Chapter 11: Q. 50 (page 890)

Find the curvature on the graph of the elliptical helix defined by $r (t) = (a \cos t, b \sin t, c t),$ where a, b, and c are positive constants.

Short Answer

Expert verified

The curvature on the graph of the elliptical helix defined by r(t)is $κ = \frac{\sqrt{b^{2} c^{2} \sin^{2} t + a^{2} c^{2} \cos^{2} t + a^{2} b^{2}}}{{(a^{2} \sin^{2} t + b^{2} \cos^{2} t + c^{2})}^{\frac{3}{2}}} .$

Step by step solution

Step 1. Given Information.

The given elliptical helix is $r (t) = (a \cos t, b \sin t, c t) .$

Step 2. Find the curvature.

To find the curvature on the graph of the elliptical helix, we will use the formula for the Curvature of a Space Curve $κ = \frac{||r' (t) \times r'' (t)||}{{||r' (t)||}^{3}} .$

So,

$r (t) = <a \cos t, b \sin t, c t> r' (t) = <- a \sin t, b \cos t, c> r'' (t) = <- a \cos t, - b \sin t, 0> Now, r' (t) \times r'' (t) = |\begin{matrix} i & j & k \\ - a \sin t & b \cos t & c \\ - a \cos t & - b \sin t & 0 \end{matrix}| = i (b c \sin t) - j (a c \cos t) + k (a b) = <b c \sin t, - a c \cos t, a b> Let' s find ||r' (t) \times r'' (t)|| = \sqrt{{(b c \sin t)}^{2} + {(- a c \cos t)}^{2} + {(a b)}^{2}} = \sqrt{b^{2} c^{2} \sin^{2} t + a^{2} c^{2} \cos^{2} t + a^{2} b^{2}} . . . . . . . . . (a) And ||r' (t)|| = \sqrt{{(- a \sin t)}^{2} + {(b \cos t)}^{2} + c^{2}} = \sqrt{a^{2} \sin^{2} t + b^{2} \cos^{2} t + c^{2}} . . . . . . . . (b)$

Step 3. Calculate.

Put the values of (a) and (b) in the formula of Curvature of a space curve,

$κ = \frac{\sqrt{b^{2} c^{2} \sin^{2} t + a^{2} c^{2} \cos^{2} t + a^{2} b^{2}}}{{(\sqrt{a^{2} \sin^{2} t + b^{2} \cos^{2} t + c^{2}})}^{3}} κ = \frac{\sqrt{b^{2} c^{2} \sin^{2} t + a^{2} c^{2} \cos^{2} t + a^{2} b^{2}}}{{(a^{2} \sin^{2} t + b^{2} \cos^{2} t + c^{2})}^{\frac{3}{2}}}$

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Find the curvature on the graph of the elliptical helix defined by $r (t) = (a \cos t, b \sin t, c t),$ where a, b, and c are positive constants.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Find the curvature.

Step 3. Calculate.

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Find the curvature on the graph of the elliptical helix defined byr(t)=acost,bsint,ct,where a, b, and c are positive constants.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Find the curvature.

Step 3. Calculate.

One App. One Place for Learning.

Most popular questions from this chapter

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Find the curvature on the graph of the elliptical helix defined by $r (t) = (a \cos t, b \sin t, c t),$ where a, b, and c are positive constants.