http://www.ck12.org Chapter 5. Motion in Two Dimensions
The position shown in the illustration is theequilibrium position. This position is the middle, where the spring is
not exerting any force either to the left or to the right. If the object is pulled to the right, the spring will be stretched
and exert arestoring forceto return to the weight to the equilibrium position. Similarly, if the object is pushed to
the left, the spring will be compressed and will exert a restoring force to return the object to its original position.
The magnitude of the restoring force,F, in either case must be directly proportional to the distance,x, the spring has
been stretched or compressed. (A spring must be chosen that obeys this requirement.)
F=−kx
In the equation above, the constant of proportionality is called thespring constant. The spring constant is repre-
sented by k and its units are N/m. This equation is accurate as long as the spring is not compressed to the point that
the coils touch nor stretched beyond elasticity.
Suppose the spring is compressed a distancex=A, and then released. The spring exerts a force on the mass pushing
it toward the equilibrium position. When the mass is at the maximum displacement position, velocity is zero because
the mass is changing direction. At the position of maximum displacement, the restoring force is at its greatest - the
acceleration of the mass will be greatest. As the mass moves toward the equilibrium position, the displacement
decreases, so the restoring force decreases and the acceleration decreases. When the mass reaches the equilibrium
position, there is no restoring force. The acceleration, therefore, is zero, but the mass is moving at its highest velocity.
Because of its inertia, the mass will continue past the equilibrium position, and stretch the string. As the spring is
stretched further, the displacement increases, the restoring force increases, the acceleration toward the equilibrium
position increases, and the velocity decreases. Eventually, when the mass reaches its maximum displacement on
this side of the equilibrium position, the velocity has returned to zero and the restoring force and acceleration have
returned to the maximum. In a frictionless system, the mass would oscillate forever, but in a real system, friction
gradually reduces the motion until the mass returns to the equilibrium position and motion stops.
Imagine an object moving in uniform circular motion. Remember the yo-yo we spin over our heads? In your mind,
turn the circle so that you are looking at it on edge; imagine you are eight feet tall, and the yo-yo’s circle is exactly
at eye level. The object will move back forth in the same way that a mass moves in SHM. It moves consistently
from the far left to the far right until you stop spinning the yo-yo. Another example is to imagine a glowing light
bulb riding a merry-go-round at night. You are sitting in a chair at some distance from the merry-go-round so that
the only part of the system that is visible to you is the light bulb. The movement of the light will appear to you to be
back and forth in simple harmonic motion. Circular motion and simple harmonic motion have a lot in common.
The greatest displacement of the mass from the equilibrium position is called theamplitudeof the motion. One
cyclerefers to the complete to-and-fro motion that starts at some position, goes all the way to one side, then all the
way to the other side, and returns to the original position. Theperiod ,T,is the time required for one cycle and the
frequency,f, is the number of cycles that occur in exactly 1.00 second. The frequency, in this case, is the reciprocal
of the period.