Units and Algebra: Maths Diagnostic

$\rm {The \ length \ of \ one \ side \ of \ a \ cube \ is \ 5.00 \ cm. \\ What \ is \ the \ volume \ in \ m^3 \ ?}$

$\rm {The \ density \ of \ aluminium \ is \ 2.70 \ g cm^{−3}, \\ convert \ this \ density \ to \ kg m^{−3}}.$

$\rm{Calculate \ {\it a} \ in \ ms^{-2}, \ where \ {\it V} = 72 \ km/h \ and \ {\it R} = 2000 \ m, \\ from \ the \ equation \ {\it a} =\frac{ V^2}{R} }$

${\rm Given} \ \pi r^2v = 5.2 \times 10^{-6} \ \rm {m^3 s^{-1}}, \\ find \ the \ value \ of \ { \it v} \ if \ {\it r} = 3.6 \ mm.$

$\rm The \ mass \ of \ a \ proton \ can \ be \ expressed \ as \ 1.0 {\it \ u}, \ where \ {\it u} \ is \ the \ Atomic \ Mass \ Unit \ and \ 1 \ {\it u} = 1.67\times 10^{−27} kg. \\ If \ a \ proton \ has \ a \ mass \ 2,000 \ times \ the \ mass \ of \ an \ electron, \ determine \ the \ energy, \ {\it E}, \ of \ an \ electron \ from \ its \ mass, \ {\it m}, \\ and \ the \ speed \ of \ light \ {\it c} = 3.0 \times 10^8 m s^{−1}, \ using \ the \ relationship \ between \ energy \ and \ mass \ given \ by \ \it E = mc^2.$

$P_2 = \rho g h + P_1 \ {\rm and} \ Q = \frac{\pi R^4(P_2−P_1)}{8\eta L}. \\ \rm Eliminate \ {\it P_1} \ and \ {\it P_2} \ and \ write \ an \ equation \ for \ {\it h} \ in \ terms \ of \ \it Q, \ R, \ ρ, \ g, \ \eta \ {\rm and} \ L.$

$\rm Determine \ the \ uncertainty \ in \ density, \sigma_{\rho}, \ in \ kg m^{−3}, \ of \ a \ cube \ of \ Manganese, \ if \ for \ a \ given \ set \ of \ measurements \\ length \ {\it l} = 3 \ cm, \ mass \ {\it m} = 2 \times 10^{-1} \ kg, \ uncertainty \ in \ length, \ \sigma_{\it l} = 5 \times 10^{-3} m, \ and \ uncertainty \ in \ mass, \ \sigma_{\it m} = 2 \ g, \\ using \ the \ equation$ $$\sigma_\rho = \frac{\it m}{\it l^3}\sqrt{ \left(\frac{\sigma_{\it m}}{\it m}\right)^2 + 3 \left(\frac{\sigma_{\it l}}{\it l} \right)^2}$$