Chapter 6Chapter 6

Linear MomentumLinear Momentum

MomentumMomentum

Momentum is defined as the product of Momentum is defined as the product of mass and velocity.mass and velocity.

pp = m = m··vv Momentum is measured in [kgMomentum is measured in [kg··m/s]m/s] Momentum is a vector quantityMomentum is a vector quantity The momentum of a system of particles is The momentum of a system of particles is

the vector SUM of the individual momenta of the vector SUM of the individual momenta of each particle.each particle.

ExampleExample

Comparison of a bullet, a cruise ship, and a Comparison of a bullet, a cruise ship, and a glacier…glacier…

Qualitative Reasoning…Qualitative Reasoning…

Quantitative Reasoning…Quantitative Reasoning…

Newton’s 2Newton’s 2ndnd Law Law

Write Newton’s 2Write Newton’s 2ndnd Law another way Law another way

FFnetnet = m = maa

FFnetnet = m( = m(vvff – – vvi i /t)/t)

FFnetnet = ( = (ppff – – ppii)/t)/t

FFnetnet = = ΔΔpp/t/t

Impulse – Momentum RelationshipImpulse – Momentum Relationship FFavav = = ΔΔpp/t/t

FFavav··ΔΔt = t = ΔΔpp

FFavav··ΔΔt = t = ppff – – ppii

Impulse = Impulse = FFavav··ΔΔt measured in [Ns]t measured in [Ns]

When a force acts on an object for a particular When a force acts on an object for a particular amount of time, the impulse it imparts is equivalent to amount of time, the impulse it imparts is equivalent to the change in momentum of the object.the change in momentum of the object.

Examples Examples

A golfer drives a 0.046 kg ball from an A golfer drives a 0.046 kg ball from an elevated tee, giving the ball a horizontal elevated tee, giving the ball a horizontal speed of 40m/s. What is the magnitude of speed of 40m/s. What is the magnitude of the average force delivered by the club the average force delivered by the club during this time? The contact time is during this time? The contact time is approximately 1 millisecond.approximately 1 millisecond.

ExamplesExamples

A 70.0 kg worker jumps stiff-legged from a A 70.0 kg worker jumps stiff-legged from a height of 1 meter onto a concrete floor. height of 1 meter onto a concrete floor. What is the magnitude of the force he feels What is the magnitude of the force he feels on landing, assuming a sudden stop in 8.0 on landing, assuming a sudden stop in 8.0 milliseconds.milliseconds.– Two parts to the problem: the fall to the floor Two parts to the problem: the fall to the floor

and the stop by the floorand the stop by the floor

Kinetic EnergyKinetic Energy

KE = ½ mvKE = ½ mv22

KE = pKE = p22/(2m)/(2m)

Conservation of MomentumConservation of Momentum Conservation of Momentum is a Conservation of Momentum is a

fundamental concept in physics that allows fundamental concept in physics that allows for analysis of many systems.for analysis of many systems.

It is commonly used to analyze collisions.It is commonly used to analyze collisions. Conservation of Momentum can only be Conservation of Momentum can only be

applied if no external forces act on a applied if no external forces act on a system. Internal forces do not change to system. Internal forces do not change to overall momentum of a system.overall momentum of a system.

In a closed system, the total momentum of In a closed system, the total momentum of the system is conserved.the system is conserved.

Conservation of MomentumConservation of Momentum

In a closed system:In a closed system:

ppii = = ppff

pp1i1i + p + p2i2i + p + p3i3i + … = p + … = p1f1f + p + p2f2f + p + p3f3f + … + …

ExamplesExamples

Two masses mTwo masses m11 = 1.0 kg and m = 1.0 kg and m22 = 2.0 kg, = 2.0 kg,

are held on either side of a light compressed are held on either side of a light compressed spring by a light string joining them. The spring by a light string joining them. The string is burned (negligible external force) string is burned (negligible external force) and the masses move apart on a frictionless and the masses move apart on a frictionless surface, with msurface, with m11 having a velocity of 1.8 m/s. having a velocity of 1.8 m/s.

What is the velocity of m What is the velocity of m22??

