General Physics (calculus based) Class Notes

Dr. Rakesh Kapoor,

University of Alabama at Birmingham, Birmingham, AL 35294

Torque and Angular Momentum

In this chapter we will learn about torque and angular momentum.  
After finishing this chapter we should know following topics:

Torque  for rotation (symbol  τ)

Newton’ s second law for rotational motion.

Work and rotational kinetic energy.

General definition of torque for a particle that moves along any path relative to a fixed point.

Angular Momentum of single particles and systems or particles.

Relation between angular momentum and torque (Newton’s Second Law in angular Form).

Rotational and translational angular momentum.

Relation between Angular Momentum and Angular velocity

Conservation of angular Momentum.

Applications of the conservation of angular momentum.

How a Gyroscope works?

What is Torque?

Torque measures the ability of a force to rotate an object.

Let us try to open (rotate) the door by applying a force TorqueAndAngularMomentum_1.gif.

Will the force in its present direction be able to open (rotate) the door?

Click "Open door" button and see the answer.

Now rotate the force by 90 °, will it be able to open (rotate) the door?

Click "Open door" button and see the answer.

Switch to the top view and see the direction of force vector with respect to rotating path (blue dots).

We observed that when a force acts tangentially to the rotating path of an object, it can rotate the object.

Now change the position of the force.

When the force is acting closer to the rotating axis, will it be easy or more difficult to rotate (open) the door?

From this simulation we can conclude.

Applied force TorqueAndAngularMomentum_3.gif is able to rotate the door only when it acts tangentially to the path of rotation.

Ability of the force to rotate the object depends on the position of the force from the rotating axis.

Torque

Consider a simple rigid body, with a particle of mass m attached to a mass less rod, free to rotate about an axis through O.

TorqueAndAngularMomentum_4.gif

A force TorqueAndAngularMomentum_5.gif is applied to the particle at a position TorqueAndAngularMomentum_6.gif.

Angle between TorqueAndAngularMomentum_7.gif and TorqueAndAngularMomentum_8.gif is φ.

Only tangential component TorqueAndAngularMomentum_9.gif is capable of rotating the object.

Ability to rotate also depends on the position TorqueAndAngularMomentum_10.gif of action.

This ability to rotate is called torque and its symbol is τ.

TorqueAndAngularMomentum_11.gif

Another equivalent way of writing torque is

TorqueAndAngularMomentum_12.gif

where TorqueAndAngularMomentum_13.gif is the perpendicular distance between the rotation axis at O and an extended line running through TorqueAndAngularMomentum_14.gif. This extended line is called the line of action of TorqueAndAngularMomentum_15.gif, and TorqueAndAngularMomentum_16.gifis called the moment arm of TorqueAndAngularMomentum_17.gif.

Torque TorqueAndAngularMomentum_18.gif is vector quantity and can be written as

TorqueAndAngularMomentum_19.gif

SI units of Torque:

The SI unit of torque is the Newton-meter (N·m).

Caution: The Newton-meter is also the unit of work.

Torque and work, however, are quite different quantities and must not be confused.

Work is often expressed in joules (1 J = 1 N m), but torque is never expressed in joules.

Problem 1: (Work and rotational kinetic energy)

A rod is pivoted about its center. A 5 N force is applied 4m from the pivot and another 5 N force is applied 2m from the pivot, as shown.
What is the magnitude of the total torque about the pivot (in N·m)?

TorqueAndAngularMomentum_20.gif

Solution :

TorqueAndAngularMomentum_21.gif

TorqueAndAngularMomentum_22.gif

Newton’s Second law for Rotation

Torque can cause rotation of a rigid body as force can accelerate a body.

Net torque TorqueAndAngularMomentum_24.gif acting on a rigid body produces angular acceleration α and these two quantities can be related by Newton's second law for rotation.

TorqueAndAngularMomentum_25.gif

Let us prove the above relation. Consider the system shown in figure.

