# Configuring the restoring force for a pendulum equation

1. Apr 19, 2010

### outdoorjunkie

1. The problem statement, all variables and given/known data

In order to configure the equation for the restoring force of a pendulum, my physics book uses this equation:
sinθ ≈ θ when the angle is less than π/10 (or about 20 degrees.) I've tried to figure this out on my calculator, but cannot figure out how they make this assumption.

2. Relevant equations

The above equation is then put in to Fr = (w)(sinθ) to create this equation:
Fr = (w)θ

3. The attempt at a solution

Nothing, I have used radians and degrees to try to figure out how they get this equation, but I cannot. Thanks so much for your help!

2. Apr 19, 2010

### Staff: Mentor

π/10 ≈ .314 radians
sin(.314) ≈ .309

Seems pretty close to me. And it gets better for smaller angles.

3. Apr 19, 2010

### karkas

Well, trying out some rad values at Wolfram Alpha gives me the following results:

$sin(\pi /20)\approx 0,156$ when
$\pi/20 \approx 0,157$

so I guess it is a valid approximation, at least it holds up to 2 decimals in this case.

edit: You got me Doc Al!

4. Apr 19, 2010

### outdoorjunkie

Okay, now I understand. I was figuring the problem wrong, I was doing sin(π/10) instead of π/10 and then putting that answer into the sine. Thank you so much!

5. Apr 20, 2010

### Jolsa

If you do a taylor series expansion of the sine function the second order term is proportional to the angle cubed. If the angle is in radians the term becomes insignificant for small angles.

Google "Taylor series" or "Small angle approximation" for more information :).

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