Irodov Problem 1.153 – Torque about Different Points

Problem Statement

A force F = aî + bĵ is applied at point r = xî + yĵ. Find the torque about the origin and about point (x₀, y₀).

Given Information

• Force F = (a, b)
• Point of application r = (x, y)

Physical Concepts & Formulas

$$\boldsymbol{\tau}=\mathbf{r}\times\mathbf{F}$$

Step-by-Step Solution

Step 1: τ about origin = r × F = (xb − ya)k̂.

Step 2: Displacement from (x₀,y₀): Δr = (x−x₀)î + (y−y₀)ĵ.

Step 3: τ about (x₀,y₀) = [(x−x₀)b − (y−y₀)a]k̂.

Worked Calculation

τ_origin = (xb−ya)k̂

Answer

$$\boxed{\boldsymbol{\tau}=(xb-ya)\hat{k}\text{ about origin}}$$

Physical Interpretation

Torque depends on the choice of reference point. The cross product selects the component of force perpendicular to the moment arm.