More Examples YAY More Examples YAY A 30 g bullet with speed of A 30 g bullet with speed of

400 m/s strikes a glancing 400 m/s strikes a glancing blow to a target brick of blow to a target brick of mass 1.0kg. The brick mass 1.0kg. The brick breaks into two fragments. breaks into two fragments. The bullet deflects at an The bullet deflects at an angle of 30angle of 30° above the x-° above the x-axis with speed of 100 axis with speed of 100 m/s. One piece of the brick m/s. One piece of the brick , with mass of 0.75 kg, , with mass of 0.75 kg, goes off to the right with a goes off to the right with a speed of 5.0 m/s. speed of 5.0 m/s. Determine the speed and Determine the speed and direction of the other direction of the other piece.piece.

A physics teacher is A physics teacher is lowered from a helicopter lowered from a helicopter to the middle of a smooth to the middle of a smooth level frozen lake. She is level frozen lake. She is challenged to make her challenged to make her way off the ice. Walking is way off the ice. Walking is out of the question. (Why?) out of the question. (Why?) She decides to throw her She decides to throw her identical, heavy mittens, identical, heavy mittens, which will provide the which will provide the momentum to get off the momentum to get off the ice. Should she throw ice. Should she throw them together or them together or separately?separately?

Inelastic vs. Elastic CollisionsInelastic vs. Elastic Collisions

For an isolated system, momentum is For an isolated system, momentum is alwaysalways conserved conserved whether a collision is elastic or inelastic.whether a collision is elastic or inelastic.

In an inelastic collision, KE is not conserved. Inelastic In an inelastic collision, KE is not conserved. Inelastic collisions involve deformation or coupling of individual collisions involve deformation or coupling of individual parts. Some energy goes into the deformation, so parts. Some energy goes into the deformation, so mechanical energy is not conserved. Example: Train cars mechanical energy is not conserved. Example: Train cars joining, car accidents.joining, car accidents.

In an elastic collision, KE is conserved in addition to In an elastic collision, KE is conserved in addition to momentum. In an elastic collision, there is no deformation. momentum. In an elastic collision, there is no deformation. KE is transferred from one object to another. Example: KE is transferred from one object to another. Example: billiard ballsbilliard balls

Example – Inelastic CollisionExample – Inelastic Collision

A 1.0 kg ball with a speed of 4.5 m/s strikes A 1.0 kg ball with a speed of 4.5 m/s strikes a 2.0 kg stationary ball. If the collision is a 2.0 kg stationary ball. If the collision is completely inelastic (the balls stick together completely inelastic (the balls stick together after the collision) find the final velocity after after the collision) find the final velocity after the collision. How much Kinetic Energy is the collision. How much Kinetic Energy is lost in this collision?lost in this collision?

Center of MassCenter of Mass

The center of mass is the point at which all The center of mass is the point at which all of the mass of an object or system may be of the mass of an object or system may be considered to be concentrated.considered to be concentrated.

FFnetnet = MA = MAcm cm where M is total mass of system where M is total mass of system

and Aand Acmcm is acceleration of the center of mass is acceleration of the center of mass

Calculating Center of MassCalculating Center of Mass

ExamplesExamples

Three masses – 2.0 kg, 3.0 kg and 6.0 kg Three masses – 2.0 kg, 3.0 kg and 6.0 kg are located at positions (3.0,0), (6.0,0) and are located at positions (3.0,0), (6.0,0) and (-4.0,0) respectively, in meters, from the (-4.0,0) respectively, in meters, from the origin. Find the center of mass.origin. Find the center of mass.

A dumbbell has a connecting bar of A dumbbell has a connecting bar of negligible mass. Find the location of the negligible mass. Find the location of the center of mass if mcenter of mass if m11 (located at 0.2 meters) (located at 0.2 meters) and mand m22 (located at 0.9 meters) are each 5.0 (located at 0.9 meters) are each 5.0 kg. What if mkg. What if m11 is 5.0 kg and m is 5.0 kg and m22 is 10.0 kg? is 10.0 kg?

Center of GravityCenter of Gravity

Center of Gravity is similar to Center of Mass – it is Center of Gravity is similar to Center of Mass – it is the point on an object where the force of gravity is the point on an object where the force of gravity is considered to be concentrated.considered to be concentrated.

Many times the location of the center of gravity Many times the location of the center of gravity can be determined by symmetry (circles, squares) can be determined by symmetry (circles, squares)

For flat irregularly shaped objects, the center of For flat irregularly shaped objects, the center of gravity can be found by suspending the shape gravity can be found by suspending the shape from two different points and looking for the from two different points and looking for the intersection (see example in text)intersection (see example in text)