Skip to content Skip to navigation Skip to collection information

OpenStax-CNX

You are here: Home » Content » College Physics » Gyroscopic Effects: Vector Aspects of Angular Momentum
Content endorsed by: OpenStax College

Navigation

Table of Contents

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Endorsed by Endorsed (What does "Endorsed by" mean?)

This content has been endorsed by the organizations listed. Click each link for a list of all content endorsed by the organization.
  • OpenStax College

    This collection is included in aLens by: OpenStax College

    Click the "OpenStax College" link to see all content they endorse.

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • Pierpont C & TC display tagshide tags

    This module is included inLens: Pierpont Community & Technical College's Lens
    By: Pierpont Community & Technical CollegeAs a part of collection: "College Physics -- HLCA 1104"

    Click the "Pierpont C & TC" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

  • Featured Content display tagshide tags

    This collection is included inLens: Connexions Featured Content
    By: Connexions

    Comments:

    "This introductory, algebra-based, two-semester college physics book is grounded with real-world examples, illustrations, and explanations to help students grasp key, fundamental physics concepts. […]"

    Click the "Featured Content" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

Gyroscopic Effects: Vector Aspects of Angular Momentum

Module by: OpenStax College. E-mail the author

Summary:

  • Describe the right-hand rule to find the direction of angular velocity, momentum, and torque.
  • Explain the gyroscopic effect.
  • Study how Earth acts like a gigantic gyroscope.

Angular momentum is a vector and, therefore, has direction as well as magnitude. Torque affects both the direction and the magnitude of angular momentum. What is the direction of the angular momentum of a rotating object like the disk in Figure 1? The figure shows the right-hand rule used to find the direction of both angular momentum and angular velocity. Both LL size 12{L} {} and ωω size 12{ω} {} are vectors—each has direction and magnitude. Both can be represented by arrows. The right-hand rule defines both to be perpendicular to the plane of rotation in the direction shown. Because angular momentum is related to angular velocity by L=IωL=Iω size 12{L=Iω} {}, the direction of LL size 12{L} {} is the same as the direction of ωω size 12{ω} {}. Notice in the figure that both point along the axis of rotation.

Figure 1: Figure (a) shows a disk is rotating counterclockwise when viewed from above. Figure (b) shows the right-hand rule. The direction of angular velocity ω ω size and angular momentum L L are defined to be the direction in which the thumb of your right hand points when you curl your fingers in the direction of the disk’s rotation as shown.
In figure a, a disk is rotating in counter clockwise direction. The direction of the angular momentum is shown as an upward vector at the centre of the disk. The vector is labeled as L is equal to I-omega. In figure b, a right hand is shown. The fingers are curled in the direction of rotation and the thumb is pointed vertically upward in the direction of angular velocity and angular momentum.

Now, recall that torque changes angular momentum as expressed by

net τ= Δ L Δ t .net τ= Δ L Δ t . size 12{"net "τ= { {ΔL} over {Δt} } } {}
(1)

This equation means that the direction of Δ L Δ L size 12{ΔL} {} is the same as the direction of the torque ττ size 12{τ} {} that creates it. This result is illustrated in Figure 2, which shows the direction of torque and the angular momentum it creates.

Let us now consider a bicycle wheel with a couple of handles attached to it, as shown in Figure 3. (This device is popular in demonstrations among physicists, because it does unexpected things.) With the wheel rotating as shown, its angular momentum is to the woman's left. Suppose the person holding the wheel tries to rotate it as in the figure. Her natural expectation is that the wheel will rotate in the direction she pushes it—but what happens is quite different. The forces exerted create a torque that is horizontal toward the person, as shown in Figure 3(a). This torque creates a change in angular momentum LL size 12{L} {} in the same direction, perpendicular to the original angular momentum LL size 12{L} {}, thus changing the direction of LL size 12{L} {} but not the magnitude of LL size 12{L} {}. Figure 3 shows how ΔLΔL size 12{ΔL} {} and LL size 12{L} {} add, giving a new angular momentum with direction that is inclined more toward the person than before. The axis of the wheel has thus moved perpendicular to the forces exerted on it, instead of in the expected direction.

Figure 2: In figure (a), the torque is perpendicular to the plane formed by rr size 12{r} {} and FF size 12{F} {} and is the direction your right thumb would point to if you curled your fingers in the direction of FF size 12{F} {}. Figure (b) shows that the direction of the torque is the same as that of the angular momentum it produces.
In figure a, a plane is shown. Force F, lying in the same plane, is acting at a point in the plane. At a point, at distant-r from the force, a vertical vector is shown labeled as tau, the torque. In figure b, there is a child on a horse on a merry-go-round. The radius of the merry-go-round is r units. At the foot of the horse, a vector along the plane of merry-go-round is shown. At the centre, the direction of torque tau, angular velocity omega, and angular momentum L are shown as vertical vectors.

Figure 3: In figure (a), a person holding the spinning bike wheel lifts it with her right hand and pushes down with her left hand in an attempt to rotate the wheel. This action creates a torque directly toward her. This torque causes a change in angular momentum Δ L Δ L in exactly the same direction. Figure (b) shows a vector diagram depicting how Δ L Δ L and L L add, producing a new angular momentum pointing more toward the person. The wheel moves toward the person, perpendicular to the forces she exerts on it.
In figure a, a lady is holding the spinning bike wheel with her hands. The wheel is rotating in counter clockwise direction. The direction of the force applied by her left hand is shown downward and that by her right hand in upward direction. The direction of angular momentum is along the axis of rotation of the wheel. In figure b, addition of two vectors L and delta-L is shown. The resultant of the two vectors is labeled as L plus delta L. The direction of rotation is counterclockwise.

