OBJECTIVE:  I will be able to evaluate the motion of a pendulum using SHM principles after today's class.

WORDS/FORMULAE FOR TODAY:

• (0) F = -kx
• (1) x(t) = Acos (ωt + φ)
• (2) ω = angular frequency = (k/m)^1/2 measured in radians/second
• (3) T = period = 2π/ω = 1/f
• (4) f = frequency 1/T = measured in cycles per seconds or waves/sec
• (5) v = dx/dt = -ωAsin(ωt + φ) (notice the similarity to rω)
• (6) a = dx/dt = -ω²Acos(ωt + φ)(notice the similarity to rω²)
• (7) KE = 1/2mv² = 1/2m[-ωAsin(ωt + φ)]² = 1/2mω²A²sin²(ωt + φ)
• (8) U = 1/2kx² = 1/2kA²cos²(ωt + φ)]
• (9) TOTAL E = 1/2kA²
• (10) ω = (g/L)^1/2 (for a simple pendulum)
• (11) T = 2π/ω = 2π(L/g)^1/2 (for a simple pendulum)

WORK O' THE DAY: 

Mr Chase will lead us in a discussion of our homework

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Let's start by digesting more formulae: there are two more today!

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Recap:

Our discussions of SHM will primarily deal with one dimension (x) so we'll typically ASSUME that any motion is constrained to X.

We can accurately predict the motion of an object experiencing SHM by the following formula:

x(t) = Acos (ωt + φ)

That formula characterizes the mathematical (graphic) representation of an object experiencing SHM

• x = x direction
• (t) = with respect to time
• A = the AMPLITUDE of the resulting wave when the motion of the object is being graphed (measured in meters)
• ω = angular frequency (measured in radians/sec)
• t = time (in seconds)
• φ (spelled "phi" and pronounced "fee") = the "phase constant" or "initial phase angle" and is measured in radians

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Introduction to a Simple Pendulum:

We can use SHM to model the back forth swing of a pendulum if (AND ONLY IF) the angle the pendulum makes with the verticle is less than 10 degrees or .2 radians)

There is a principle (more like a guideline I suppose) that creeps into physics from time to time called the SMALL ANGLE APPROXIMATION.

SAA Definition:

sinθ ~ = θ when θ is small (less than 10 degrees = .2 radians)

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The motion of a simple pendulum (see above) is found by the following expressions:

• (10) ω = (g/L)^1/2 (for a simple pendulum)
• (11) T = 2π/ω = 2π/(L/g)^1/2 (for a simple pendulum)

Take a look at quick quiz on the bottom of page 465 and discuss with your groupies.

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Now take a look at the table of sines and angles at the bottom of page 465. Convince yourself that the small angle approximation is appropriate.

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Have some fun with the example 15.5

Homework: 35, 39 & 41