Question

The torque of a force \(5\^{i}+3\^{j}−7\^{k}\) about the origin is τ. If the force acts on a particle whose position vector is\( 2\^{i}+2\^{j}+\^{k}\), then the value of τ will be

• A. \(11\^{i}+19\^{j}−4\^{k}\)
• B. \(-11\^{i}+9\^{j}−16\^{k}\)
• C. \(-17\^{i}+19\^{j}−4\^{k}\)
• D. \(17\^{i}+9\^{j}+16\^{k}\)

Answer 1

The Correct Option is C

To determine the torque \(\tau\) produced by a force \(\vec{F} = 5\hat{i} + 3\hat{j} - 7\hat{k}\) about the origin, given that the force acts on a particle located at the position \(\vec{r} = 2\hat{i} + 2\hat{j} + \hat{k}\), we can use the formula for torque: 

\(\vec{\tau} = \vec{r} \times \vec{F}\)

This is the cross product of the position vector with the force vector. We calculate the cross product by setting up the determinant:

	\(\hat{i}\)	\(\hat{j}\)	\(\hat{k}\)
Row 1	2	2	1
Row 2	5	3	-7

Using the determinant, the cross product is:

\(\vec{\tau} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 2 & 1 \\ 5 & 3 & -7 \end{vmatrix}\)

Expanding this determinant:

• \(\hat{i} (2 \times (-7) - 1 \times 3) = \hat{i} (-14 - 3) = -17\hat{i}\)
• \(\hat{j} (1 \times 5 - (-7) \times 2) = \hat{j} (5 + 14) = 19\hat{j}\)
• \(\hat{k} (2 \times 3 - 5 \times 2) = \hat{k} (6 - 10) = -4\hat{k}\)

Thus, the torque \(\vec{\tau}\) is:

\(\vec{\tau} = -17\hat{i} + 19\hat{j} - 4\hat{k}\)

This torque matches the option \(-17\hat{i} + 19\hat{j} - 4\hat{k}\), which is the correct answer.