Principles of Momentum

Principles of MomentumPrinciples of Momentum

• defined: a measure of an object’s inertia of motion

• What physical properties affect momentum?• mass• velocity

MomentumMomentum

• defined mathematically:

MomentumMomentum

p = mv

• units: kg·m/s • Linear momentum is a

vector in the same direction as the velocity vector.

• since the object’s mass is usually constant:

MomentumMomentum

Δp = mΔv• after doing some algebra:

ΣF = Δp Δt

• Momentum results from an unbalanced force.

• This is how Newton originally stated his second law of motion.

MomentumMomentum

ΣF = Δp Δt

• The longer an unbalanced force is applied, the greater its change in momentum.

• defined: the product of a force and the time the force is exerted

ImpulseImpulse

• defined mathematically:

ImpulseImpulse

I = ΣFΔt

or

I = Δp

impulse vector has same magnitude and direction as Δp vector

• “Impulsive force” generally refers to forces on a system that are large in comparison to the other forces.

• Example 11-2 shows the importance of time to a change of momentum.

ImpulseImpulse

• When no unbalanced external forces act on a system, the momentum of a system stays the same.

• The recoil of a firearm demonstrates conservation of momentum.

Conservation of Momentum Conservation of Momentum

• The mass can vary, such as in a space vehicle.

• The change in momentum of the rocket has equal magnitude and opposite direction as the momentum of the propellant exhaust.


• Thrust: Fexh x


Fexh x = mrar x

Δmprop Fexh x = - vexh x = mrar x ( ) Δt

CollisionsCollisions

• gliding masses on an air track—momentum is conserved

Linear CollisionsLinear Collisions

ptotal bfr = p1 bfr + p2 bfr

ptotal aft = p1 aft + p2 aft

Δp1 = -Δp2

• only conservative forces act

• total mechanical energy conserved

Elastic CollisionsElastic Collisions

• one or more colliding masses is deformed

• total mechanical energy not conserved

• some kinetic energy converted to other forms

Partially Elastic Collisions

Partially Elastic Collisions

• colliding masses stick together

• kinetic energy is not conserved

Inelastic CollisionsInelastic Collisions

• kinetic energy is conserved• two important equations:


v1 aft x = v1 bfr x

m1 - m2

m1 + m2( )( )( )v2 aft x = v1 bfr x

2m1

m1 + m2m1 + m2

• equations are for an elastic collision where a moving m1 collides with a resting m2

• if v1 aft x is negative then m1 has rebounded in the opposite direction from which it came


• Example 11-4: equal masses on an air track

• Example 11-5: large object collides with small object

• Example 11-6: small object collides with large object


• only one equation needed:

Inelastic CollisionsInelastic Collisions

m1 v1 bfr x = (m1 + m2) vaft x

• Example 11-7 illustrates that kinetic energy is not conserved in such collisions.

• used to determine the energy of projectiles such as bullets

• kinetic energy is almost completely converted to potential energy in the pendulum

Ballistic PendulumsBallistic Pendulums

• Most collisions occur in two (or three) dimensions.

• Air tables are often used to illustrate these.

• Δp1 = -Δp2 is still true.

Two-Dimensional Collisions


• Since momentum is still conserved...• x-component of the

momentum before the collision must equal the sum of the x-components afterward

• also true of y-components



• When two particle-like objects collide elastically, the post-collision angle between the object’s paths is exactly 90°.

• The angle decreases as the deformation in the objects increases.



Center of Mass and Angular Momentum

Center of Mass and Angular Momentum

• defined: the point in space where the mass of the entire system is assumed to be located for the purpose of determining the momentum of the system

Center of MassCenter of Mass

• in the explosion example:

Center of MassCenter of Mass

psys bfr = Σpi aft

• Σpi is the vector sum of the momenta of all of the individual particles.

• momentum is conserved• total kinetic energy is not

conserved

• A mass moving in a circle has momentum associated with its circular motion.

• For a single orbiting particle:

Angular MomentumAngular Momentum

L = mr²ω

• L = angular momentum• m = mass• r = radius of circular path• ω = angular speed


L = mr²ω

• If the mass stays constant and the radius reduces to half, then the angular speed quadruples.


L = mr²ω

• is a vector quantity• direction is found using the

rotational right-hand rule


Principles of Momentum

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