Linear Momentum

Linear Momentum

Momentum is a measure of how hard it is to stop or turn a moving object.

p = mv (single particle) P = Σpi (system of particles)

• Problem: How fast must an electron move to have the same momentum as a proton moving at 300 m/s?

• Problem: A 90-kg tackle runs north at 5.0 m/s and a 75-kg quarterback runs east at 8.0 m/s. What is the momentum of the system of football players?

Impulse (J)

Impulse is the integral of force over a period of time.

J = Fdt The change in momentum of a particle is

equal to the impulse acting on it. Dp = J

Problem: Restate Newton’s 2nd Law in terms of impulse.

time

Force Impulsive forces are generally of high magnitude and short duration.

Problem: A 150-g baseball moving at 40 m/s 15o below the horizontal is struck by a bat. It leaves the bat at 55 m/s 35o above the horizontal. What is the impulse exerted by the bat on the ball?

If the collision took 2.3 ms, what was the average force?

• Problem: An 85-kg lumberjack stands at one end of a floating 400-kg log that is at rest relative to the shore of a lake. If the lumberjack jogs to the other end of the log at 2.5 m/s relative to the shore, what happens to the log while he is moving?

• Problem: Two blocks of mass 0.5 kg and 1.5 kg are placed on a horizontal, frictionless surface. A light spring is compressed between them. A cord initially holding the blocks together is burned; after this, the block of mass 1.5 kg moves to the right with a speed of 2.0 m/s. A) What is the speed and direction of the other block? B) What was the original elastic energy in the spring?

Conservation of Linear Momentum

The linear momentum of a system is conserved unless the system experiences an external force.

Collision Review

Collisions in 1 and 2 Dimensions

Collision review

In all collisions, momentum is conserved. Elastic Collisions: No deformation occurs

Kinetic energy is also conserved. Inelastic Collisions: Deformation occurs

Kinetic energy is lost. Perfectly Inelastic Collisions

Objects stick together, kinetic energy is lost. Explosions

Reverse of perfectly inelastic collision, kinetic energy is gained.

• 1 Dimensional Problem: A 1.5 kg cart traveling at 1.5 m/s collides with a stationary 0.5 kg cart and sticks to it. At what speed are the carts moving after the collision?

• 1 Dimensional Problem: A 1.5 kg cart traveling at 1.5 m/s collides elastically with a stationary 0.5 kg cart. At what speed are each of the carts moving after the collision?

• 1 Dimensional Problem: What is the recoil velocity of a 120-kg cannon that fires a 30-kg cannonball at 320 m/s?

For 2-dimensional collisions

Use conservation of momentum independently for x and y dimensions.

You must resolve your momentum vectors into x and y components when working the problem

•2 Dimensional Problem: A pool player hits a cue ball in the x-direction at 0.80 m/s. The cue ball knocks into the 8-ball, which moves at a speed of 0.30 m/s at an angle of 35o angle above the x-axis. Determine the angle of deflection of the cue ball.

Collision Problems Practice Day

Problem 1 A proton, moving with a velocity of vii, collides elastically with another

proton initially at rest. If the two protons have equal speeds after the collision, find a) the speed of each in terms of vi and b) the direction of the velocity of each.

Center of Mass

Center of Mass

The center of mass of a system is the point at which all the mass can be assumed to reside.

Sometimes the system is an assortment of particles and sometimes it is a solid object.

Mathematically, you can think of the center of mass as a “weighted average”.

Center of Mass – system of points

Analogous equations exist for velocity and acceleration, for example

i icm

x mx

M

i icm

y my

M

i icm

z mz

M

,,

x i ix cm

v mv

M

Sample Problem Determine the Center of Mass.

2 4-2 0

0

2

-2

x

y

2 kg

3 kg1 kg

Center of Mass-- solid objects

If the object is of uniform density, you pick the geometric center of the object.

xx

x For combination objects,

pick the center of each part and then treat each center as a point.

x

x

Center of Mass Problem Determine the

center of mass of this shape.

x

y

2R

Center of Mass for more complicated situations If the shapes are points or simple geometric

shapes of constant density, then

cm

xdmx

M

i icm

x mx

M

Anything else is more complicated, and

• Problem: Find the x-coordinate of the center of mass of a rod of length L whose mass per unit length varies according to the expression l = ax.

Lx

y

• Problem: A thin strip of material of mass M is bent into a semicircle of radius R. Find its center of mass.

x

y

R

R

-R

Motion of a System of Particles

Motion of a System of Particles If you have a problem involving a system of

many particles, you can often simplify your problem greatly by just considering the motion of the center of mass.

If all forces in the sytem are internal, the motion of the center of mass will not change.

The effect of an external force that acts on all particles can be simplified by considering that it acts on the center of mass; that is:

SFext = Macm

Homework problem #47 – collaborative work (15 minutes) This problem is a toughie! However, it gets easier if you

use the idea that the center of mass of the system of Romeo, Juliet, and canoe does not change because all forces are internal. The question is paraphrased below:

Rome (77 kg) sits in the back of a canoe 2.70 m away from Juliet (55 kg) who sits in the front. The canoe has a mass of 80 kg. Juliet moves to the rear of the canoe to kiss Romeo. How far does the canoe move forward when she does this? (Assume a canoe that is symmetrical).