
This same idea can be shown on a graph. The graph of force and
stretch of the spring is a straight line. The slope of this line
is k, the "spring constant", which describes how strong
or weak the spring is. The spring constant k is large for a strong
spring and k is small for a weak spring.
Figure 10.8 A graph of the stretch of a spring and the
external force causing the stretch is a straight line. The slope
of the line is the spring constant, k.
Since a mass attached to a spring is a simple harmonic oscillator,
we know the amplitude does not affect the period. If the mass
is moved only a little from equilibrium and released, it will
oscillate back and forth gently with some period. If we pull the
mass a larger distance from equilibrium and release it, it will
move back and forth with more speed but the period will remain
the same. If we now pull it an even greater distance from equilibrium
and release it, it will rush back and forth with greater speed
but the period will still be the same.
This is a characteristic of all simple harmonic oscillators.
What, then, does determine the period? You knowboth from your own practical experience and from Newton's second lawthat a larger mass will be more difficult to move. The period will increase as the mass increases.
More masswith the same springwill mean a larger period.
What else determines the period? How strong or weak the spring
is will certainly affect the motion. A stronger springwith a
larger value of kwill move the same mass more quickly for a smaller
period. As the spring constant k increases, the period decreases.
These two ideas are contained in the equation
which gives the period T for a mass m attached to a spring with
spring constant k.
Q: Thinking in terms of Newton's second law, why would a greater mass lead to a longer period?
A: Newton's second law, F = ma or a = F/m , tells
us that a larger mass will have a smaller acceleration (for the
same force) so that a greater mass will simply move slower and,
therefore, take a longer time to complete its motion.
Q: Thinking in terms of Newton's second law, why would a stiffer spring lead to a shorter period?
A: A stiffer or stronger spring means a greater force in Newton's second law, F = ma. For a given mass, that means a greater acceleration so the mass will move faster and, therefore, complete its motion quicker or in a shorter period.
