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10.4 - Tangents, normals, and curvature
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10.4 Unit tangents, unit normals, and curvature
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<ol>
<li>Let $\myv r(t)=3\cos t\,\uv i+3\sin t\,\uv j+2t\,\uv k$. (It's a spiral: Moving in what direction on average? What's the radius of the spiral?)
<p class="justme answer">The particle is moving on the surface of a cylinder of radius 3, spiralling in the $\hat k$ direction.
<ol type="a"><li>Calculate the unit tangent vector, $\uv T(t)$.
<p class="justme answer">
$\uv T(t)=\myv r'(t)/|\myv r'(t)|$.  So... 
$$\myv r'(t)=\langle -3\sin t, 3\cos t, 2\rangle$$
$$|\myv r'(t)|=\sqrt{(-3\sin t)^2+(3\cos t)^2 + 2^2}=\sqrt{9+4}=\sqrt 13.$$
$$\uv T(t)=\frac{1}{\sqrt 13}\langle -3\sin t, 3\cos t, 2\rangle.$$</p>
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<li>Calculate the principle unit normal vector, $\uv N(t)$.
<p class="justme answer">
$\uv N(t)= \uv T'(t)/|\uv T'(t)|$.  So... 
$$\uv T'(t)=\frac{1}{\sqrt 13}\langle -3\cos t, -3\sin t, 0\rangle.$$
$$|\uv T'(t)|=\frac{1}{\sqrt 13}\sqrt{ (-3\cos t)^2+ (3\sin t)^2\rangle}=\frac 3{\sqrt 13}.$$
$$\uv N(t) = \langle -\cos t, -\sin t \rangle$$

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</ol>
<li>Given the following position function $\myv r(t)=(2+3t+3t^2)\uv i
+(4t+4t^2)\uv j+(-6\cos t)\uv k$, write <br>
$\myv a=a_T \uv T+a_N\uv N$ at $t=0$.  You will <b>not</b> need to do really complicated calculations.
<p class="justme answer">
$$\myv r'(t)=\langle 3+6t,4+8t, +6\sin t\rangle$$
$$\myv a(t) = \myv r''(t)=\langle 6,8, +6\cos t\rangle$$
At $t=0$, $\myv r'=\langle 3,4,0\rangle$, and $|\myv r'|=5$.  So $\uv T=\langle \frac 35, \frac 45,0\rangle$. 
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At $t=0$ $\myv a=\langle 6,8,1\rangle=\langle 6,8,0\rangle + \langle 0,0,1\rangle=10\uv T+\langle 0,0,1\rangle$.  $\langle 0,0,1\rangle$ is a unit vector, and is perpendicular to $\uv T$, so this must be $\uv N$.  
<br><br>So our acceleration vector has tangential and normal components $a_T=10$ and $a_N=1$.
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<li style="page-break-before:always">A particle moves along the curve given by $\myv r(t)=
t\,\uv i+\frac{\sqrt 6}{2}t^2\,\uv j+t^3\,\uv k$.
<ol type="a">
<li>Find the speed of the particle.
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<li>Find the tangential component of acceleration.
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<li>Find the normal component of acceleration.
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<li>Find the curvature of the curve.
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</ol>