TorqueAndAngularMomentum_26.gif

Tangential force TorqueAndAngularMomentum_27.gif is related to tangential acceleration TorqueAndAngularMomentum_28.gif as TorqueAndAngularMomentum_29.gif. Therefore torque τ can be written as

TorqueAndAngularMomentum_30.gif

Tangential acceleration TorqueAndAngularMomentum_31.gif is related to angular acceleration α as

TorqueAndAngularMomentum_32.gif

Therefore

TorqueAndAngularMomentum_33.gif

Checkpoint-1: (Newton’s Law for rotation)

A uniform disk, a thin hoop, and a uniform sphere, all with the same mass and same outer radius, are each free to rotate about a fixed axis through its center. Assume the hoop is connected to the rotation axis by light spokes. With the objects starting from rest, identical forces are simultaneously applied to the rims, as shown.
Rank the objects according to their angular accelerations, least to greatest.

TorqueAndAngularMomentum_34.gif

Key Idea:

TorqueAndAngularMomentum_35.gif

TorqueAndAngularMomentum_36.gif

Work and Rotational Kinetic Energy

Suppose a torque τ change the angular velocity of a rotating object from its initial value TorqueAndAngularMomentum_37.gif to TorqueAndAngularMomentum_38.gif.

Therefore we can say that the torque has changed the rotational kinetic energy of the rotating object.

The change in rotational kinetic energy ΔK can be given as

TorqueAndAngularMomentum_39.gif

Change in kinetic energy is equal to the work W done by the torque.

TorqueAndAngularMomentum_40.gif

The work dW done due to tangential component of force TorqueAndAngularMomentum_41.gif, in moving the particle by a distance ds along the circular path is

TorqueAndAngularMomentum_42.gif

By substituting the value of ds in terms of angular displacement , we get

TorqueAndAngularMomentum_43.gif

Therefore work done by torque in moving the object from its initial angular position TorqueAndAngularMomentum_44.gif to final angular position TorqueAndAngularMomentum_45.gif.

TorqueAndAngularMomentum_46.gif

Similarly, the rate at which the work (Power) is done by torque is given as is power

TorqueAndAngularMomentum_47.gif

Problem 2: (Work and rotational kinetic energy)

A disk has a rotational inertia of TorqueAndAngularMomentum_48.gif and a constant angular acceleration of TorqueAndAngularMomentum_49.gif.
If it starts from rest, what will be  the work done during the first 5.0 s by the net torque acting on it?

Solution :

Initial angular velocity TorqueAndAngularMomentum_50.gif, therefore initial kinetic energy is zero.
We need to calculate final angular velocity, final kinetic energy to compute work done.

TorqueAndAngularMomentum_51.gif

TorqueAndAngularMomentum_52.gif

TorqueAndAngularMomentum_53.gif

Torque (A general Definition)

For an object B rotating in a plane, we have defined torque TorqueAndAngularMomentum_55.gif  as

TorqueAndAngularMomentum_56.gif

TorqueAndAngularMomentum_57.gif

Here TorqueAndAngularMomentum_58.gif is the position from the center of rotation and TorqueAndAngularMomentum_59.gif is the force acting on that object.

This definition can be expanded to any individual particle A that moves along any path (circle, ellipse or straight line) due to influence of a force TorqueAndAngularMomentum_60.gif.

TorqueAndAngularMomentum_61.gif

The torque on A about any point O is again defined as

TorqueAndAngularMomentum_62.gif

Magnitude of the torque TorqueAndAngularMomentum_63.gif of the particle, relative to any point O is given as

TorqueAndAngularMomentum_64.gif

TorqueAndAngularMomentum_65.gif

TorqueAndAngularMomentum_66.gif

We have three formulas to compute magnitude of torque.

φ is the smaller angle between TorqueAndAngularMomentum_67.gif and TorqueAndAngularMomentum_68.gif.

TorqueAndAngularMomentum_69.gif is the perpendicular distance from point O to the line of action (dashed blue line). It is called moment arm.

TorqueAndAngularMomentum_70.gif is the tangential component of the force with respect to TorqueAndAngularMomentum_71.gif.

Torque has meaning only with respect to a specified location. If a different location (O) is chosen, TorqueAndAngularMomentum_72.gif will change, the value of the torque is usually changed too.

