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0593_C10_fm Page 337 Monday, May 6, 2002 2:57 PM

Introduction to Energy Methods 337

To answer these questions using the work–energy principle, recall from Eq. (10.2.22)
that when a linear spring is stretched (or compressed) a distance δ, the corresponding
2
work W done by the stretching (or compressing) force is (1/2)kδ . Because the force exerted
on the spring is equal, but oppositely directed, to the force exerted on B, the work done
on B as the spring is relaxed is also (1/2)kδ . (That is, the work on B is positive because
2
the force of the spring on B is in the same direction as the movement of B.)
Because B is released from rest, its initial kinetic energy is zero. The kinetic energy at
the equilibrium position is:
K = ( ) mv = ( ) mx ˙ 2 (10.9.1)
2
12
12
Then, from the work–energy principle, we have:

k )
(
W = ∆ K or 12 δ 2 = ( ) mv − 0 (10.9.2)
2
12
or

v = ˙ x = k m δ (10.9.3)

where the minus sign is selected because B is moving to the left.
Next, as B continues to move to the left past the equilibrium position, the spring force
will be directed opposite to the movement of B. Therefore, the work W done on B as B
moves to the left a distance d from the equilibrium position is:

W =−( ) kd 2 (10.9.4)
12

When B moves to its leftmost position, its kinetic energy is zero. From Eqs. (10.9.1) and
(10.9.2), the kinetic energy of B at the equilibrium position is:

K = ( ) mv = ( ) kδ 2 (10.9.5)
2
12
12
The work–energy principle then produces:

W = ∆ K or −(1 kd ) 2 2 =−(1 k ) 2 δ 2 (10.9.6)
0

or

d =δ (10.9.7)

That is, the block moves to the left by precisely the same amount as it was originally
displaced to the right.
The usual explanation of this phenomenon is that when the spring is stretched (or
compressed) the work done by the stretching (or compressing) force stores energy (potential
energy) in the spring. This stored energy in the spring is derived from the kinetic energy
of the block. Then, as the spring is relaxing, its potential energy is transferred back to
kinetic energy of the block. There is thus a periodic transfer of energy between the spring