Chapter 7

Rotational Motion

and

The Law of Gravity

IB Test Objectives
for Chapters 7 & 8

nDraw a vector diagram to show that the acceleration of a particle moving with uniform speed in a circle is directed toward the center of the circle

State the expression for centripetal acceleration

nIdentify the force producing circular motion in various situations
n    Examples include gravitational force (acting on the moon) and friction (acting sideways on the tires of a car turning acorner
nSolve problems for particles moving in circles with uniform speed
nGravitational force and field
nState Newton’s Law of Gravitation
n    Students should be aware that the masses in the force law are point masses not extended masses, but that the interaction   between two spherical masses is the same as if the mass were concentrated at the centers of the spheres.
nDefine Gravitational field strength
n    Students should recognize the vector nature of gravitational fields.
n
nDerive an expression for the gravitational field at the surface of a planet.
n    Students should also understand how the gravitational field strength and the acceleration due to gravity at the surface are related.
nSolve problems involving gravitational forces and fields
nVector addition may be required to find the gravitational field strength due to more than one mass
nGravitational energy and potential
nDefine gravitational potential energy and gravitational potential
n    Students should understand that the work done in moving a mass between two points in a gravitational field independent of the path taken and that gravitational potential energy is taken to be zero at infinity.
nState the expression for gravitational potential due to a point mass
nExplain the concept of escape speed.
nDerive an expression for the escape speed of an object from the surface of a planet.
Solve problems involving gravitational potential energy and gravitational potential
n    These should include problems on escape speed.
n
nOrbital Motion
nState that gravitation provides the centripetal force for circular orbital motion
nState Kepler’s third law: the law of periods
nDerive Kepler’s third law.
nThis derivation is for the case of circular orbits and assumes Newton’s Law of Universal Gravitation
nDerive expressions for the kinetic, potential and total energy of an orbiting satellite.
nDraw graphs showing the variation of the kinetic energy, gravitational potential energy and total energy with orbital radius of a satellite.
nDiscuss the concept of weightlessness in orbital motion and freefall.
nSolve problems involving orbital motion.
nStatics
nDefine torque (moment of force)
nThe vector nature of torque need not be addressed but students should include the sense (eg clockwise or counterclockwise) of a torque
nState the conditions for translational and rotational equilibrium
n
nDescribe the concept of center of gravity.
nStudents are not required to calculate the center of gravity of objects. However, they should be aware that the weight of an object may be taken as concentrated at the center of gravity for determination of gravitational torques.
nSolve problems involving extended objects in equilibrium.

 

AP Test Objectives
for Chapters 7 & 8

Uniform Circular Motion:

Students should understand the uniform circular motion of a particle so they can:

a) Relate the radius of the circle and the speed or rate of revolution of the particle to the magnitude of the centripetal acceleration.

b) Describe the direction of the particle's velocity and acceleration at any instant during the motion.

c) Determine the components of the velocity and acceleration vectors at any instant, and sketch or identify graphs of these quantities.

AP Test Objectives
for Chapters 7 & 8 (cont)

Angular Momentum and Its Conservation

Students should understand angular momentum conservation so they can recognize the conditions under which the law of conservation is applicable and relate this law to one and two-particle systems such as satellite orbits.

AP Test Objectives
for Chapters 7 & 8 (cont)

Torque and Rotational Statics

Students should understand the concept of torque so they can:

Calculate the magnitude and sense of the torque associated with a given force

Calculate the torque on a rigid body due to gravity

AP Test Objectives
for Chapters 7 & 8 (cont)

Students should be able to analyze problems in statics so they can:

State the conditions for translational and rotational equilibrium of a rigid body

Apply these conditions in analyzing the equilibrium of a rigid body under the combined influence of a number of coplanar forces applied at different locations.

AP Test Objectives
for Chapters 7 & 8 (cont)

Gravitation

Students should know Newton's Law of Universal Gravitation so they can:

Determine the force that one spherically symmetrical mass exerts on another.

Determine the strength of the gravitational field at a specified point outside a spherically symmetrical mass

AP Test Objectives
for Chapters 7 & 8 (cont)

Students should understand the motion of a body in orbit under the influence of gravitational forces so they can:

For a circular orbit:

Recognize that the motion does not depend on the body's mass

describe qualitatively how the velocity, period of revolution, and centripetal acceleration depend upon the radius of the orbit

derive expressions for the velocity and period of revolution in such an orbit.

AP Test Objectives
for Chapters 7 & 8 (cont)

For a general orbit:

Apply conservation of angular momentum to determine the velocity and radial distance at any point in the orbit.

Apply angular momentum conservation and energy conservation to relate the speeds of a body at the two extremes of an elliptic orbit

Angular Displacement

Axis of rotation is the center of the disk

Need a fixed reference line

During time t, the reference line moves through angle θ

 

Angular Displacement, cont.

Every point on the object undergoes circular motion about the point O

Angles generally need to be measured in radians

 

s is the length of arc and r is the radius

 

More About Radians

 

Comparing degrees and radians

 

Converting from degrees to radians

 

Angular Displacement, cont.

The angular displacement is defined as the angle the object rotates through during some time interval

Every point on the disc undergoes the same angular displacement in any given time interval

 

Angular Speed

The average angular speed, ω, of a rotating rigid object is the ratio of the angular displacement to the time interval

 

Angular Speed, cont.

