Tangential and normal acceleration

Question

The electron again

The electron is joined by a tiny rubber blob of length and width each equal to $0.1$. Usual motion:

$$\mathbf f(t)=\langle \cos(t)(2-\cos(4t)),\sin(t)(2-\cos(4t)),\sin(4t)\rangle$$

Headwind (resp. tiny monkey) causes rubber blob to change length (resp. width) in proportion to the acceleration $a$ along the path (resp. the acceleration $b$ normal to the path).

$$\Delta(\textrm{length})=-0.01a\qquad \Delta(\textrm{width})=0.01b$$

Examples

What is acceleration along and perpendicular to these paths?

Examples

First: $(t^{45},0,0), -1\leq t\leq 1$ loop

Examples

First: $(t^{45},0,0)$

Motion on a string!

The acceleration in the direction of motion is just the vector $\langle 1980 t^{43},0,0\rangle$. It always points in the direction of the motion (including negative scalars, so could point in opposite direction).

How is the sign change at $t=0$ reflected in the motion?

Examples

Second: $(t^2-1,t(t^2-1),0), -2\leq t\leq 2$ loop

acceleration

tangential component

normal component

Examples

Second: $(t^2-1,t(t^2-1),0)$, observations

Examples

Second: $(t^2-1,t(t^2-1),0)$

To calculate: find the unit tangent and normal vectors, then calculate components of acceleration.

Examples

Do the third: the helix $(\cos(t),\sin(t),t)$

To calculate: find the unit tangent and normal vectors, then calculate components of acceleration.

Examples

Do the third: the helix $(\cos(t),\sin(t),t)$

To calculate: find the unit tangent and normal vectors, then calculate components of acceleration.

General theory

The procedure is always the same: find unit tangent and unit normal, calculate components.

The book has a discussion of various formulas, deductions using curvature, the product rule, etc.

The upshot: given a path $\mathbf r(t)$ with unit tangent $\mathbf T(t)$ and unit normal $\mathbf N(t)$, we can write the acceleration $\mathbf a=a_T\mathbf T+a_N\mathbf N$ where

$$a_T=\frac{\mathbf r'(t)\cdot\mathbf r''(t)}{|\mathbf r'(t)|}\qquad\quad a_N=\frac{|\mathbf r'(t)\times\mathbf r''(t)|}{|\mathbf r'(t)|}$$

Do one: the electron/rubber blob $$\mathbf f(t)=\langle \cos(t)(2-\cos(4t)),\sin(t)(2-\cos(4t)),\sin(4t)\rangle$$

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