Calculus A simple pendulum. An idealized simple pendulum is given by a mass m ha

Question

Calculus

A simple pendulum. An idealized simple pendulum is given by a mass m hanging from a massless string of length L. Its motion is described by the angle theta(t), where t is time. The distance travelled by the pendulum is the arclength s(t) - L theta(t). According to Newton's 2nd law, the pendulum mass x acceleration equals the restoring force F net acting on it. mL theta"(t) = F net where F net = mg sin theta (see picture). For small angles one often uses the approximation sin theta ? theta to replace this differential equation by the simpler equation mL theta"(t)=mg theta (which is easy to solve, as you'll find out in Math 316).Question: If the angle swings with ?pi/10 ? theta ? pi/10, what is an upper bound for the error made in the approximation sin theta ? theta ?

Explanation / Answer

Error = Mg(sin(theta) -theta)/[mg*sin(theta)]

=[1 - theta/(sin(theta)]

If theta = pi/10

then error = [ 1-( pi/10)/sin(pi/10)] =0.016

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