# Math 293

# Lecture 13

### Given by Subrata Mukherjee on 2/20/95 at 10:10 in Kimball B11

# Properties of Curves (continued)

Lecture 12 examined some properties
of curves on which we will now make a further expansion, looking at
tangential and normal components of acceleration, oscillatory and
exponential motion, and the distance from a point to a line.

**v** = d**R**/dt = d**R**/ds * ds/dt =
**T** * ds/dt

**a** = d**v**/dt = d**T**/dt * ds/dt +
**T** * =

= d**T**/dt * (ds/dt)^2 + **T** *

d**T**/ds = K***N**, where
`K` is the radius of curvature.

**a** = K(ds/dt)^2 * **N** +
* **T**

**a** = aN * **N** +
aT * **T**

aN = K*(ds/dt)^2 =
K |**v**|^2 = (|**v**|^2)/rho, where rho is the radius.

`aT =
= d|`**v**|/dt

aN from above is the
centripetal acceleration.

**Example:**

Circle:

for constant speed:

### Diff. Equations with complex roots.

`x'' + 2ax' + bx = 0`

x = C1 exp(r1*t) + C2 exp(r2*t), where r1, and r2 are both real

or,` x = (C1 + C2*x) exp(r*t), `where, r1 = r2 =
r

or the oscillatory solution:

`x = exp(-r*t) (C1 cos(w*t) + C2 sin(w*t))`

This would look similar to the following:

**Example:**

`
m*x'' + C*x' + k*x = 0`

x'' + (C/m)x' + (k/m)x = 0

We will now define w^2 = k/m, and the damping factor, C = 2 m z w,
where z represents the Greek letter Zeta.

So,` x'' + 2zw * x' + w^2 * x = 0`

In order to solve a differential equation we can use the same method as
earlier:

`(D^2 + 2zw*D + w^2) x = 0 `

or, make the assumption that x = exp(r*t)

=> `(r^2 + 2zw*r + w^2) exp(r*t) = 0`

These two equations are equivalent.

>From this we can solve for the different values of r:

For z^2 < 1,

`=> r1 = - zw + i w(1-z^2)^(1/2)`

r2 = - zw - i w(1-z^2)^(1/2)

Let` wd = w (1-z^2)^(1/2)`

x = d1 exp(-z w t) exp(i wd t) +

+ d2 exp(-z w t) exp(-i wd t) =

= exp(-z w t) [d1 (cos(wd t) + i sin(wd t)) +

+ d2 (cos(wd t) - i sin(wd t))] =

= exp(-z w t) [C1 cos(wd t) + C2 sin(wd t)]

The Natural Frequency (w) can be represented as:

and the Damped Frequency (wd) as:

In the case where z = 1, the system is called critically damped,
and the equation takes the form:

`x = (A + Bt) exp(-z w t)`

It is called overdamped for z>1.

**Example 2:**

Find the distance (d) from a point (S), to a line (PB).

**u** is the unit vector in the direction of the line.

**Example 3:**

Find the distance between the point S(1,1,1), and
the line: x = -t, y = 1 + t, and z = t

**v** = -**i** + **j** + **k**

**u** = (-**i** + **j** + **k**)/3^(1/2)

Chose t = 0

P = (0,1,0), => **PS** = **i** + **k**

=> d = 2^(1/2)

Created by Milos Borojevic on 3/8/95

Edited by on Lawrence C. Weintraub 3/13/95

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