Physics 101: Lecture 15, Pg 1

Physics 101: Physics 101: Lecture 15Lecture 15Impulse and MomentumImpulse and Momentum

Today’s lecture will be a review of Chapters 7.1 - 7.2 and New material: Collisions and Center of Mass, Chapters 7.3-7.5 Rotational Motion and Angular Displacement, Chapter 8.1

Physics 101: Lecture 15, Pg 2

Conservation of Linear MomentumConservation of Linear Momentum

Consider a system of two colliding objects with masses m1 and m2

and initial velocities v01 and v02 and final velocities vf1 and vf2 :

If the sum of the average external forces acting on the two objects is zero ( = isolated system), the total momentum of the system is conserved:

Fave,ext t = Pf - P0 => Pf = P0 if Fave,ext = 0 Pf and Po are the total momenta of the system:

Pf = pf1 + pf2 and P0 = p01 + p02

This is true for any number of colliding objects.

Physics 101: Lecture 15, Pg 3

Applying the Principle of Momentum Applying the Principle of Momentum ConservationConservation

Decide which objects are included in the system. Identify external and internal forces acting on the

system. Verify that the system is isolated. Initial and final momenta of the isolated system can

be considered to be equal.

Example for an application:

Determination of velocities of colliding objects after

collision.

Physics 101: Lecture 15, Pg 4

Impulse and Momentum SummaryImpulse and Momentum Summary

Fave t J = pf – p0 = p

For a single object….

Fave = 0 momentum conserved (p = 0)

For collection of objects …

Fave,ext = 0 total momentum conserved (P = 0)

Physics 101: Lecture 15, Pg 5

CollisionsCollisions

If colliding objects constitute an isolated system (= no average external force), the total linear momentum is conserved. Sometimes also the kinetic energy is conserved.

Elastic collision: Total kinetic energy before and after the collision is the same. Inelastic collision: Total kinetic energy is not conserved, i.e. part (or all) of the kinetic energy of the objects is converted into another form of energy.

Collisions in two dimensions: Fave,ext,x t = Pf,x - P0,x => Pf,x = P0,x if Fave,ext,x = 0 Fave,ext,y t = Pf,y - P0,y => Pf,y = P0,y if Fave,ext,y = 0 x and y components of the total linear momentum are separately conserved.

Physics 101: Lecture 15, Pg 6

Center of MassCenter of Mass The center of mass of a system of objects is defined as the average location of the total mass.

Consider two interacting objects (in 1-dim.) with masses m1 and m2

at the positions x1 and x2:

xcm = (m1 x1 + m2 x2)/(m1+m2)

Displacement of center of mass:

xcm = (m1 x1 + m2 x2)/(m1+m2)

Velocity of center of mass:

vcm = (m1 v1 + m2 v2)/(m1+m2)

In an isolated system vcm does not change.

Physics 101: Lecture 15, Pg 7

Rotational KinematicsRotational Kinematics

The motion of a rigid body about a fixed axis is described by using the same concept as for linear motion (see C&J Chapter 2):

Displacement, Velocity, Acceleration

Angular Displacement:

Identify the axis of rotation and choose a line

perpendicular to this axis. Observe the motion of a point

on this line. How can one define the change of position of

this point during rotation about an axis ?

Answer:

Change of angle the line makes with a reference line: