Mathematical Introduction for Physics and Engineering

by Samuel Dagan (Copyright © 2007-2020)

**
The force of a non-linear spring is:
**

$F=-q{x}^{3}$ |
---|

We'll follow the notation for the 3 basic physical dimensions already used.

*Μ*denotes the dimension of a mass*m*: [*m*]*Θ*denotes the dimension of time*t*: [*t*]*Λ*denotes the dimension of length*l*: [*l*]

The dimensions of *q* can be expressed by the dimensions of the force *F* and the elongation *l* according to the above given relation:

The rest of the variables have obviously the dimensions:

- [
*T*]=*Θ* - [
*m*]=*Μ* - [
*l*]=*Λ*

We assume that the period *T* is expressed by the other variables by

$T=c{m}^{\alpha}{l}^{\beta}{q}^{\gamma}$ ,
where *c* is a dimensionless constant.
## Part 3

From part 1 and part 2 one obtains

$\Theta ={{\rm M}}^{\alpha}{\Lambda}^{\beta}{\left({\rm M}{\Lambda}^{-2}{\Theta}^{-2}\right)}^{\gamma}$ .
## Part 4

## Part 5

## Score

The comparison of the powers of part 3 gives:

*α*+*γ*=0*β*−2*γ*=0- −2
*γ*=1

The solution of part 4 is

- $\alpha ={\scriptstyle \frac{1}{2}}$
- $\beta =-1$
- $\gamma =-{\scriptstyle \frac{1}{2}}$

and finally

**$T=\frac{c}{l}\sqrt{\frac{m}{q}}$
** .

Part 1 is worth 3 points.

Parts 2 is worth 1 point.

Parts 3,4 and 5 are worth 2 points each.