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Question: A body is rotating with angular velocity \(\overrightarrow{\omega} = \left( 3\widehat{i} - 4\widehat...

A body is rotating with angular velocity ω→=(3i^−4j^+k^)\overrightarrow{\omega} = \left( 3\widehat{i} - 4\widehat{j} + \widehat{k} \right) The linear velocity of a point having position vector r→=(5i^−6j^+6k^)\overrightarrow{r} = \left( 5\widehat{i} - 6\widehat{j} + 6\widehat{k} \right) is

A

6i^+2j^−3k^6\widehat{i} + 2\widehat{j} - 3\widehat{k}

B

18i^+3j^−2k^18\widehat{i} + 3\widehat{j} - 2\widehat{k}

C

−18i^−13j^+2k^- 18\widehat{i} - 13\widehat{j} + 2\widehat{k}

D

6i^−2j^+8k^6\widehat{i} - 2\widehat{j} + 8\widehat{k}

Answer

−18i^−13j^+2k^- 18\widehat{i} - 13\widehat{j} + 2\widehat{k}

Explanation

Solution

Here, ω→=3i^−4j^+k^\overrightarrow{\omega} = 3\widehat{i} - 4\widehat{j} + \widehat{k}

r^=5j^−−6j^+6k^\widehat{r} = 5\widehat{j} - - 6\widehat{j} + 6\widehat{k}

As v→=ω→×r→\overrightarrow{v} = \overrightarrow{\omega} \times \overrightarrow{r}

\widehat{i} & \widehat{j} & \widehat{k} \\ 3 & - 4 & 1 \\ 5 & - 6 & 6 \end{matrix} \right|$$ =$\widehat{i}( - 24 - ( - 6)) + \widehat{j}(5 - 18) + \widehat{k}( - 18 - ( - 20))$ $$= - 18\widehat{i} - 13\widehat{j} + 2\widehat{k}$$