The Angular Momentum

Step 1: Angular Momentum Formula The angular momentum L of a particle with respect to the origin is given by the cross product of its position vector r and its linear momentum vector p = m v: L = r \times p = r \times (m v). Step 2: Position and Velocity Vectors The position of the particle is given as (0, R), so the position vector is: r = 0 î + R ĵ = R ĵ. The velocity of the particle is given as: v = -v î. The linear momentum vector is: p = m (-v î) = -m v î. Step 3: Compute the Cross Product Now, we compute the cross product: L = (R ĵ) \times (-m v î) = -m v R (ĵ \times î). Step 4: Cross Product of Unit Vectors We know that the cross product of unit vectors follows the cyclic order: î \times ĵ = k̂, ĵ \times k̂ = î, k̂ \times î = ĵ. Also, ĵ \times î = - (î \times ĵ) = - k̂. Step 5: Final Expression for Angular Momentum Substituting this into the expression for L: L = -m v R (-k̂) = m v R k̂. Therefore, the angular momentum of the particle with respect to the origin is: L = m v R k̂.

Chapter Questions
1 MCQs