Angular Momentum

When a particle is moving with a linear momentum TorqueAndAngularMomentum_73.gif, the angular momentum TorqueAndAngularMomentum_74.gif of the particle, relative to a location A is defined as

TorqueAndAngularMomentum_75.gif

TorqueAndAngularMomentum_76.gif

Here TorqueAndAngularMomentum_77.gif is the position vector of the particle measured from location A.

Magnitude of the angular momentum TorqueAndAngularMomentum_78.gif of the particle, relative to a location A is given as

TorqueAndAngularMomentum_79.gif

TorqueAndAngularMomentum_80.gif

TorqueAndAngularMomentum_81.gif

We have three formulas to compute magnitude of angular momentum.

φ is the smaller angle between TorqueAndAngularMomentum_82.gif and TorqueAndAngularMomentum_83.gif.

TorqueAndAngularMomentum_84.gif is the perpendicular distance from location A to the line of motion (dashed blue line). It is called moment arm.

TorqueAndAngularMomentum_85.gif is the tangential component of the momentum with respect to TorqueAndAngularMomentum_86.gif.

Angular momentum has meaning only with respect to a specified location. If a different location (A) is chosen, TorqueAndAngularMomentum_87.gif will change, the value of the angular momentum is usually changed too.

SI units for angular momentum

SI units for angular momentum are written as

TorqueAndAngularMomentum_88.gif

Since angular momentum TorqueAndAngularMomentum_89.gif of the particle, relative to a location A is given as

TorqueAndAngularMomentum_90.gif

Units for position r is meter (m) and units for linear momentum (p=m v) are kilogram meter/second (kg m/s).

Checkpoint-2: (    Angular Momentum)

In part a of the figure, particles 1 and 2 move around point O in opposite directions, in circles with radii 2 m and 4 m. In part b, particles 3 and 4 travel in the same direction, along straight lines at perpendicular distances of 4 m and 2 m from point O. Particle 5 moves directly away from O. All five particles have the same mass and the same constant speed.
(a) Rank the particles according to the magnitudes of their angular momentum relative to point O, greatest first.
(b) Which particles have negative angular momentum relative to point O?

TorqueAndAngularMomentum_91.gif

Hint : TorqueAndAngularMomentum_92.gif

Newton’s Second Law in Angular Form

As per Newton's second law, time derivative of linear momentum is equal to net external force:

TorqueAndAngularMomentum_93.gif

Let us see what the time derivative of angular momentum TorqueAndAngularMomentum_94.gif relative to point A, turn out to be?

TorqueAndAngularMomentum_95.gif

Now TorqueAndAngularMomentum_96.gif, therefore the first term on right hand side in above equation can be written as

TorqueAndAngularMomentum_97.gif

Therefore the time rate of change of angular momentum is given as

TorqueAndAngularMomentum_98.gif

TorqueAndAngularMomentum_99.gif is defined as the net torque acting on the object relative to location A.

TorqueAndAngularMomentum_100.gif

Time derivative of angular momentum of an object, relative to point A, is equal to net external torque relative to same point A:

Another way of writing SI units of Angular Momentum :

As per above relation, SI units for angular momentum can also be written as

Newton-meter (N·m) × s.

TorqueAndAngularMomentum_101.gif

Since

TorqueAndAngularMomentum_102.gif

Angular Momentum of multi-particle systems

Total angular momentum TorqueAndAngularMomentum_103.gif of a system of particles relative to a point A, is the vector sum of the angular momentum TorqueAndAngularMomentum_104.gif of the individual particles relative to same point A.

TorqueAndAngularMomentum_105.gif

The change in total angular momentum with time will be equal to the net sum of external torques (relative to point A) acting on individual particles.

TorqueAndAngularMomentum_106.gif

Rotational Angular Momentum*

Consider a multi-particle system of three particles with linear momentum TorqueAndAngularMomentum_107.gif, TorqueAndAngularMomentum_108.gif and TorqueAndAngularMomentum_109.gif.

