Question

Use the alternative curvature formula K=|a x v|/|v|^3 to find the curvature of the following parameterized curves. to find the curvature of the following parameterized curve. r(t)=<6+5t^2, t, 0>

Answer 1

Answer:

$k=\frac{10}{(100t^2+1)^{\frac{3}{2}}}$

Or

$k=\frac{10\sqrt{100t^2+1}}{(100t^2+1)^{2}}$

Step-by-step explanation:

We want to compute the curvature of the parameterized curve, $r(t)=\:\:$ using the alternative formula:

$k=\frac{|a\times v|}{|v|^3}$.

We first compute the required ingredients.

The velocity vector is $v=r'(t)=$

The acceleration vector is given by $a=r''(t)=$

The magnitude of the velocity vector is $|v|=\sqrt{(10t)^2+1^2+0^2}=\sqrt{100t^2+1}$

The cross product of the velocity vector and the acceleration vector is

$a\times v=\left|\begin{array}{ccc}i&j&k\\10&0&0\\10t&1&0\end{array}\right|=10k$

We now substitute ingredients into the formula to get:

$k=\frac{|10k|}{(\sqrt{100t^2+1})^3}$.

$k=\frac{10}{(100t^2+1)^{\frac{3}{2}}}$

Or

$k=\frac{10\sqrt{100t^2+1}}{(100t^2+1)^{2}}$