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catkin

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## Homework Statement

This is from Advanced Physics by Adams and Allday. Spread 3.34, Q 3.

A friction-free trolley of mass 2 kg is tethered to rigid supports at each end by two identical springs of spring constant 20 N/m each. The springs obey Hooke's law and remain in tension as the trolley is displaced 2 cm to one side and released. What is the effective spring constant of the system tethering the trolley?

## Homework Equations

[itex]F = - k x[/itex]

Where

- F is restoring force
- k is spring (stiffness) constant
- x is displacement from equilibrium position

## The Attempt at a Solution

In the equilibrium position let each spring be stretched by x generating a restoring force F(each x and F are identical by symmetry).

Move the trolley [itex]\delta x[/itex] to the right ([itex]\delta x < x[/itex]).

The extension of the left spring is now [itex]x + \delta x[/itex] and the right spring [itex]x - \delta x[/itex].

The restoring force from the left spring is now [itex]F_L = - k ( x + \delta x)[/itex]

The restoring force from the right spring is now [itex]F_R = - k ( x - \delta x)[/itex]

The net force (on the trolley), acting to restore the previous equilibrium position is

[itex]F_L - F_R[/itex]

[itex]= - k ( x + \delta x) - (- k ( x - \delta x)[/)[/itex]

[itex]= - 2 k \delta x[/itex]

Thus the spring constant of the system is twice the spring constant of the individual springs, that is 40 N/m.

The answer given in the textbook is 10 N/m.

What is the right answer?