<< Chapter < Page Chapter >> Page >

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.
In figure (a), the torque is perpendicular to the plane formed by r size 12{r} {} and F size 12{F} {} and is the direction your right thumb would point to if you curled your fingers in the direction of F 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 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.
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 in exactly the same direction. Figure (b) shows a vector diagram depicting how Δ L and 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.

Applying science practices: angular momentum and torque

You have seen that change in angular momentum depends on the average torque applied and the time interval during which the torque is applied. Plan an experiment similar to the one shown in Figure 10.30 to test the relationship between the change in angular momentum of a system and the product of the average torque applied to the system and the time interval during which the torque is exerted. What would you use as your test system? How could you measure applied torque? What observations could you make to help you analyze changes in angular momentum? Remember that, since angular momentum is a vector, changes can relate to its magnitude or its direction.

This same logic explains the behavior of gyroscopes. [link] 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 L size 12{L} {} . If the gyroscope is not spinning, it acquires angular momentum in the direction of the torque ( L = Δ L 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.

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.
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 Δ L size 12{L} {} that is also horizontal. In figure (b), Δ L size 12{L} {} and 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.

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

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.

Got questions? Get instant answers now!

Test prep for ap courses

A globe (model of the Earth) is a hollow sphere with a radius of 16 cm. By wrapping a cord around the equator of a globe and pulling on it, a person exerts a torque on the globe of 120 N • m for 1.2 s. What angular momentum does the globe have after 1.2 s?

Since the globe is stationary to start with,

τ = Δ L Δ t

τ Δ t = Δ L

By substituting,

120 N•m • 1.2 s = 144 N•m•s.

The angular momentum of the globe after 1.2 s is 144 N•m•s.

Got questions? Get instant answers now!

How could you use a fishing reel to test the relationship between the torque applied to a system, the time for which the torque was applied, and the resulting angular momentum of the system? How would you measure angular momentum?

Got questions? Get instant answers now!

Section summary

  • Torque is perpendicular to the plane formed by r size 12{r} {} and F 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 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 size 12{L} {} . If the gyroscope is not spinning, it acquires angular momentum in the direction of the torque ( L = Δ L 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

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.

Got questions? Get instant answers now!

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?

Got questions? Get instant answers now!

Problem exercises

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 [link] , 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?

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.
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 size 12{L} {} is unchanged.

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

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

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

Got questions? Get instant answers now!
Practice Key Terms 1

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College physics for ap® courses' conversation and receive update notifications?

Ask