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Chapter 9 - PowerPoint PPT Presentation

Chapter 9. Linear Momentum and Collisions. Intro. Consider bowling: Bowling ball collides with initial pin Force on/Acceleration of the Pin Force on/Acceleration of the ball Momentum- simplified way to study these moving objects. 9.1 Linear Momentum and Conservation.

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Presentation Transcript

Chapter 9

Linear Momentum and Collisions

• Consider bowling:

• Bowling ball collides with initial pin

• Force on/Acceleration of the Pin

• Force on/Acceleration of the ball

• Momentum- simplified way to study these moving objects.

Consider two particles (isolated) m1 and m2 moving at v1 and v2

From Newton’s 3rd Law

• And if the the masses are constant

• If the derivative of a function is 0 it is constant (conserved)

• Linear Momentum- the product of the mass and velocity of a moving particle.

• Momentum is a vector quantity

• Has dimensions MLT-1 and SI units kg.m/s

• All momentum is conserved

• Three Components of Momentum

• Directly related to Newton’s 2nd Law

• Instead of Net Force equals mass times accel.

• Can be described as Net Force equals time rate of change of momentum

• Conservation of Momentum

• For isolated systems the time derivative of the total momentum is 0

• The total momentum is therefore constant or conserved.

Law of Conservation of Momentum

• And in 3 components

• Quick Quizzes p 254-255

• Examples 9.1-9.2

• The momentum of an object changes when a net force acts on it

or

• Integrating this gives

• Impulse-Momentum Theorem- the impulse of the force F acting on a particle equals the change in momentum of the particle

• Force Varies, impulse time is short impulse can generally be calculated with the average force.

• Quick Quizzes p. 258

• Examples 9.3-9.4

• Momentum is conserved

• Three types of Collisions

• Inelastic

• Perfectly Inelastic

• Elastic

• Inelastic Collision- Momentum is conserved but kinetic energy is not.

• Inelastic- objects collide and separate, some K is lost

• Perfectly Inelastic- objects collide and stick together (moving as one), some K is lost

• Elastic Collisions-A collision in which no energy is lost to (surroundings / internal / potential)

• Both momentum and kinetic energy are conserved

• By combining the Conservation of p and K equations

• In all collision types careful attention to the direction (and sign) of velocities must be paid.

• Quick Quizzes p. 262

• Examples 9.5 - 9.9

• Momentum is conserved on each axis

• Examples 9.10 – 9.12

• We can describe the overall motion of a mechanical system by tracking its center of mass

• System could be a group of particles

• System could be a large extended object

• A force applied to the center of mass will cause no rotation to the system

• To find the center of mass in 3-D space for a number (i) particles

• Or in terms of the position vector of each particle

• For extended objects that have a continuous mass distribution

• Consider them an infinite number of closely spaced particles

• The sum becomes an integral

• Or in terms of the position vector

• For symmetrical objects, the center of mass lies on the axis/plane of symmetry

• Examples: uniform rod,

sphere,

cube,

donut?

• For extended objects, the force of gravity acts individually on each small piece of mass (dm)

• The net effect of all these forces is equivalent to the single force Mg, through a point called the center of gravity.

• If the gravitational field is uniform across all dm, the center of gravity and center of mass are one and the same.

• Quick quiz p 272

• Examples 9.13, 9.14, 9.15

• If the mass of a system remains constant (no particles entering/leaving) then we can track the motion of the center of mass, rather than the individual particles.

• Also assumes any forces on the system are internal (isolated)

• Velocity of the center of mass

• Acceleration of the center of mass

• If there is a net force on the system, it will move equivalent to the way a single M with the same net force would move.

• And if the net force is zero

• Quick Quizzes p. 276

• Examples 9.17, 9.18

• Most forms of vehicular motion result from action/reaction friction.

• A rocket has nothing to push against so its motion/control depend on conservation of motion of the system.

• The system includes the rocket body (and payload) plus the ejected fuel

• The rocket burns fuel and oxidizer creating expanding gases that are directed through the nozzle.

• Each gas molecule has a mass (that was once part of the rockets total mass) and velocity, therefore a downward momentum.

• The rocket receives the same compensating momentum upward.

• Looking a rocket initially with mass M + Δm, moving with velocity v…

• And some time, Δt, later...

• The rocket now has mass, M and velocity v + Δv, compensating the momentum of the exhausted mass, Δm.

• The conservation of momentum expression for this change…

• Can be simplified to…

• A rocket motor produces a continuous flow of exhaust gas a fairly constant speed, through the burn

• For continually changing values…

Δv dv

Δm dm

So… 

• Because the increase in exhaust mass = the decrease in rocket mass…

• Then integrate this expression

• Discuss integral of M-1

• Evaluating from vi to vf gives the basic expression for rocket propulsion.

• Mi is the total mass of the rocket/payload plus fuel

• Mf is the mass of the rocket/payload

• Mi – Mf is the mass of fuel needed to achieve a certain speed (eg. Escape speed to power down rocket)

• Thrust- the actual force on the rocket at any given time is

• Thrust is proportional to exhaust speed and also the rate of change of mass (burn rate).

• Examples 9.19 p. 279