Angular momentum of each particle, relative to a location A is defined as

TorqueAndAngularMomentum_110.gif

Here TorqueAndAngularMomentum_111.gif, TorqueAndAngularMomentum_112.gif and TorqueAndAngularMomentum_113.gif are positions of particles 1, 2 and 3 with respect to point A.

Total angular momentum TorqueAndAngularMomentum_114.gif of the system relative to point A will be vector sum of angular momentum of the individual three particles relative to same point A.

TorqueAndAngularMomentum_115.gif

TorqueAndAngularMomentum_116.gif

If TorqueAndAngularMomentum_117.gif is the position of center of mass of the system with respect to point A, the position vectors TorqueAndAngularMomentum_118.gif, TorqueAndAngularMomentum_119.gif and TorqueAndAngularMomentum_120.gif can be written in terms of TorqueAndAngularMomentum_121.gif.

TorqueAndAngularMomentum_122.gif

Here TorqueAndAngularMomentum_123.gif, TorqueAndAngularMomentum_124.gif and TorqueAndAngularMomentum_125.gif are positions of particles with respect to center of mass (com) of the system.

By substituting the values of  TorqueAndAngularMomentum_126.gif, TorqueAndAngularMomentum_127.gif and TorqueAndAngularMomentum_128.gif, the expression of total angular momentum TorqueAndAngularMomentum_129.gif of the system can be rewritten as

TorqueAndAngularMomentum_130.gif

Here first part in above equation is called rotational angular momentum TorqueAndAngularMomentum_131.gif of the system relative to its center of mass. Second part is called translational angular momentum TorqueAndAngularMomentum_132.gif of the center of mass of the system relative to point A.

TorqueAndAngularMomentum_133.gif

TorqueAndAngularMomentum_134.gif

Here TorqueAndAngularMomentum_135.gifTorqueAndAngularMomentum_136.gif, is the total linear momentum of the system or linear momentum of the center of mass.

Therefore total angular momentum TorqueAndAngularMomentum_137.gif of a system relative to any point A, is equal to the vector sum of rotational angular momentum TorqueAndAngularMomentum_138.gif of the system, and the translational angular momentum TorqueAndAngularMomentum_139.gif of the center of mass of the system relative to the point A.

TorqueAndAngularMomentum_140.gif

For a stationary system with total linear momentum TorqueAndAngularMomentum_141.gif, the total angular momentum  TorqueAndAngularMomentum_142.gif relative to any point A in space will be equal to its rotational angular momentum TorqueAndAngularMomentum_143.gif.

When linear momentum of a system TorqueAndAngularMomentum_144.gif, the total angular momentum  TorqueAndAngularMomentum_145.gif relative to any point A in space is always equal to its rotational angular momentum TorqueAndAngularMomentum_146.gif.

TorqueAndAngularMomentum_147.gif

Will an object with finite angular Momentum always rotate?

A system with finite rotational angular momentum TorqueAndAngularMomentum_148.gif will always rotate or spin about an axis passing through its center of mass.

Examples of Rotational angular momentum:

Spinning of Earth bout its axis.

Spinning of electrons about their axis.

A system with finite translational angular momentum TorqueAndAngularMomentum_149.gif about a point A, may or may not rotate or revolve around point A.

In the following simulation there are two disks with same translational angular momentum TorqueAndAngularMomentum_150.gif relative to point A. Release (click play) and watch, one revolves around A, while other goes along a straight line.

Examples of Translational angular momentum:

Revolving of Earth around sun (an axis passing through a point in sun).

Orbiting of electrons around the nucleus in an atom.

Linear motion of a rocket with respect to a point on Earth or anywhere in space.

Direction of Angular Momentum and Angular velocity.

Angular velocity TorqueAndAngularMomentum_152.gif of a rotating object is always parallel to its axis of rotation.

Rotational Angular momentum TorqueAndAngularMomentum_153.gif of a rigid body is not necessarily parallel to its axis of rotation or parallel to its angular velocity TorqueAndAngularMomentum_154.gif.

This is because, moment of inertia/rotational inertia of a rigid body changes with the choice of the rotation axis (moment of inertia is a 3×3 tensor).

A rigid body of any shape, possesses three mutually perpendicular axes through the center of mass. These axes are called principal axes of the body, and they have an important property.

