Irodov Problem 1.152 – Angular Momentum of Projectile

Problem Statement

A stone is thrown at angle θ with speed v₀. Find the angular momentum about the launch point as a function of time.

Given Information

• Launch angle θ, speed v₀
• Origin at launch point

Physical Concepts & Formulas

$$L_z=m(xv_y-yv_x)$$

Step-by-Step Solution

Step 1: x = v₀cosθ·t, y = v₀sinθ·t − ½gt².

Step 2: vₓ = v₀cosθ, vᵧ = v₀sinθ − gt.

Step 3: L = m(xvᵧ − yvₓ) = m·v₀cosθ·t·(v₀sinθ−gt) − m·(v₀sinθ·t−½gt²)·v₀cosθ = ½mgv₀t²cosθ.

Worked Calculation

L = ½mgv₀t²cosθ

Answer

$$\boxed{L=\frac{1}{2}mgv_0t^2\cos\theta}$$

Physical Interpretation

Angular momentum grows as t² because gravity exerts a torque mgx = mg·v₀cosθ·t that grows linearly with time, causing L to increase as t².