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Lecture PowerPoint

Physics for Scientists and Engineers, 3rd edition

FishbaneGasiorowicz

Thornton

Chapter 8

Linear Momentum, Collisions, and the Center of Mass

Main Points of Chapter 8• Definition of momentum

• Conservation of momentum

• Collisions – elastic and inelastic

• Impulse

• Explosions

• Center of mass

• Rocket motion

8-1 Momentum and its ConservationDefinition of linear momentum:

Writing force and kinetic energy in terms of momentum:

and

(8-2)

(8-3) (8-4)

8-1 Momentum and its ConservationConservation of momentum:

for zero net external force,(8-9)

• All internal forces are action-reaction pairs – momentum is conserved

8-2 Collisions and ImpulseImpulsive forces

• A force often acts on an object for a limited time

• The integral of the force over time gives the change in momentum, and is called the impulse

Definition of impulse: (8-14)

(8-15)

8-2 Collisions and Impulse

Change in momentum:

• The same change in momentum can be created by a large force acting for a short time, or a smaller force acting for a longer time

8-2 Collisions and ImpulseClassification of Collisions

1. Objects can stick together after collision

2. Objects can remain unchanged

3. Mass can be transferred from one object to another

4. One or both objects can shatter

8-3 Perfectly Inelastic Collisions; ExplosionsPerfectly Inelastic Collisions:

• Objects stick together after colliding

• Only one final velocity

(8-20)

8-3 Perfectly Inelastic Collisions; Explosions

Energy Loss in Perfectly Inelastic Collisions

• Momentum is conserved

• Kinetic energy is not

(8-24)

8-3 Perfectly Inelastic Collisions; Explosions

Explosions

• Perfectly inelastic collision “run in reverse”

• Most easily analyzed in reference frame where initial object is at rest

• In that case, total momentum is always zero:

(8-25)

8-4 Elastic Two-Body Collisions in One Dimension

• Elastic collisions conserve both momentum and kinetic energy• The relative velocity of the colliding objects changes sign but does not change magnitude.

8-4 Elastic Two-Body Collisions in One Dimension

When Object 2 is initially at rest:

(8-35a) (8-35b)

When the initial total momentum is zero:

(8-38)

8-5 Elastic Collisions in Two and Three Dimensions

In the general case, given masses and initial velocities, there are more variables than equations – no unique solution

Conservation of Momentum:

Conservation of Energy:

(8-41)

(8-42)

8-5 Elastic Collisions in Two and Three Dimensions

• Specific cases can be solved, given enough information

8-6 Center of Mass

• Importance: objects move under Newton’s second law as though all mass were at center of mass

• How to find it: Center of mass is “average” position of mass in all three dimensions

• Balance point

• Independent of coordinate system

8-6 Center of Mass

Center of mass of three pointlike objects:

8-6 Center of MassCenter of Mass Motion in the Absence of External Forces

(8-50)

In the absence of external forces the center of mass

moves with constant velocity.

8-6 Center of MassCenter of Mass Motion in the Presence of External Forces

(8-55)

The object moves under Newton’s second law as though all its mass were at center of mass.

8-6 Center of MassCenter of Mass of a Continuous

Mass Distribution

(8-63)

For a continuous distribution, integrate over the whole mass:

8-6 Center of MassHow to deal with holes:

1. Find the center of mass of the piece with no hole.

2. Find the center of mass of the hole.

3. Subtract.

8-7 Rocket Motion

(8-69)

• Similar to an explosion except that mass is continually lost

• Derivative accounts for lost mass

Summary of Chapter 8• Momentum:

• Momentum is conserved in the absence of external forces

• Impulse is force integrated over time; is also change in momentum

• Perfectly inelastic collision: objects stick together afterwards. Conserves momentum but not energy.

• Explosion: perfectly inelastic collision run backwards

(8-2)

Summary of Chapter 8, cont.• Elastic collisions: conserve momentum and energy

• Center of mass: object obeys Newton’s 2nd law as though all mass were there

• Sum of mass x distance from a given point

• Center of mass of a continuous object: find by integrating

Summary of Chapter 8, cont.

• Center of mass of object with hole: treat hole as having “negative mass”

• Rocket motion: mass is expelled continuously; need to find speed at any given time by differentiating