the potential energy stored in spring = 0.5 k x2 = 0.5*200*0.12= 1 Joul

3) total spring constant (slope) from point from B to C is 40/0.05 = 800N/m hence the potential energy stored in spring is= 0.5 k x2 = 0.5*800*0.052= 1 Joul

4) since the total Potential energy stored in string is 1+1 = 2 Jouls (from 2 and 3 answers) that would be equals to initial kinetic energy . KE = 0.5mv2 = 2 , substute m = 5Kg we get velociy v= 0.4472m/s

5) we know the total spring constant combine parallel is k1+k2 = 800N/m (we got it in 3) and we know k1=200N/m that implies the outer spring's spring constant is 600N/m

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