Question

A 5-kilogram object initially slides with speed v_o in a hollow friction less pipe. The end of the pipe contains a spring as shown. The object makes contact with the spring at point A and moves 0.1 meter before coming to rest at point B. The graph shows the magnitude of the force exerted on the object by the spring as a function of the objects distance from point A.

a. Calculate the spring constant for the spring.

b. Calculate the decrease in kinetic energy of the object as it moves from point A to point B.

c. Calculate the initial speed v_o of the object

Answer 1

(a) The graph is one of force vs ∆x so the slope of this graph is the spring constant. Slope = 200N/m

(b) Since there is no friction, energy is conserved and the decrease in kinetic energy will be equal to the gain in spring potential 

|∆K| = U_sp(f) = ½k∆x² = ½(200)(0.1)² = 1J

Note: This is the same as the area under the line since the area would be the work done by the conservative spring force and the work done by a conservative force is equal to the amount of energy transferred.

(c) Using energy conservation. 

K_i = U_sp(f)

1/2mv_o² = 1J

1/2(5)v_o² = 1

v_o = 0.63m/s