This same logic explains the behavior of gyroscopes. Figure 4 shows the two forces acting on a spinning gyroscope. The torque produced is perpendicular to the angular momentum, thus the direction of the torque is changed, but not its magnitude. The gyroscope precesses around a vertical axis, since the torque is always horizontal and perpendicular to LL size 12{L} {}. If the gyroscope is not spinning, it acquires angular momentum in the direction of the torque (L=ΔLL=ΔL size 12{L=ΔL} {}), and it rotates around a horizontal axis, falling over just as we would expect.

Earth itself acts like a gigantic gyroscope. Its angular momentum is along its axis and points at Polaris, the North Star. But Earth is slowly precessing (once in about 26,000 years) due to the torque of the Sun and the Moon on its nonspherical shape.

Figure 4: As seen in figure (a), the forces on a spinning gyroscope are its weight and the supporting force from the stand. These forces create a horizontal torque on the gyroscope, which create a change in angular momentum ΔLΔL size 12{L} {} that is also horizontal. In figure (b), ΔL ΔL size 12{L} {} and L L size 12{L} {} add to produce a new angular momentum with the same magnitude, but different direction, so that the gyroscope precesses in the direction shown instead of falling over.
In figure a, the gyroscope is rotating in counter clockwise direction. The weight of the gyroscope is acting downward. The supportive force is acting at the base. The line of action of the weight and supportive force are different. The torque is acting along the radius of the horizontal circular part of gyroscope. In figure b, the two vectors L and L plus delta L are shown. The vectors start from a point at the bottom of the figure and terminate at two points on a horizontal dotted circle, directed in counter clockwise direction, at the top of the figure. Another vector delta L starts from the head of vector L and terminates at the head of vector L plus delta L.

Check Your Understanding

Rotational kinetic energy is associated with angular momentum? Does that mean that rotational kinetic energy is a vector?

Solution

No, energy is always a scalar whether motion is involved or not. No form of energy has a direction in space and you can see that rotational kinetic energy does not depend on the direction of motion just as linear kinetic energy is independent of the direction of motion.

Section Summary

  • Torque is perpendicular to the plane formed by rr size 12{r} {} and FF size 12{F} {} and is the direction your right thumb would point if you curled the fingers of your right hand in the direction of F F size 12{F} {}. The direction of the torque is thus the same as that of the angular momentum it produces.
  • The gyroscope precesses around a vertical axis, since the torque is always horizontal and perpendicular to L L size 12{L} {}. If the gyroscope is not spinning, it acquires angular momentum in the direction of the torque (L =ΔL L =ΔL size 12{L=ΔL} {}), and it rotates about a horizontal axis, falling over just as we would expect.
  • Earth itself acts like a gigantic gyroscope. Its angular momentum is along its axis and points at Polaris, the North Star.

Conceptual Questions

Exercise 1

While driving his motorcycle at highway speed, a physics student notices that pulling back lightly on the right handlebar tips the cycle to the left and produces a left turn. Explain why this happens.

Exercise 2

Gyroscopes used in guidance systems to indicate directions in space must have an angular momentum that does not change in direction. Yet they are often subjected to large forces and accelerations. How can the direction of their angular momentum be constant when they are accelerated?

Problem Exercises

Exercise 1

Integrated Concepts

The axis of Earth makes a 23.5° angle with a direction perpendicular to the plane of Earth’s orbit. As shown in Figure 5, this axis precesses, making one complete rotation in 25,780 y.

(a) Calculate the change in angular momentum in half this time.

(b) What is the average torque producing this change in angular momentum?

(c) If this torque were created by a single force (it is not) acting at the most effective point on the equator, what would its magnitude be?

Figure 5: The Earth’s axis slowly precesses, always making an angle of 23.5° with the direction perpendicular to the plane of Earth’s orbit. The change in angular momentum for the two shown positions is quite large, although the magnitude L L size 12{L} {} is unchanged.
In the figure, the Earth’s image is shown. There are two vectors inclined at an angle of twenty three point five degree to the vertical, starting from the centre of the Earth. At the heads of the two vectors there is a circular shape, directed in counter clockwise direction. An angular momentum vector, directed toward left, along its diameter, is shown. The plane of the Earth’s orbit is shown as a horizontal line through its center.

Solution

(a) 5.64×1033kgm2/s5.64×1033kgm2/s size 12{5 "." "65" times "10" rSup { size 8{"33"} } `"kg" "." m rSup { size 8{2} } "/s"} {}

(b) 1.39×1022Nm1.39×1022Nm size 12{1 "." "39" times "10" rSup { size 8{"22"} } `N cdot m} {}

(c) 2.17×1015N2.17×1015N size 12{2 "." "18" times "10" rSup { size 8{"15"} } `N} {}

Glossary

right-hand rule:
direction of angular velocity ω and angular momentum L in which the thumb of your right hand points when you curl your fingers in the direction of the disk’s rotation

Collection Navigation

Content actions

Download module as:

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks