Irodov Problem 1.212

Problem Statement

A particle moves in the xy-plane. Its position is r = (at, bt²) where a and b are constants. Find the angular momentum about the origin.

Given

r(t) = (at, bt²). m = mass.

Concepts & Formulas

L = m(r × v). v = dr/dt = (a, 2bt). L_z = m(x·vy − y·vx).

Step-by-Step Solution

Step 1: x = at, y = bt², vx = a, vy = 2bt.
Step 2: L_z = m(x·vy − y·vx) = m(at·2bt − bt²·a) = m(2abt² − abt²) = mabt².
Step 3: dL_z/dt = 2mabt = torque.

Worked Calculation

L_z = mabt².

Boxed Answer

L_z = mabt²

Physical Interpretation

The angular momentum grows with time, confirming a non-zero torque acts — the trajectory curves under a force with a moment about the origin.