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Unformatted text preview: Equation 6? If
you give the pendulum a “kick” as you let it go, will this affect ω ? Why or
why not?
A photograph of the experimental setup for this part of the laboratory is shown in Figure 5. LoggerPro will record the times at which the pendulum bob passes through the photogate and use them
to calculate the period of the pendulum’s oscillation. Figure 5: A picture of the apparatus used for the simple pendulum experiment. Experimental Background: Ball in Dish
A ball rolling in a circular dish with a sphericallyrounded bottom is another example of a system
that undergoes simple harmonic oscillation. Its motion is slightly more complicated than that of
the simple pendulum, but we can analyze it in a similar way.
First, consider the approximation shown in Figure 6, where the bowl bottom is treated as a plane
inclined at a shallow angle θ , where θ also happens to be the ball’s angular displacement from the
bottom point of the dish.
The linear acceleration of the ball down the “plane” is given by the equation of motion
m at = Ff − m g sin(θ ),
90 (9) Figure 6: The more complicated curvature of the bowl bottom is approximated as a simple inclined
plane. where Ff is the force of friction. The force of friction also produces a torque on the ball, which
causes it to undergo a constant angular acceleration α . The magnitude of the torque the ball
experiences is given by
τ = I α = −r Ff ,
(10)
where I is the ball’s moment of inertia and r is the ball’s radius. As long as the ball rolls without
slipping, its linear and angular acceleration are related by
at = α r. (11) Combining Equations 9, 10, and 11 yields
at = − g sin(θ )
.
1 + I /mr2 (12) Problem 10.6 Can you derive Equation 12 using Equations 9, 10, and 11?
You should know how to do so.
If we consider θ , the ball’s angular displacement from the bottom of the dish, as our natural
coordinate, and we use the relationship
at = (R − r) αθ , 91 (13) where αθ is the ball’s angular acceleration about the center of curvature of the dish, we obtain
(R − r)αθ = − gsin(θ )
.
1 + I /mr2 (14) Note that αθ means something very differ...
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 Spring '11
 Gerson

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