Physics 105
Hooke's Law states that the
force F exerted by a spring is proportional and in the opposite direction to the
displacement x of the end of the spring. The constant of
proportionality k is called the spring constant.
Fspring = -kx
When the force is in the
opposite direction from displacement, this is an example of a restoring force.
When you have a restoring force, you get oscillatory motion called harmonic
motion: if you attach a mass to the end of the spring, pull it away from
equilibrium and let it go, it will oscillate about its equilibrium position
(equally in both directions). It is called simple harmonic motion (SHM) if
the force is directly proportional to the displacement (as it is in this case).
Notice that a body
undergoing harmonic motion is not influenced by a constant force, so if you
apply Newton's Law you find the acceleration is not constant. This
means that the kinematic equations that we have been using all along for
constant acceleration do not hold, and we will have to develop a new set of
kinematic relations. We will explore the equations describing SHM in a
later chapter.
Equilibrium/Non-Equilibrium
In equilibrium, a mass on a
spring will hang in a position where all the other forces balance the spring
force. What is the net force on the mass at this position? You can draw a free
body diagram to help determine what forces are acting on the spring. How would
one calculate the equilibrium position with a mass versus the equilibrium
position with no mass (the "unstretched" position, when measuring, be sure to
always measure from the bottom of the spring)?
If you pull the mass away
from its equilibrium position, it will oscillate back and forth about that
position. The oscillation cannot be described from our constant
acceleration equations of motion since the acceleration varies. We can
characterize the oscillation with a few variables like amplitude (A), frequency
(f) and period (T).
The amplitude of
oscillation A is the maximum displacement from the equilibrium position.
This means that when a mass on a spring oscillates, it goes from a position of A
away from equilibrium, stops, turns around, goes through equilibrium to -A.
It continues this type of oscillation if there are no dissipative forces.
The period of oscillation T
is the time it takes for one complete oscillation. It is the time elapsed
between any two instants at which it completely reproduces its motion.
The frequency f is the
number of oscillations per unit time, and is the inverse of period: f = 1/T.
Apparatus
Spring
Weight hanger
Mass set
Ruler
Stopwatch
Outline
Determine a procedure to
verify the linear relation between the spring force Fspring
and x, and to measure the spring constant. Be sure to report k
with 90% confidence.
The amplitude A of the
motion is the distance you pull the mass from equilibrium. Using a fixed
value of mass, measure the periods (T) of oscillation for different amplitudes
of motion and try to determine the relationship between A and T for
constant mass. Try to measure the period as precisely as you can. You are not
supplied with an equation for this relationship. A normal way to try to
determine a relationship for two variables is to try plotting them in various
ways. Hint: try to plot T on the y-axis and A on the x-axis. Does it
appear to be linear? Pay attention to the y-axis scale. What is the resulting
slope with 90% confidence?
Note: Keep the amplitude of
oscillation small enough that the oscillation stays reasonably uniform.