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the x or y-axis oscillates in simple harmonic motion (see Figure 2). From this perspective, the
angular speed of the object around the circle is equal to ω , the angular frequency of the simple
harmonic oscillation. y y v
x t Figure 2: The relationship between uniform circular motion and simple harmonic motion. The
wavy lines at right represent the y-position of the object that is moving around the circle, and the
colored dots correspond to its y-position at the similarly-colored points on the circle. Experimental Background: Mass Hanging From A Spring
Consider a mass hanging from a “massless” spring of spring constant k, as shown in Figure 3. Figure 3: Schematic diagram of a mass hanging from a spring. 87 Let y = 0 be the point at which the spring is in equilibrium; then the equation of motion for the
m ay = −k y − m g,
which looks suspiciously like Equation 1 except for the extra g term on the right. This makes this
equation a little more difﬁcult to solve, but the end result is that
y = A sin(ω t + δ ) − mg
k (4) The mass therefore oscillates in simple harmonic motion about the equilibrium point y = mg/k.
This makes sense because when the mass is in equilibrium, the spring will be slightly stretched to
counteract the force of gravity. The angular frequency of oscillation for this system is
ω = k/m.
Problem 10.1 What y value do you obtain when you set the force of the
spring, −k y, equal to the force of gravity, −m g, which is what occurs when
the mass is in equilibrium. Does this resemble anything in the equations
Problem 10.2 What advantage do we gain by measuring k via the period
and not the displacement of the pendulum bob from equilibrium? In other
words, why is our measurement of k more precise if we use the period? Experimental Background: Simple Pendulum
One example of a situation in which an object undergoes simple harmonic motion is the simple
pendulum, which is shown in Figure 4. Figure 4: Diagram of a simple pendulum. The pendulum bob has mass m, the distance from the
pivot point t...
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This note was uploaded on 09/13/2011 for the course ECONOMICS 101 taught by Professor Gerson during the Spring '11 term at University of Michigan.
- Spring '11