The instantaneous angular speed is defined as the limit of the average speed as the time interval approaches zero

Units of angular speed are radians/sec

rad/s

Speed will be positive if θ is increasing (counterclockwise)

Speed will be negative if θ is decreasing (clockwise)

 

Angular Acceleration

The average angular acceleration, ,

of an object is defined as the ratio of the change in the angular speed to the time it takes for the object to undergo the change:

 

 

More About Angular Acceleration

Units of angular acceleration are rad/s²

When a rigid object rotates about a fixed axis, every portion of the object has the same angular speed and the same angular acceleration

 

Problem Solving Hints

Similar to the techniques used in linear motion problems

With constant angular acceleration, the techniques are much like those with constant linear acceleration

There are some differences to keep in mind

For rotational motion, define a rotational axis

The object keeps returning to its original orientation, so you can find the number of revolutions made by the body

 

Analogies Between Linear and Rotational Motion

 

 

 

 

Relationship Between Angular and Linear Quantities

Displacements

Speeds

Accelerations

Every point on the rotating object has the same angular motion

Every point on the rotating object does not have the same linear motion

 

Centripetal Acceleration

An object traveling in a circle, even though it moves with a constant speed, will have an acceleration

The centripetal acceleration is due to the change in the direction of the velocity

Centripetal Acceleration, cont.

Centripetal refers to "center-seeking"

The direction of the velocity changes

The acceleration is directed toward the center of the circle of motion

 

Centripetal Acceleration and Angular Velocity

The angular velocity and the linear velocity are related (v = ωr)

The centripetal acceleration can also be related to the angular velocity

 

Total Acceleration

The tangential component of the acceleration is due to changing speed

The centripetal component of the acceleration is due to changing direction

Total acceleration can be found from these components

 

 

Vector Nature of Angular Quantities

Assign a positive or negative direction in the problem

A more complete way is by using the right hand rule

Grasp the axis of rotation with your right hand

Wrap your fingers in the direction of rotation

Your thumb points in the direction of ω

 

Forces Causing Centripetal Acceleration

Newton’s Second Law says that the centripetal acceleration is accompanied by a force

F = maC

F stands for any force that keeps an object following a circular path

Tension in a string

Gravity

Force of friction

 

Problem Solving Strategy

Draw a free body diagram, showing and labeling all the forces acting on the object(s)

Choose a coordinate system that has one axis perpendicular to the circular path and the other axis tangent to the circular path

 

 

Problem Solving Strategy, cont.

Find the net force toward the center of the circular path (this is the force that causes the centripetal acceleration)

Solve as in Newton’s second law problems

The directions will be radial and tangential

The acceleration will be the centripetal acceleration

Applications of Forces Causing Centripetal Acceleration

Many specific situations will use forces that cause centripetal acceleration

Level curves

Banked curves

Horizontal circles

Vertical circles

Level Curves

Friction is the force that produces the centripetal acceleration

Can find the frictional force, µ, v

Banked Curves

A component of the normal force adds to the frictional force to allow higher speeds

 

Horizontal Circle

The horizontal component of the tension causes the centripetal acceleration

Vertical Circle

Look at the forces at the top of the circle

The minimum speed at the top of the circle can be found

 

Forces in Accelerating Reference Frames

Distinguish real forces from fictitious forces

Centrifugal force is a fictitious force

Real forces always represent interactions between objects

 

Newton’s Law of Universal Gravitation

Every particle in the Universe attracts every other particle with a force that is directly proportional to the product of the masses and inversely proportional to the square of the distance between them.

 

 

Law of Gravitation, cont.

G is the constant of universal gravitational

G = 6.673 x 10-11 N m² /kg²

This is an example of an inverse square law

 

Gravitation Constant

Determined experimentally

Henry Cavendish

1798

The light beam and mirror serve to amplify the motion

 

Applications of Universal Gravitation

Mass of the earth

Use an example of an object close to the surface of the earth

r ~ RE

 

Applications of Universal Gravitation

Acceleration due to gravity

g will vary with altitude

 

Gravitational Potential
Energy

PE = mgy is valid only near the earth’s surface

For objects high above the earth’s surface, an alternate expression is needed

 

Zero reference level is infinitely far from the earth

 

Escape Speed

The escape speed is the speed needed for an object to soar off into space and not return

 

For the earth, vesc is about 11.2 km/s

Note, v is independent of the mass of the object

 

Kepler’s Laws

All planets move in elliptical orbits with the Sun at one of the focal points.

A line drawn from the Sun to any planet sweeps out equal areas in equal time intervals.

The square of the orbital period of any planet is proportional to cube of the average distance from the Sun to the planet.

 

 

Kepler’s Laws, cont.

Based on observations made by Brahe

Newton later demonstrated that these laws were consequences of the gravitational force between any two objects together with Newton’s laws of motion

 

Kepler’s First Law

All planets move in elliptical orbits with the Sun at one focus.

Any object bound to another by an inverse square law will move in an elliptical path

Second focus is empty

 

Kepler’s Second Law

A line drawn from the Sun to any planet will sweep out equal areas in equal times

Area from A to B and C to D are the same

 

Kepler’s Third Law

The square of the orbital period of any planet is proportional to cube of the average distance from the Sun to the planet.

For orbit around the Sun, KS = 2.97x10-19 s2/m3

K is independent of the mass of the planet

 

Kepler’s Third Law application

Mass of the Sun or other celestial body that has something orbiting it

Assuming a circular orbit is a good approximation