Springs

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A ideal spring has an equilibrium length.  If a spring is compressed, then a force with magnitude proportional to the decrease in length from the equilibrium length is pushing each end away from the other.  If a spring is stretched, then a force with magnitude proportional to the increase in length from the equilibrium length is pulling each end towards the other.

The force exerted by a spring on objects attached to its ends is proportional to the spring's change in length away from its equilibrium length and is always directed towards its equilibrium position.

Assume one end of a spring is fixed to a wall or ceiling and an object pulls or pushes on the other end.  The object exerts a force on the spring and the spring exerts a force on the object.  The force the spring exerts on the object is in a direction opposite to the displacement of the free end.  If the x-axis of a coordinate system is chosen parallel to the spring and the equilibrium position of the free end of the spring is at x = 0, then the F = -kx. The proportional constant k is called the spring constant.  It is a measure of the spring’s stiffness.

 

When a spring is stretched or compressed, so that its length changes by an amount r from its equilibrium length, then it exerts a force F = -kr in a direction towards its equilibrium position.  The force a spring exerts is a restoring force, it acts to restore the spring to its equilibrium length.

 

Problem:

A stretched spring supports a 0.1 N weight.  Adding another 0.1 N weight, stretches the string by an additional 3.5 cm.  What is the spring constant k of the spring?

Solution:

  • k = |F/x| = (0.1 N)/ (0.035 m) = 2.85 N/m.