Skip to main content
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 42
Chapter 11, Problem 42

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.]
Two perpendicular rods with 480 g masses at each end, spaced 42 cm apart, mounted on a rotating axle.

Verified step by step guidance
1
Step 1: Understand the problem. The system consists of two rods with masses at their ends, rotating about an axle. The goal is to calculate (a) the component of the total angular momentum along the axle and (b) the angle the angular momentum vector makes with the axle. The rods are perpendicular to the axle, and the masses are rotating at an angular velocity of 4.5 rad/s.
Step 2: Calculate the moment of inertia for each mass. The moment of inertia for a point mass is given by the formula: I = mr2, where m is the mass and r is the distance from the axis of rotation. For each mass, the distance from the axis is 24 cm (converted to meters: 0.24 m). The mass is 480 g (converted to kilograms: 0.48 kg). Compute the moment of inertia for each mass.
Step 3: Calculate the angular momentum of each mass. The angular momentum is given by the formula: L = Iω, where I is the moment of inertia and ω is the angular velocity (4.5 rad/s). Compute the angular momentum for each mass.
Step 4: Determine the total angular momentum along the axle. Since the two masses are at 180° to each other, their angular momentum components along the axle will add up. Use the geometry of the system to resolve the angular momentum vector of each mass into components along the axle and perpendicular to the axle. Add the components along the axle to find the total angular momentum along the axle.
Step 5: Calculate the angle the angular momentum vector makes with the axle. Use the relationship between the total angular momentum vector and its components. The angle θ can be found using the formula: θ = tan-1(LperpendicularLaxial). Compute the angle using the resolved components of angular momentum.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
13m
Was this helpful?

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 inertia and rotational velocity of an object. It is calculated as the product of the moment of inertia and the angular velocity. For point masses, it can be expressed as the cross product of the position vector and the linear momentum vector. Understanding angular momentum is crucial for analyzing rotational motion and its conservation in systems.
Recommended video:
Guided course
06:18
Intro to Angular Momentum

Moment of Inertia

Moment of inertia is a measure of an object's resistance to changes in its rotational motion about an axis. It depends on the mass distribution relative to the axis of rotation. For point masses, it is calculated as the sum of the products of each mass and the square of its distance from the axis. This concept is essential for determining the angular momentum of the system in the given problem.
Recommended video:
Guided course
11:47
Intro to Moment of Inertia

Vector Components

Vector components are the projections of a vector along the axes of a coordinate system. In the context of angular momentum, it is important to resolve the total angular momentum into components to analyze its direction and magnitude relative to the axle. This involves using trigonometric functions to find the angles and applying the right-hand rule to determine the orientation of the angular momentum vector.
Recommended video:
Guided course
07:30
Vector Addition By Components
Related Practice
Textbook Question

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.

Textbook Question

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. If they now pull on each other’s hands, reducing their radius to half its original value, what is their common angular speed after reducing their radius?

1
views
Textbook Question

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.

2
views
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′.

1
views
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?

2
views
Textbook Question

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. They now pull on each other’s hands, reducing their radius to half its original value. Calculate the change in kinetic energy for this process.

2
views