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Ch. 11 - Angular Momentum; General Rotation
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 11 - Angular Momentum; General RotationProblem 37
Chapter 11, Problem 37

Two identical particles have equal but opposite momenta, p→\(\overrightarrow{p}\) and −p→-\(\overrightarrow{p}\), but they are not traveling along the same line. Show that the total angular momentum of this system does not depend on the choice of origin.

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Start by recalling the definition of angular momentum for a particle. The angular momentum of a particle relative to a chosen origin is given by the vector cross product: L⃗ = r⃗ × p⃗, where r⃗ is the position vector of the particle relative to the origin, and p⃗ is its linear momentum.
For the system of two particles, the total angular momentum is the vector sum of the angular momenta of the two particles: L⃗ _{total} = L⃗ _1 + L⃗ _2 = (r⃗ _1 × p⃗ _1) + (r⃗ _2 × p⃗ _2), where subscripts 1 and 2 refer to the two particles.
Substitute the given momenta into the equation. The first particle has momentum p⃗ _1 = p⃗, and the second particle has momentum p⃗ _2 = -p⃗. Thus, the total angular momentum becomes: L⃗ _{total} = (r⃗ _1 × p⃗) + (r⃗ _2 × (-p⃗)).
Simplify the expression by factoring out p⃗: L⃗ _{total} = (r⃗ _1 - r⃗ _2) × p⃗. Notice that the total angular momentum depends only on the relative position vector r⃗ _1 - r⃗ _2 and the momentum p⃗, not on the choice of origin.
Conclude that since the relative position vector r⃗ _1 - r⃗ _2 is independent of the origin, the total angular momentum L⃗ _{total} does not depend on the choice of origin. This completes the proof.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Angular Momentum

Angular momentum is a vector quantity that represents the rotational motion of an object. It is defined as the product of the position vector and the momentum of the particle. For a system of particles, the total angular momentum is the vector sum of the angular momenta of each particle, which depends on both their momenta and their positions relative to a chosen origin.
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Conservation of Angular Momentum

The conservation of angular momentum states that if no external torque acts on a system, the total angular momentum remains constant. This principle is crucial in analyzing systems of particles, as it implies that the angular momentum calculated from different origins will yield the same total value, provided the internal interactions are unchanged.
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Choice of Origin

The choice of origin in a coordinate system can affect the calculation of position vectors but does not affect the physical laws governing the system. When analyzing angular momentum, shifting the origin will change the position vectors of the particles, but the total angular momentum remains invariant due to the symmetrical nature of the momenta involved, particularly when they are equal and opposite.
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Related Practice
Textbook Question

An engineer estimates that under the most adverse expected weather conditions, the total force on the highway sign in Fig. 11–33 will be F→\(\overrightarrow{F}\) = (± 2.4 î - 4.1 ĵ) kN, acting at the cm. What torque does this force exert about the base O?

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Two ice skaters, both of mass 68 kg, approach on parallel paths 1.6 m apart. Both are moving at 3.5 m/s with their arms outstretched. They join hands as they pass, still maintaining their 1.6-m separation, and begin rotating about one another. Treat the skaters as particles with regard to their rotational inertia. Calculate the change in kinetic energy for this process.

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Textbook Question

Calculate the angular momentum of a particle of mass m moving with constant velocity υ for two cases (see Fig. 11–34): about O′.

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Textbook Question

A particle is at the position (x, y, z) = (1.0, 2.0, 3.0)m. It is traveling with a vector velocity (-5.0 ,+ 2.8, -3.1)m/s. Its mass is 4.3 kg. What is its vector angular momentum about the origin?

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Textbook Question

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Textbook Question

Two lightweight rods 24 cm in length are mounted perpendicular to an axle and at 180° to each other (Fig. 11–35). At the end of each rod is a 480-g mass. The rods are spaced 42 cm apart along the axle. The axle rotates at 4.5 rad/s.

(a) What is the component of the total angular momentum along the axle?

(b) What angle does the vector angular momentum make with the axle? [Hint: Remember that the vector angular momentum must be calculated about the same point for both masses, which could be the cm.]

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