Transcript Chapter 6 - Linear Momentum

Chapter 6
Linear Momentum
Momentum
 Momentum is defined as the product of
mass and velocity.
 p = m·v
 Momentum is measured in [kg·m/s]
 Momentum is a vector quantity
 The momentum of a system of particles is
the vector SUM of the individual momenta of
each particle.
Example
 Comparison of a bullet, a cruise ship, and a
glacier…
 Qualitative Reasoning…
 Quantitative Reasoning…
Newton’s 2nd Law
 Write Newton’s 2nd Law another way
Fnet = ma
Fnet = m(vf – vi /t)
Fnet = (pf – pi)/t
Fnet = Δp/t
Impulse – Momentum Relationship

Fav = Δp/t
Fav·Δt = Δp
Fav·Δt = pf – pi
 Impulse = Fav·Δt
measured in [Ns]
 When a force acts on an object for a particular
amount of time, the impulse it imparts is equivalent
to the change in momentum of the object.
Examples
 A golfer drives a 0.046 kg ball from an
elevated tee, giving the ball a horizontal
speed of 40m/s. What is the magnitude of
the average force delivered by the club
during this time? The contact time is
approximately 1 millisecond.
Examples
 A 70.0 kg worker jumps stiff-legged from a
height of 1 meter onto a concrete floor.
What is the magnitude of the force he feels
on landing, assuming a sudden stop in 8.0
milliseconds.
– Two parts to the problem: the fall to the floor
and the stop by the floor
Kinetic Energy

KE = ½ mv2
KE = p2/(2m)
Conservation of Momentum
 Conservation of Momentum is a
fundamental concept in physics that allows
for analysis of many systems.
 It is commonly used to analyze collisions.
 Conservation of Momentum can only be
applied if no external forces act on a
system. Internal forces do not change to
overall momentum of a system.
 In a closed system, the total momentum of
the system is conserved.
Conservation of Momentum
 In a closed system:
pi = pf
p1i + p2i + p3i + … = p1f + p2f + p3f + …
Examples
 Two masses m1 = 1.0 kg and m2 = 2.0 kg,
are held on either side of a light compressed
spring by a light string joining them. The
string is burned (negligible external force)
and the masses move apart on a frictionless
surface, with m1 having a velocity of 1.8 m/s.
What is the velocity of m2?
More Examples YAY 
 A 30 g bullet with speed of
400 m/s strikes a glancing
blow to a target brick of
mass 1.0kg. The brick
breaks into two fragments.
The bullet deflects at an
angle of 30° above the xaxis with speed of 100
m/s. One piece of the brick
, with mass of 0.75 kg,
goes off to the right with a
speed of 5.0 m/s.
Determine the speed and
direction of the other
piece.
 A physics teacher is
lowered from a helicopter
to the middle of a smooth
level frozen lake. She is
challenged to make her
way off the ice. Walking is
out of the question. (Why?)
She decides to throw her
identical, heavy mittens,
which will provide the
momentum to get off the
ice. Should she throw
them together or
separately?
Inelastic vs. Elastic Collisions
 For an isolated system, momentum is always conserved
whether a collision is elastic or inelastic.
 In an inelastic collision, KE is not conserved. Inelastic
collisions involve deformation or coupling of individual
parts. Some energy goes into the deformation, so
mechanical energy is not conserved. Example: Train cars
joining, car accidents.
 In an elastic collision, KE is conserved in addition to
momentum. In an elastic collision, there is no deformation.
KE is transferred from one object to another. Example:
billiard balls
Example – Inelastic Collision
 A 1.0 kg ball with a speed of 4.5 m/s strikes
a 2.0 kg stationary ball. If the collision is
completely inelastic (the balls stick together
after the collision) find the final velocity after
the collision. How much Kinetic Energy is
lost in this collision?
Center of Mass
 The center of mass is the point at which all
of the mass of an object or system may be
considered to be concentrated.
 Fnet = MAcm where M is total mass of
system and Acm is acceleration of the center
of mass
Calculating Center of Mass
Examples
 Three masses – 2.0 kg, 3.0 kg and 6.0 kg
are located at positions (3.0,0), (6.0,0) and
(-4.0,0) respectively, in meters, from the
origin. Find the center of mass.
 A dumbbell has a connecting bar of
negligible mass. Find the location of the
center of mass if m1 (located at 0.2 meters)
and m2 (located at 0.9 meters) are each 5.0
kg. What if m1 is 5.0 kg and m2 is 10.0 kg?
Center of Gravity
 Center of Gravity is similar to Center of Mass – it is
the point on an object where the force of gravity is
considered to be concentrated.
 Many times the location of the center of gravity
can be determined by symmetry (circles, squares)
 For flat irregularly shaped objects, the center of
gravity can be found by suspending the shape
from two different points and looking for the
intersection (see example in text)