Question

A force \( \vec{F}=(\hat{i}+2 \hat{j}+3 \hat{k}) N \) acts at a point \( (4 \hat{i}+3 \hat{j}-\hat{k}) m \). Then the magnitude of torque about the point \( (\hat{i}+2 \hat{j}+\hat{k}) \) \( m \) will be \( \sqrt{x} N-m \). The value of \( x \) is [5 Sep, 2020 (Shift-1)] (a) 195 (b) 165 (c) 105 (d) 135

Answer 1

Correct option is (a) 195

Force, \(F = (i + 2j + 3k)N\)

Torque, \(\tau = \sqrt x N-m\)

\(r= (4i - i) + (3j - 2j) + (-k-k)\)

\(r = (3i + j - 2k)m\)

\(\tau = r \times F\)

\(\tau = (3i + j - 2k) \times (i + 2j + 3k)\)

\(\tau = \begin{vmatrix} i& j&k\\3&1&-2\\1&2&3\end{vmatrix}\)

\(= i(3 + 4) - j(9 + 2) + k(6-1)\)

\(= 7i - 11 j + 5k\)

\(|\tau| =\sqrt{7^2 +11^2+5^2}\)

\(= \sqrt{195}\)

\(|\tau | = \sqrt x N - m\)

\(\sqrt x = \sqrt{195}\)

\(\Rightarrow x = 195\)