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

Giancoli Douglas 5th edition

Ch. 11 - Angular Momentum; General Rotation

Problem 23b

Chapter 11, Problem 23b

Show that î x ĵ = k̂ , î x k̂ = - ĵ, and ĵ x k̂ = î.

Verified step by step guidance

Understand the problem: The task is to verify the cross products of unit vectors in a Cartesian coordinate system. The unit vectors î, ĵ, and k̂ represent the x, y, and z directions, respectively. The cross product of two vectors results in a vector perpendicular to both, following the right-hand rule.

Recall the right-hand rule: To determine the direction of the cross product, point your right-hand fingers in the direction of the first vector (e.g., î), curl them toward the second vector (e.g., ĵ), and your thumb will point in the direction of the resulting vector (e.g., k̂).

Verify î x ĵ = k̂: Using the right-hand rule, point your fingers in the direction of î (x-axis) and curl them toward ĵ (y-axis). Your thumb points in the direction of k̂ (z-axis), confirming that î x ĵ = k̂.

Verify î x k̂ = -ĵ: Point your fingers in the direction of î (x-axis) and curl them toward k̂ (z-axis). Your thumb points in the negative y-direction, confirming that î x k̂ = -ĵ.

Verify ĵ x k̂ = î: Point your fingers in the direction of ĵ (y-axis) and curl them toward k̂ (z-axis). Your thumb points in the direction of î (x-axis), confirming that ĵ x k̂ = î.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Was this helpful?

Key Concepts

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

Vector Cross Product

The vector cross product is a binary operation on two vectors in three-dimensional space, resulting in a third vector that is perpendicular to the plane formed by the original vectors. The magnitude of the cross product is given by the product of the magnitudes of the two vectors and the sine of the angle between them. This operation is denoted by the symbol '×' and is essential for determining torque, angular momentum, and other vector quantities.

Unit Vectors

Unit vectors are vectors with a magnitude of one, used to indicate direction. In three-dimensional space, the standard unit vectors are î, ĵ, and k̂, which represent the x, y, and z axes, respectively. They are fundamental in vector analysis, allowing for the representation of any vector as a combination of these unit vectors, simplifying calculations involving direction and magnitude.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the resultant vector in a cross product. By aligning the fingers of the right hand in the direction of the first vector and curling them towards the second vector, the thumb points in the direction of the cross product. This rule is crucial for visualizing and understanding the orientation of vectors in three-dimensional space.

Recommended video:

Guided course

19:11

Force on Moving Charges & Right Hand Rule

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 $\vec{F}$ = (± 2.4 î - 4.1 ĵ) kN, acting at the cm. What torque does this force exert about the base O?

views

Textbook Question

A woman of mass m stands at the edge of a solid cylindrical platform of mass M and radius R. At t = 0, the platform is rotating with negligible friction at angular velocity ω₀ about a vertical axis through its center, and the woman begins walking with speed υ (relative to the platform) toward the center of the platform. Determine the angular velocity of the system as a function of time.

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′.

views

Textbook Question

A woman of mass m stands at the edge of a solid cylindrical platform of mass M and radius R. At t = 0, the platform is rotating with negligible friction at angular velocity ω₀ about a vertical axis through its center, and the woman begins walking with speed υ (relative to the platform) toward the center of the platform. What will be the angular velocity when the woman reaches the center?

views

Textbook Question

A person of mass 75 kg stands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 920. kg·m². The platform rotates without friction with angular velocity 0.95 rad/s. The person walks radially to the edge of the platform. Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.

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 origin O.

views