How do you Derive a formula for the maximum speed?

Question

Derive a formula for the maximum speed Vmax of a simple pendulum bob in terms of grams, the length L, and the angle of the Swing?

Answer 1

You first determine the period of the pendulum. Assuming a massless arm (all mass concentrated at radius r) the period T = 2*pi*sqrt(r/g), where g is gravitational acceleration. (Note the period is independent of the mass.) From T you calculate the pendulum's radian frequency w = 2*pi/T. (Actually this means w = sqrt(g/r).) The maximum speed, which is achieved when the pendulum passes through zero displacement, is theta*w, where theta is the angle of the swing. Note that the pendulum is only approximately constant in period. As amplitude increases the period tends to become longer since the restoring torque is not exactly proportional to displacement but rather to the sine of displacement. See the ref.
Example: For a 1 m pendulum under earth gravity (9.80665 m/s^2), T = 2.00641 s and w = 3.1316 rad/s. If the pendulum amplitude is 10 deg, the maximum velocity is 31.316 deg/s.

Answer 2

surely, deriving the right in this subject is extra of a geometry situation. imagine that you've a triangle - one aspect is the pendulum string and one aspect is the "vertical." (the different aspect is the floor, yet it really is unimportant at the moment.) The pendulum string kinds the hypotenuse of the triangle, and the attitude you've - 12 degrees - the between the hypotenuse the vertical, or adjacent, aspect. From our popular politically incorrect acronym, SOH-CAH-TOA, we'd want to continually be using the cosine function. The cosine of 12 degrees is a similar because the size of the adjacent aspect over the hypotenuse. sparkling up for the size of the adjacent aspect. Why do you pick the adjacent length? because that's the vertical perfect of the pendulum, as measured from the pivot element. if you're defining ability ability from another element, this may a minimum of provide you with a initiate.