Chapter 11: Q. 47 (page 890)

Show that the curvature is constant at every point on the circular helix defined by $r (t) = (a \cos t, a \sin t, b t),$ where a and b are positive constants.

Short Answer

Expert verified

The curvature is constant at every point on the circular helix defined by $r (t) .$

Step by step solution

Step 1. Given Information.

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

Step 2. Showing.

To show that the curvature is constant at every point on the circular helix defined by $r (t) = (a \cos t, a \sin t, b t),$ we will use the formula for the Curvature of a Space Curve $κ = \frac{||r' (t) \times r'' (t)||}{{||r' (t)||}^{3}} .$

So,

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

Step 3. Calculate.

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

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

Thus, kis constant.

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Show that the curvature is constant at every point on the circular helix defined by $r (t) = (a \cos t, a \sin t, b t),$ where a and b are positive constants.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Showing.

Step 3. Calculate.

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Show that the curvature is constant at every point on the circular helix defined by r(t)=acost,asint,bt, where a and b are positive constants.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Showing.

Step 3. Calculate.

One App. One Place for Learning.

Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

Show that the curvature is constant at every point on the circular helix defined by $r (t) = (a \cos t, a \sin t, b t),$ where a and b are positive constants.