When axis of rotational is one of the principal axis, rotational angular momentum TorqueAndAngularMomentum_155.gif is parallel to axis of rotation or parallel to the angular velocity TorqueAndAngularMomentum_156.gif.

TorqueAndAngularMomentum_157.gif

TorqueAndAngularMomentum_158.gif is the moment of inertia about the principal axis.

For a body having axes of symmetry, the principal axes are along the symmetry axes.

What if TorqueAndAngularMomentum_159.gif is not parallel to one of the principal axes?

We can choose a coordinate system with x, y and z-axes along the principal axes of a rigid body with corresponding moment of inertia TorqueAndAngularMomentum_160.gif, TorqueAndAngularMomentum_161.gif and TorqueAndAngularMomentum_162.gif the components of the rotational angular momentum TorqueAndAngularMomentum_163.gif about these axes can be related to corresponding components of angular velocity TorqueAndAngularMomentum_164.gif as below.

TorqueAndAngularMomentum_165.gif

Above relations are only valid when x, y, z-axes are the principal axes of the rigid body.

TorqueAndAngularMomentum_166.gif

Above relation clearly indicates that directions of TorqueAndAngularMomentum_167.gif and TorqueAndAngularMomentum_168.gif are not necessarily parallel to each other as moment of inertia TorqueAndAngularMomentum_169.gif, TorqueAndAngularMomentum_170.gif and TorqueAndAngularMomentum_171.gif along each principal axes, can have different values.

Rotational kinetic energy TorqueAndAngularMomentum_172.gif of a rotating rigid object can be related to its rotational angular momentum TorqueAndAngularMomentum_173.gif and its angular velocity TorqueAndAngularMomentum_174.gif.

TorqueAndAngularMomentum_175.gif

TorqueAndAngularMomentum_176.gif

Examples of rotation axis.

If the rotation axis is not the symmetric axis or one of the principal axes, as shown in the simulation, the angular momentum TorqueAndAngularMomentum_177.gif and angular velocity TorqueAndAngularMomentum_178.gif will not be in the same direction.

You can see that angular momentum vector TorqueAndAngularMomentum_180.gif continually change direction (wobbling) and requires a non-zero torque (In a rotating object, this is applied to the axle by the bearings).

To make TorqueAndAngularMomentum_181.gif parallel to TorqueAndAngularMomentum_182.gif, we can add another identical object to balance it, now the rotation axis becomes a symmetric axis and is parallel to one of the principal axes, (select balance).

You can see that the components of the two angular momentums, perpendicular to rotation axis (red arrows),   cancel each other while components of the two angular momentums, parallel to rotation axis (blue arrows), adds up to give total angular momentum.

Net angular momentum now points in the direction of TorqueAndAngularMomentum_183.gif or points along the rotation axis and does not change direction.

Car tires must be carefully balanced to make axis of rotation axis one of the principal axes (symmetric axis). This makes rotation more efficient, prevents wobbling and damage of bearings.

Checkpoint-3: (Angular momentum of a rotating rigid object)

In the figure, a disk, a hoop, and a solid sphere are made to spin about fixed central axes (like a top) by means of strings wrapped around them, with the strings producing the same constant tangential force TorqueAndAngularMomentum_184.gif on all three objects. The three objects have the same mass and radius, and they are initially stationary.  
(a) Rank the objects according to their angular momentum about their central axes and
(b) Rank the objects according to their angular speed, greatest first, when the strings have been pulled for a certain time t.

TorqueAndAngularMomentum_185.gif

Hint : TorqueAndAngularMomentum_186.gif  and L=I ω

TorqueAndAngularMomentum_187.gif

Quiz- 1
The position vector of a particle is directed along the positive y axis.  What is the direction of the net force acting on the particle if the net torque is directed along the negative x direction?

negative x direction

positive x direction

negative y direction

positive z direction

negative z direction

Conservation of Angular Momentum

According to Newton’s second law.

TorqueAndAngularMomentum_188.gif

When the applied external net torque TorqueAndAngularMomentum_189.gif.             .

