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9.1 impulse and momentum
How is velocity affected by force?
1.Newton’s first Law of motion says if
no net force acts on a body its velocity
is constant
?What would happen to a rolling ball
in the absence of friction?
?How could you move to the shore if
you were stuck in the middle of a icy
lake with a frictionless surface?
2.Newton’s 2nd Law of motion
describes how the velocity of a body is
changed by a net force acting on it
Impulse
-product of the average net force
exerted on an object and the time
interval over which the force exerts
and equals
and has the unit N s
Linear momentum-product of mass
and velocity of an object (p=mv)
?what is an example of linear
momentum?
Impulse-momentum theory
p=mv where p = momentum
-the impulse of an object is = to the
change in momentum that it causes
Problem:
What is the change in momentum of a
tennis ball that was hit by a tennis
racket?
-We know that the area under the
curve is 1.4N s ?What does this mean?
-the change in the momentum of the
ball is 1.4 N s
-remember that N s=kg m/s
?How can we rearrange the impulsemomentum theorem to solve for the
momentum of the ball after it was hit?
-so the ball’s final momentum is the
sum of the initial momentum and the
impulse
-we know that the mass of the ball is
.06kg and that the ball was originally
at rest
?How can we solve this problem?
-the answer is = to its impulse
?How can we find its velocity?
-large change in momentum occurs
only when there is a large impulse
-a large impulse can result either from
a large force acting over a short period
of time or from a smaller force acting
over a longer period of time
?What happens to a driver when a car
crash suddenly stops a car?
-an impulse is needed to bring the
driver’s momentum to zero
?What in the car can exert a large
force during a short period of time?
-a steering wheel works great for this!
?how can an air bag change this?
-an air bag reduces the force exerted
on the driver by greatly increasing the
length of the time the force is exerted
Problem:
A 2200 kg SUV traveling at 94km/h
can be stopped in 21 s by applying the
brakes. If the SUV is stopped in .22s
after hitting a wall, what is the average
force exerted by the SUV in each stop?
Know
m=2200kg
v1=26m/s
v2=0m/s
Change in T=21s and .22s
Unknown
F=?
Formula: F t = p2-p1
Demo-What happens to a person when
they jump from a chair onto the floor
in regards to their knees?
-think of the impulse-momentum
theorem
-how would knee locking vs bending
the knees affect the force AND why?
9.2 Conservation of momentum
When is the momentum of the system
of two balls conserved?
-closed system-no balls are lost and no
balls are gained(don’t knock a pool ball
off the table!)
-the forces involved are internal forces
meaning that there are no forces acting
on the system by objects outside it
Isolated system-when the net external
force on a closed system is zero
No system on Earth can be said to be
absolutely isolated because there will
always be some interactions between a
system and its surroundings
Law of conservation of momentumstates that the momentum of any
closed, isolated system does not
change
Recoil
-the momentum of a baseball changes
when the external force of a bat is
exerted on it so the baseball is not an
isolated system
-after clashing with each other, both
skaters are moving, making this
situation similar to that of an explosion.
Because the push was an internal force,
you can use the law of conservation of
momentum to find the skater’s relative
velocities
-the total momentum of the system
was zero before the push therefore, it
must be zero after the push
-the coordinate system was chosen so
that the positive direction is to the left
-the momenta of the skaters after the
push are equal in magnitude but
opposite in direction. The backward
motion of skater C is an example of
recoil.
-the velocities depend on the skaters’
relative masses. The less massive skater
moves at the greater velocity.
Two-dimensional collisions
-after the collision, both billiard balls
are moving and have momenta
-as long as the friction with the
tabletop can be ignored, the system is
closed and isolated
-the initial momentum equals the
vector sum of the final momenta:
-the equality of the momenta before
and after the collision also means that
the sum of the components of the
vectors before and after the collision
must be equal
-if the x-axis in the direction of the
initial momentum then the ycomponent of the initial momentum is
equal to zero
-the sum of the final y-component must
be zero
-the y-components are equal in
magnitude but are in the opposite
direction and have opposite signs. The
sum of the horizontal components also
is equal.
