A force of $-\mathrm{P} \hat{\mathrm{k}}$ acts on the origin of the coordinate system. The torque about the point $(2,-3)$ is $\mathrm{P}(\mathrm{a} \hat{\mathrm{i}}+\mathrm{b} \hat{\mathrm{j}})$, the ratio of $\frac{\mathrm{a}}{\mathrm{b}}$ is $\frac{\mathrm{x}}{2}$. The value of $x$ is

JEE Main · 2023 · 10 Apr 2023 · Shift 2 · Rotational Mechanics

A force of

- P \hat{k}

acts on the origin of the coordinate system. The torque about the point

(2, - 3)

P (a \hat{i} + b \hat{j})

, the ratio of

\frac{a}{b}

\frac{x}{2}

. The value of

x

Answer: 3

Solution

The torque about a point is

τ = r \times F

, where

r

runs from that point to where the force acts. From

(2, - 3, 0)

to the origin,

r = - 2 \hat{i} + 3 \hat{j}

, and the force is

F = - P \hat{k}

. Hence

τ = (- 2 \hat{i} + 3 \hat{j}) \times (- P \hat{k}) = - 3 P \hat{i} - 2 P \hat{j} = P (- 3 \hat{i} - 2 \hat{j})

. Matching with

P (a \hat{i} + b \hat{j})

gives

a = - 3

and

b = - 2

, so

\frac{a}{b} = \frac{3}{2} = \frac{x}{2}

. The value of

x

3

JEE Main · 2023 · 10 Apr 2023 · Shift 2 · Integer answer

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