A massless spring of elastic constant k = 0.2 N/m has a length of 10 cm if no fo

Question

A massless spring of elastic constant k = 0.2 N/m has a length of 10 cm if no force is acting on it. Now a toy car is pushed against the spring in such a way that the length of the spring 7 cm. Suppose you hold the car so that it doesn't move. There is no friction in this problem.

a. What is the potential energy stored in the spring? What is the Force that you have to exert on the car in order to keep it in its position? Choose a positive direction for the x axis and indicate direction and magnitude of the force.

Now suppose you instaneously remove your hand.

b. At this instant of time, what is the direction of the accleration of the car? What is the magnitude of the acceleration?

c. The car starts, in position x = 0m, and accelerates, say, towards the right. Will the force on the car be constant or will it change with position while the car is pushed to the right? At what value of the x coordinate will the force on the car be zero?

d. What is the value of the acceleration of the car when the length of the spring reaches 10 cm? What is the speed of the car at this point?

e. Suppose that the car can freely disconnect from the spring. At what position will it lose contact with the spring? Why? [Hint: consider the velocity of the car and of the end of the spring during the motion.]

Explanation / Answer

a.

U=0.5kx^2 =0.5×0.2×(107)×10^2

U=0.003 J

F= kx = 0.2×0.03 =0.006 N ( along +X)

b.

Direction of accelration is along -X axis

a= -kX/ m =-0.006 / m

c.

Acceleration will be changing in proportional to length.

Force will be zero when length is Zero (X=0)

d.

Acceleration a=0 m/s^2

Speed v= Aw = 0.03*sqrt(k/m) =0.03×0.2 /m

v=0.013 / m m/s

e.

It will disconnect at equilibrium as both will be having opposite variation in velocities.

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