Conservation of angular momentum
-states that if no net external torque
acts on an object, then its angular
momentum does not change
-an object’s initial angular momentum
is equal to its final angular
momentum
-Earth spins on its axis with no
external torque so its angular
momentum is constant and conserved
so the length of a day does not
change
-if a torque-free object starts with no
angular momentum, it must continue
to have no angular momentum
-if part of an object rotates in one
direction, another part must rotate in
the opposite direction
Example-if you switch on a loosely
held electric drill, the drill body will
rotate in the direction opposite to the
rotation of the motor and bit
-because of the conservation of the
angular momentum, the direction of
rotation of a spinning object can be
changed only by applying a torque
-when a top is vertical, there is no
torque on it and the direction of its
rotation does not change
-if the top is tipped, torque tries to
rotate it downward. Rather than
tipping over, the upper end of the top
revolves or precesses slowly about
the vertical axis.
-a gyroscope is a wheel or disk that
spins rapidly around one axis while
being free to rotate around one or
two other axes
-the direction of its large angular
momentum can be changed only by
applying an appropriate torque.
Without such a torque, the direction
of the axis of rotation does not
change.
-gyroscopes are used in airplanes,
submarines, and space crafts to keep
unchanging reference in direction
9.2 conservation of momentum
Two particle collisions
-during the collisions of two balls, each
briefly exerts a force on the other
-the forces that they exert on each other
are equal and opposite in reaction
?How do the impulses compare?
-they are equal in magnitude and opposite
in direction
?How can we find the momentum?
*F t=p2-p1
Ball 1 pA2=FBonA t + pA1
Ball 2 pB2=FAonB t + pB1
Conservation of momentum
pA2 + pB2 = pA1 + pB1
?what does this mean?
-the sum of the momentum of the balls is
the same before and after the collision
Momentum in a closed system
Closed system-a system that does not gain
or lose mass
Internal forces-all the forces within a
closed system
Isolated system-when the net external
force on a closed system is zero
Law of conservation of momentum
States that the momentum of any closed
system with no net external force does not
change
Problem: car collisions
A 2275 kg car is going 28m/s and rear
ends a 875 kg compact car on ice going
16m/s in the same direction. The two cars
stick together. How fast does the
wreckage move immediately after the
collision?
mA = 2275 kg
vA1 = 28m/s
mB = 875 kg
vB1 = 16 m/s
v2=unknown
remember that momentum = impulse
remember that F t = m v
?what are some calculations to use?
Conservation of momentum
p1=p2
pA1 + pB1 = pA2 + pB2
mAvA1 + mBvB1 = mAvA2 + mBvB2
velocities are equal
vA2=vB2=v2
mAvA1 +mBvB1=(mA + mB)v2
v2= mAvA1 +mBvB1
mA + m B
v2=(2275kg)(28m/s) + (875kg)(16m/s)
2275 kg + 875kg
V2=25m/s
Explosions
Example
Skater A gives skater B a “push”
with the total momentum being 0
before and after the push. The
momentum of the skaters after the
push are equal in magnitude BUT
opposite in direction. The backward
motion of the skater after the push
is an example of recoil.
?How does a rocket in space change
its velocity?
-after chemicals are mixed
producing hot gases that leave the
exhaust nozzle at high speeds
-the law of conservation of
momentum can be applied
Problem
An astronaut at rest in space fires a
thruster pistol that expels 35g of hot
gas at 875m/s. The combined mass
of the astronaut and pistol is 84kg.
How fast and in what direction is
the astronaut moving after firing
the pistol?
(look at picture p212)
mA=84kg
mB=.035kg
vA1 = vB1=0m/s
vB2=-875m/s
vA2=?
Calculations
p1=pA1+pB1=0
pA1 + pA2 = pA2 + pB2
0 =pA2 + pB2
pA2 = -pB2
mAvA2=-mBvB2
so solve for vA2=-(mBvB2)
mA
vA2=-(.035kg)(-875m/s) = +.36m/s
84kg
Angular Momentum
-the quantity of motion used with
objects rotating around a fixed axis
-changes when torque acts on an
object
-(review) torque-product of the
applied force and the lever arm which
is the perpendicular distance from the
axis of rotation to a line along which
the force acts
-rotational inertia-resistance to change
in angular velocity
comparison
comparison
p=mv to L=Iw
w=angular velocity
I=rotational inertia
L=angular momentum
Variations in these equations
Trends
-if the torque is 0 then change in L is 0
-no change in angular momentum over
a period of time it must be conserved
-if an object is rotating at a given
velocity about a fixed axis, its
rotational inertia must increase if its
angular velocity decreases and vice
versa