Assignment 35

course PHY201

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Physics I

11-15-2007

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23:03:47

Query introductory problem set 9, #'s 12-17 How do we find the mass of a simple harmonic oscillator if we know the restoring force constant k and how long it takes to complete a cycle?

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If we know the constant ""k"" and the period ""T"", and need to find mass ""m"", then we need to use the formula:

T = 2 * pi * sqrt(m/k)

Now we solve for m...

T = 2 * pi * sqrt(m/k)

T^2 = 4 * pi^2 * (m/k)

(T / 2pi)^2 = m/k

m = k(T / 2pi)^2

or we can use the reciprocal of T: 1 / f

m = k[ 1 / (2 * pi * f )]^2

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23:03:59

We know the basic relationship omega = sqrt(k/m), which we can solve to get m = omega^2 * k. We are given k, so if we know omega we can easily find m. We know how long it takes to complete a cycle so we can find the angular frequency omega:

From the time to complete a cycle we find the frequency, which is the reciprocal of the time required. From frequency we find angular frequency omega, using the fact that 1 complete cycle corresponds to 2 pi radians.

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23:10:55

If we know the mass and length of a pendulum how can we find its restoring force constant (assuming displacements x much less than pendulum length)?

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If we have a pendulum and know the length ""L"", mass ""m"", and need to find k, then we will use the formula:

F = kx

Now we must convert the formula and solve for m

F = kx

F = k * L * theta

mg = k * L * theta

k = mg / L * theta

k = (mg) / [Lsin(theta)]

The reason we use sine is becasue we are using the restoring force which is in the x direction.

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23:11:14

For small displacement from equilibrium vector resolution of the forces tells us that the x component of tension in the same proportion to tension as the displacement x to the length L:

x / L

Since for small angles the tension is very nearly equal to the weight mg of the pendulum this gives us

Tx / m g = x / L so that

Tx = (m g / L) * x.

Since Tx is the restoring force tending to pull the pendulum back toward equilibrium we have

restoring force = k * x

for k = m g / L. So the restoring force constant is m g / L.

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23:11:38

Query Add comments on any surprises or insights you experienced

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None.

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23:11:42

as a result of this assignment.

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23:11:45

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Assignment 35

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