v-2i-2j-5k-r-i-9j-8k-Find-I-can-do-r-v-r-2-and-i-get-61i-21j-16k-146-but-i-dont-get-w-r-v-why-

Question Number 16401 by sushmitak last updated on 21/Jun/17

$$\boldsymbol{{v}}=\mathrm{2}\boldsymbol{{i}}+\mathrm{2}\boldsymbol{{j}}+\mathrm{5}\boldsymbol{{k}} \\ $$$$\boldsymbol{{r}}=\boldsymbol{{i}}+\mathrm{9}\boldsymbol{{j}}−\mathrm{8}\boldsymbol{{k}} \\ $$$$\mathrm{Find}\:\boldsymbol{\omega} \\ $$$$\mathrm{I}\:\mathrm{can}\:\mathrm{do}\:\frac{\boldsymbol{{r}}×\boldsymbol{{v}}}{{r}^{\mathrm{2}} }=\boldsymbol{\omega} \\ $$$$\mathrm{and}\:\mathrm{i}\:\mathrm{get}\:\boldsymbol{\omega}=\:\frac{\mathrm{61}\boldsymbol{{i}}−\mathrm{21}\boldsymbol{{j}}−\mathrm{16}\boldsymbol{{k}}}{\mathrm{146}} \\ $$$$\mathrm{but}\:\mathrm{i}\:\mathrm{dont}\:\mathrm{get}\:\boldsymbol{{w}}×\boldsymbol{{r}}=\boldsymbol{{v}}. \\ $$$${why}? \\ $$

Answered by ajfour last updated on 21/Jun/17

$$\:\bar {{v}}=\bar {\omega}×\bar {{r}}\:\:\:\:\left({agreed}\right) \\ $$$$\:\bar {{r}}×\bar {{v}}\:=\:\bar {{r}}×\left(\bar {\omega}×\bar {{r}}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:=\:{r}^{\mathrm{2}} \:\bar {\omega}−\bar {{r}}\left(\bar {\omega}\:.\bar {{r}}\right)\: \\ $$$$\:\:{If}\:\:\:\bar {\omega}.\bar {{r}}\:\neq\:\mathrm{0}\:,\:\:\:\:\:\bar {{r}}×\bar {{v}}\:\neq\:{r}^{\mathrm{2}} \:\bar {\omega}\:. \\ $$

Commented by prakash jain last updated on 21/Jun/17

$$\boldsymbol{{v}}=\boldsymbol{{w}}×\boldsymbol{{r}} \\ $$$$\mathrm{This}\:\mathrm{gives}\:\mathrm{the}\:\mathrm{component}\:\mathrm{of}\:\boldsymbol{\mathrm{v}} \\ $$$$\mathrm{which}\:\mathrm{is}\:\mathrm{perpendicular}\:\mathrm{to}\:\boldsymbol{\mathrm{r}}. \\ $$$$\mathrm{In}\:\mathrm{the}\:\mathrm{given}\:\mathrm{question} \\ $$$$\boldsymbol{\mathrm{v}}\centerdot\boldsymbol{\mathrm{r}}=\left(\mathrm{2}+\mathrm{18}−\mathrm{40}\right)=−\mathrm{20}\neq\mathrm{0} \\ $$$$\mathrm{component}\:\mathrm{of}\:\boldsymbol{\mathrm{v}}\:\mathrm{parallel}\:\mathrm{to}\:\boldsymbol{\mathrm{r}} \\ $$$$=\frac{\boldsymbol{\mathrm{v}}\centerdot\boldsymbol{\mathrm{r}}}{\mathrm{r}^{\mathrm{2}} }\boldsymbol{\mathrm{r}}=−\frac{\mathrm{20}}{\mathrm{1}^{\mathrm{1}} +\mathrm{9}^{\mathrm{2}} +\left(−\mathrm{8}\right)^{\mathrm{2}} }\left(\boldsymbol{\mathrm{i}}+\mathrm{9}\boldsymbol{\mathrm{j}}−\mathrm{8}\boldsymbol{\mathrm{k}}\right) \\ $$$$=−\frac{\mathrm{20}}{\mathrm{146}}\left(\boldsymbol{\mathrm{i}}+\mathrm{9}\boldsymbol{\mathrm{j}}−\mathrm{8}\boldsymbol{\mathrm{k}}\right) \\ $$$$\mathrm{comoponent}\:\mathrm{of}\:\boldsymbol{\mathrm{v}}\:\mathrm{perpendicular} \\ $$$$\mathrm{to}\:\boldsymbol{\mathrm{r}}=\boldsymbol{\mathrm{v}}−\frac{\boldsymbol{\mathrm{v}}\centerdot\boldsymbol{\mathrm{r}}}{\mathrm{r}^{\mathrm{2}} }\boldsymbol{\mathrm{r}} \\ $$$$=\left(\mathrm{2}\boldsymbol{\mathrm{i}}+\mathrm{2}\boldsymbol{\mathrm{i}}+\mathrm{5}\boldsymbol{\mathrm{k}}\right)+\left(\frac{\mathrm{20}}{\mathrm{146}}\boldsymbol{\mathrm{i}}+\frac{\mathrm{180}}{\mathrm{146}}\boldsymbol{\mathrm{j}}−\frac{\mathrm{160}}{\mathrm{146}}\boldsymbol{\mathrm{k}}\right) \\ $$$$=\frac{\mathrm{312}\boldsymbol{\mathrm{i}}+\mathrm{472}\boldsymbol{\mathrm{j}}+\mathrm{570}\boldsymbol{\mathrm{k}}}{\mathrm{140}} \\ $$$$−−−−− \\ $$$$\boldsymbol{\mathrm{v}}=\boldsymbol{\omega}×\boldsymbol{\mathrm{r}}=\frac{\mathrm{1}}{\mathrm{146}}\left(\mathrm{61}\boldsymbol{\mathrm{i}}−\mathrm{21}\boldsymbol{\mathrm{j}}−\mathrm{16}\boldsymbol{\mathrm{k}}\right)×\left(\boldsymbol{\mathrm{i}}+\mathrm{9}\boldsymbol{\mathrm{j}}−\mathrm{8}\boldsymbol{\mathrm{k}}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{146}}\left(\mathrm{312}\boldsymbol{\mathrm{i}}+\mathrm{472}\boldsymbol{\mathrm{j}}+\mathrm{570}\boldsymbol{\mathrm{k}}\right) \\ $$$$\boldsymbol{\mathrm{which}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{expected}}\:\boldsymbol{\mathrm{result}}. \\ $$$$\mathrm{Also}\:\mathrm{note}\:\mathrm{that}\:\mathrm{only}\:\mathrm{tangential} \\ $$$$\mathrm{component}\:\mathrm{of}\:\mathrm{velocity}\:\mathrm{contributes} \\ $$$$\mathrm{to}\:\mathrm{angular}\:\mathrm{velocity}. \\ $$

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