TorqueAndAngularMomentum_190.gif

That means angular momentum TorqueAndAngularMomentum_191.gif is constant.

This is called law of conservation of momentum.

If the net external torque TorqueAndAngularMomentum_192.gif relative to a point A, acting on a system is zero, the angular momentum TorqueAndAngularMomentum_193.gif relative to point A, of the system remains constant, no matter what changes take place within the system.

TorqueAndAngularMomentum_194.gif

If the component of the net external torque  TorqueAndAngularMomentum_195.gif relative to a point A acting on a system along a certain axis is zero, then the component of the angular momentum TorqueAndAngularMomentum_196.gif relative to point A, of the system along that axis cannot change, no matter what changes take place within the system.

TorqueAndAngularMomentum_197.gif

TorqueAndAngularMomentum_198.gif

TorqueAndAngularMomentum_199.gif

The spinning person:

A rotating person changes from initial position (a) to final position (b).

TorqueAndAngularMomentum_200.gif

Angular momentum of the system TorqueAndAngularMomentum_201.gif will remain constant as no net external torque is acting on the system.

In final position (b) rotational inertia TorqueAndAngularMomentum_202.gif decreases means TorqueAndAngularMomentum_203.gif.

Since magnitude of L = I ω, therefore TorqueAndAngularMomentum_204.gif

The springboard diver:

The diver’s angular momentum TorqueAndAngularMomentum_205.gif is constant throughout the dive, being represented by the tail of an arrow that is perpendicular to the plane of the figure.

TorqueAndAngularMomentum_206.gif

When the diver brings her hands and legs closer, her rotational inertia I reduces.

Since L = I ω, therefore at that time angular velocity ω increases.

Quiz- 2
Joe has volunteered to help out in his physics class by sitting on a stool that easily rotates.  As Joe holds the dumbbells out as shown, the professor temporarily applies a sufficient torque that causes him to rotate slowly.  Then, Joe brings the dumbbells close to his body and he rotates faster.  Why does his speed increase?
TorqueAndAngularMomentum_207.gif TorqueAndAngularMomentum_208.gif

By bringing the dumbbells inward, Joe exerts a torque on the stool.

By bringing the dumbbells inward, Joe decreases the moment of inertia.

By bringing the dumbbells inward, Joe increases the angular momentum.

By bringing the dumbbells inward, Joe increases the moment of inertia.

By bringing the dumbbells inward, Joe decreases the angular momentum.

Can you answer?

Look at the video.

Why spinning wheel stays in vertical position?

How a Gyroscope works?

A spinning wheel is called a Gyroscope.

What happens when wheel is not spinning?

TorqueAndAngularMomentum_210.gif

We know according to Newton`s second law TorqueAndAngularMomentum_211.gif

When the wheel is not spinning it does not have any angular momentum along its axis (x-axis) or along any direction.

Gravitational force acting on its center of mass will produce a torque TorqueAndAngularMomentum_212.gif, which will try to rotate it along y-axis. In other words torque tries to change its angular momentum from zero to a finite value. The magnitude of TorqueAndAngularMomentum_213.gif is

TorqueAndAngularMomentum_214.gif

The wheel will fall due to this rotation around y-axis.

What happens when Gyroscope wheel is spinning?

TorqueAndAngularMomentum_215.gif TorqueAndAngularMomentum_216.gif

When the wheel is rapidly spinning, it has a finite angular momentum TorqueAndAngularMomentum_217.gif along its rotation axis (x-axis).

Torque TorqueAndAngularMomentum_218.gif due to gravitational force is acting along y-axis.

Since torque is acting normal to TorqueAndAngularMomentum_219.gif, therefore it can only change direction of the angular momentum vector TorqueAndAngularMomentum_220.gif not magnitude.

This will cause rotation of TorqueAndAngularMomentum_221.gif or precession of the wheel around z-axis.

We see now gravitational force is the cause of  precession of the wheel, instead of  its fall.

Now we can calculate the precession rate Ω.

TorqueAndAngularMomentum_222.gif

TorqueAndAngularMomentum_223.gif

TorqueAndAngularMomentum_224.gif

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