It moves along the curve r(t)=<3cost, 2sint, 0>,0<t<2pi. Find v(pi/4) & a(pi/4). At what points is the curvature maximum or minimum?
I've found the acceleration and velocity. The curvature being maximum or minimum is too confusing though?! HELP!
After a little searching, I came up with this:
For your case,
`x = 3cost`
So we have `((-3sint)(-2sint)-(2cost)(-3cost))/((9sin^2 t+4cos^2 t)^(3/2))`
So let's simplify this up a little
`k=(6sin^2 t+6cos^2t)/((9sin^2 t + 4cos^2 t)^(3/2))`
Wait a second - that numerator is just 6 (since sin^2 + cos^2 = 1)
But to find max or min k, we need the derivative.
` ` `k'=-(90sint cos t)/(9sin^2t+4cos^2t)^(5/2)`
Finding the roots just involves finding when sin(t)cos(t) = 0, which happens at `pi/2` and `pi` .
At `t=pi/2` , k = 2/9.
At `t = pi` , k = 3/4.
These must be the min and max curvatures, respectively.