14.7 - Velocity
In Concept 1, you see a graph of an object in simple harmonic motion. The graph
shows the displacement of the object versus time. At any point the slope of the graph is
the object’s instantaneous velocity. The slope equals ǻy/ǻx. In this graph, this is the
change in displacement per unit time, which is velocity.
You can consider the relationship of velocity and displacement by reviewing the role of
force in SHM. Consider an object attached to a spring, where the spring is stretched
and then the object is released. The spring force pulls the object until it reaches the
equilibrium point, increasing the object’s speed.
Once the object passes through the equilibrium point, the spring is compressed and its
force resists the object’s motion, slowing it down. Because the object speeds up as it
approaches the equilibrium point and slows down as it moves away from equilibrium, its
greatest speed is at the equilibrium point.
When the spring reaches its maximum compression, the object stops for an instant. At
this point, its speed equals zero. The spring then expands until the object returns to its
initial position, with the spring fully extended. Again, the object stops for an instant, and
its speed is zero.
In the paragraphs above, we discussed the motion in terms of speed, not velocity, so
we could ignore the sign and focus on how fast the object moves. The object’s velocity
will be both positive and negative as it moves back and forth. You see this alternating
pattern of positive and negative velocities in the graph in Equation 1.
When the displacement is at an extreme, the velocity is zero, and vice-versa. One way
to state the relationship between the displacement and velocity functions is to say they
areʌ/2 radians (90°) out of phase. An equivalent way to express this without a phase
constant is to use a cosine function for displacement and a sine function for velocity,
and this is what we do. This relationship can also be derived using calculus. In
Equation 1, you see both a velocity graph and the function for velocity.
The second equation shown in Equation 1 states that the maximum speed vmax is the
amplitude of the displacement function times the angular frequency. To understand the
source of this equation, recall that the maximum magnitude of the sine function is one.
When the sine has a value of í1 in the velocity equation, the velocity reaches its
maximum value of AȦ.
Velocity in SHM
Velocity constantly changes
·Extreme velocities at equilibrium
·Zero velocity at endpoints