1. In section 15.6, we derived an equation of motion for a simple pendulum where we assumed the mass of the pendulum bob was a point mass particle with no physical size. We will now relax that assumption. We will also consider the case where drag forces damp the motion of the pendulum.
1. In section 15.6, we derived an equation of motion for a simple pendulum where we assumed the mass of the pendulum bob was a point mass particle with no physical size. We will now relax that assumption. We will also consider the case where drag forces damp the motion of the pendulum.
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Transcribed Image Text:1. In section 15.6, we derived an equation of motion for a simple pendulum where we assumed the
mass of the pendulum bob was a point mass particle with no physical size. We will now relax that
assumption. We will also consider the case where drag forces damp the motion of the pendulum.
M
R
Consider a realistic pendulum as a uniform sphere of mass M and radius R at the end of
a massless string with length L being the distance from the pivot to the center of the sphere. Sketch
the physical situation. Label your diagram and list any knowns or unknowns for the problem.
а.
b.
Find an expression for the period Treal of the real pendulum. Check your units and show
that if L >>R, your expression reduces to the period Tsimp for the simple point mass pendulum
found in section 15.6 of the text. By L >> R, we mean that L is much, much larger than R.
Suppose L = 1.0 m, M = 25 g, R = 1.0 cm which are typical values for real pendulums.
с.
What is the numerical value of the ratio of Treal/Tsimp? Comment on the result by assessing of the
validity of the point mass assumption.
d.
In our derivation of the equation of motion for the point mass pendulum, we also
assumed there was no air resistance affecting the pendulum bob. We will now relax that assumption
by assuming a force due to air resistance to be proportional to the angular velocity with linear drag
coefficient b. This is a reasonable assumption since at low speeds air resistance is linear in velocity.
The drag coefficient is a constant number that depends on the shape and frontal cross-sectional area
of the bob. Using techniques learned in class, derive an equation of motion for the simple
pendulum including air resistance. Use 0 as the dependent variable and recall the small angle
approximation.
Deduce the solutions 0(t) for the damped pendulum's equation of motion. Deduce an
е.
expression for the period Tamp of the damped pendulum. Show that if the drag coefficient b → 0
(and hence the drag force goes to zero), your expression reduces to the period Timp for the point
mass pendulum. (Hint: Look at section 15.7)
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