Dimensions in Engineering Homework

Q$1$

Using definitions give the dimensions and units for the following quantity

$($a$)$ Velocity

$($b$)$ Acceleration

$($c$)$ Charge

$($d$)$ Frequency

$($e$)$ Density

$($f$)$ Momentum

Q$2$

Give the following derived units in terms of base units $(m,kg,s,A,K,$mol,$cd)$ using appropriate laws

$($a$)$ Pa $($Pascal Unit of Pressure$)$

$($b$)T($Tesla Unit of magnetic flux density$)$

$($c$)$ Farad $($Unit of capacitance$)$

Q$3$

Q$4$ Determine the units of the following fundamental constants

$($a$)$ The permittivity of free space $({}_0)$

$($b$)$ The permeability of free space $($

Q$5$ Check that the following dimensionless numbers actually have no units

$($a$)$ Nusselt Number

$($b$)$ Reynolds number

Nusselt Number $=\frac{hd}{k}$

Reynolds Number $Re=\frac{vd}{}$

$h$ is heat transfer coefficient and has units $W{m}^{-2}{K}^{-1}$

$k$ is thermal conductivity and has units $W{m}^{-2}{K}^{-1}$

d is a characteristic distance

$$ is density

$v$ is velocity

$$ is viscosity and has units Pas $($Pascal seconds$)$

The period of a pendulum Tis known to depend on the gravitational acceleration $g$ and the length $1$

only.

Determine a formula of the form

$T=k**g^a**l^b$

Giving the values of the unknown powers a and $b,$ but not the value of the proportionality constant

k$.($k is dimensionless$)$

Hint equate dimension or unit powers of right and left hand$-$side of equation

$7$

Check that the left hand$-$side of the Bernoulli equation is dimensionally correct

$\frac{p}{g}+\frac{v^2}{2g}+h=z$

$p=$ pressure,$=$ density,$g=$ gravitational acceleration, $v=$ velocity,$h=$ height

Given the following values calculate $z$ and give

$p=100$kPa,$g=9\mathrm{.}81m{s}^{-2},v=30kmh{r}^{-1},h=20m,=1\mathrm{.}0gc{m}^{-3}$

$8$

A simply supported beam of $S355$ strength steel is required to support a load $($W$)$ of $333$ kN $.$ The

span of the beam $(L)$ is $550$ cm $.$

Calculate the deflection $$ in mm given the following information

$=\frac{5w{L}^3}{384E!}$

where:

$E=210,0000\frac{N}{m}{m}^2$

$I=11,700c{m}^4$Kk