\(\rm{The \ acceleration, \ a \ of \ an \ object \ is \ given \ by : \ } a=\frac{v_f-v_i}{t}\)
\( \rm{where \ v_f \ is \ the \ final \ velocity, \ v_i \ is \ the \ initial \ velocity, \ and \ t \ is \ the \ time \ interval.}\)
\( \rm{ \ Rearrange \ this \ equation \ to \ make \ v_i \ the \ subject. }\)
\( v_i = at-v_f \)
\( v_i = a + t -v \)
\( v_i = a + t + v_f \)
\( v_i=\frac{at}{v_f} \)
\( v_i = v_f-at \)
\( E = mgh + \frac{1}{2}mv^2 \)
\(\rm{Rearrange \ this \ equation \ for \ v}\)
\( v = \sqrt{2 ( \frac{E}{m}-gh ) } \)
\(v = \sqrt{\frac{1}{2} (\frac{E}{m}-gh ) } \)
\( v = \sqrt{2 (E-mgh)} \)
\( v = \sqrt{\frac{2}{m}(E-gh)} \)
\( v = \sqrt{\frac{m}{2}(E-gh)} \)
\(\rm{If \ } \omega = \frac{2\pi}{\it T}, \rm{ \ and \ } \it k=\it m\omega^2, \rm{ \ what \ is \ T \ in \ terms \ of \ m \ and \ k?} \)
\( T=2\pi \sqrt{\frac{m}{k}} \)
\( T = \sqrt{2\pi\frac{m}{k}} \)
\( T = 2\pi \sqrt{\frac{k}{m}} \)
\(T= \sqrt{2\pi\frac{k}{m}} \)
\( T = 4\pi^2 \sqrt{\frac{m}{k}} \)
\( P_2 = \rho g h + P_1 \ {\rm and} \ Q = \frac{\pi R^4(P_2−P_1)}{8\eta L}. \\ \rm Eliminate \ {\it P_1} \ and \ {\it P_2} \ and \ write \ an \ equation \ for \ {\it h} \ in \ terms \ of \ \it Q, \ R, \ ρ, \ g, \ \eta \ {\rm and} \ L. \)
\( h = \frac{8 \eta L Q}{\pi R^4 \rho g} \)
\( h = \frac{8 \eta L Q}{\pi R^4 } - \rho g \)
\( h = \frac{8 \eta L Q - \pi R^4}{\pi R^4 \rho g} \)
\( h = \frac{8 \eta L Q - \pi R^4}{\rho g} \)
\( h = \frac{ \pi R^4 - 8 \eta L Q }{ \rho g} \)
\( \rm{ Rearrange:} \)
\[ \sigma_\rho = \frac{\it m}{\it l^3}\sqrt{ \left(\frac{\sigma_{\it m}}{\it m}\right)^2 + 3 \left(\frac{\sigma_{\it l}}{\it l} \right)^2} \]
\(\rm{for \ \sigma_{\it m}} \)
\[ \sigma_{\it m} = \it m\sqrt{ \left(\frac{\sigma_\rho l^3}{\it m}\right)^2 - 3 \left(\frac{\sigma_{\it l}}{\it l} \right)^2} \]
\[ \sigma_{\it m} = \it m\sqrt{ \left(\frac{\sigma_\rho l^3}{\it m}\right)^2 + 3 \left(\frac{\sigma_{\it l}}{\it l} \right)^2} \]
\[ \sigma_{\it m} = \frac{\it m}{\it l^3} \sqrt{ \left(\frac{\sigma_\rho l^3}{\it m}\right)^2 - 3 \left(\frac{\sigma_{\it l}}{\it l} \right)^2} \]
\[ \sigma_{\it m} = \sqrt{\it m \left( \left(\frac{\sigma_\rho l^3}{\it m}\right)^2 - 3 \left(\frac{\sigma_{\it l}}{\it l} \right)^2 \right)} \]
