Linear Momentum

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Linear Momentum

• Unit 5

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Lesson 1 Linear Momentum and Its Conservation
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d
(m1v1 m2v2) 0
dt
Linear momentum is a vector quantity whose
direction the same as the direction of v. Its SI
unit is kg . m/s.
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dv
SF ma m
dt
(This is the form in which Newton presented his
second law.)
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Conservation of Linear Momentum
Whenever two or more particles in an isolated
system interact, the total momentum of the system
remains constant.
The total momentum of an isolated system at all
times equals its initial momentum.
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Example 1
a) With what velocity does the archer move
across the ice after firing the arrow.
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b) What if the arrow were shot in a direction
that makes an angle q with the horizontal ? How
will this change the recoil velocity of the
archer ?
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Example 2
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a) What is the speed of the block of mass M ?
b) Find the original elastic potential energy in
the spring if M 0.350 kg.
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Example 3 AP 1981 2
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The swing continues in the same direction until
the cord makes a 45o angle with the vertical as
shown in Figure II at that point it begins to
swing in the reverse direction. With what
velocity relative to the ground did the child
leave the swing ?
(cos 45o sin 45o
, sin 30o cos 60o ½,
cos 30o sin 60o
)
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Lesson 2 Impulse and Momentum
dp F dt
Integrating F with respect to t,
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Impulse is a vector quantity with the same
direction as the direction of the change in
momentum.
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Because the force imparting an impulse can
generally vary in time, we can express impulse as
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Impulse Momentum Theorem
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Example 1
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a) If the collision lasts for 0.150 s, find the
impulse caused by the collision and the average
force exerted on the car.
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b) What if the car did not rebound from the wall
? Suppose the final velocity of the car is zero
and the time interval of the collision remains
0.150 s. Would this represent a larger or a
smaller force by the wall on the car ?
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Example 2
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Example 3
a) the impulse delivered to the ball
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b) the average force exerted on the ball
c) the peak force exerted on the ball
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Lesson 3 Collisions in One-Dimension
perfectly inelastic
inelastic
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Perfectly Inelastic Collisions
when the colliding objects stick together
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Elastic Collisions
m1v1i m2v2i m1v1f m2v2f
½ m1v1i2 ½ m2v2i2 ½ m1v1f2 ½ m2v2f2
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Example 1
An 1800 kg car stopped at a traffic light is
struck from the rear by a 900 kg car, and the two
become entangled, moving along the same path as
that of the originally moving car.
a) If the smaller car were moving at 20.0 m/s
before the collision, what is the velocity of
the entangled cars after the collision ?
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b) Suppose we reverse the masses of the cars a
stationary 900 kg car is struck by a moving
1800 kg car. Is the final speed the same as
before ?
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Example 2
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Example 3
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a) Find the velocities of the two blocks after
the collision.
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b) During the collision, at the instant block 1
is moving to the right with a velocity of 3.00
m/s, determine the velocity of block 2.
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Example 4 AP 1995 1
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The figure below shows a graph of the force
exerted on the ball by the cube as a function of
time.
a) Determine the total impulse given to the ball.
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b) Determine the horizontal velocity of the ball
immediately after the collision.
c) Determine the following for the cube
immediately after the collision.
i. Its speed
ii. Its direction of travel (right or left), if
moving
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d) Determine the kinetic energy dissipated in the
collision.
e) Determine the distance between the two points
of impact of the objects with the floor.
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Example 5 AP 1994 1
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a) Calculate the energy stored in the spring at
maximum compression.
b) Calculate the speed of the clay ball and 2 kg
block immediately after the clay sticks to the
block but before the spring compresses
significantly.
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c) Calculate the initial speed v of the clay.
In a second experiment, an identical ball of clay
is thrown at another identical 2 kg block, but
this time the stop is removed so that the 8 kg
block is free to move.
d) State whether the maximum compression of the
spring will be greater than, equal to, or less
than 0.4 m. Explain briefly.
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e) State the principle or principles that can be
used to calculate the velocity of the 8 kg block
at the instant that the spring regains its
original length. Write the appropriate
equation(s) and show the numerical
substitutions, but do not solve for the velocity.
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Lesson 4 Two-Dimensional Collisions
m1v1i m1v1f cosq m2v2f cosf
0 m1v1f sinq - m2v2f sinf
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Since this is an elastic collision, KE is also
conserved.
½ m1v1i2 ½ m1v1f2 ½ m2v2f2
If this were an inelastic collision, KE would
not be conserved, and this equation does not
apply.
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Example 1
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Example 2
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Lesson 5 The Center of Mass
Center of mass (CM) is the average position of
the systems mass
Center of mass is located on the line joining the
two particles and is closer to the particle
having the larger mass.
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How to Locate the Center of Mass
System rotates clockwise when F is applied
between the less massive particle and CM.
System rotates counterclockwise when F is applied
between the more massive particle and CM.
System moves in the direction of F without
rotating when F is applied at CM.
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Center of mass for a system of many particles
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Using a Position Vector (r) to locate CM of a
system of particles
Using a Position Vector (r) to locate CM of an
extended object
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(likewise for yCM and zCM)
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Locating CM of an irregularly shaped object
CM is where lines AB and CD intersect
If wrench is hung freely from any point, the
vertical line through this point must pass
through CM.
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Example 1
a) the piece on the right.
b) the piece on the left.
c) both have same mass.
d) impossible to determine.
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Example 2
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Example 3
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b) Suppose a rod is nonuniform such that its
mass per unit length varies linearly with x
according to the expression l ax, where a is a
constant. Find the x coordinate of the center
of mass as a fraction of L.
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Lesson 6 Motion of a System of Particles
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Total momentum of a system of particles
Newtons Second Law for a system of particles
SFext MaCM
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Example 1
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Example 2
a) Plot these particles on a grid or graph paper.
Draw their position vectors and show their
velocities.
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b) Find the CM of the system and mark it on the
grid.
c) Determine the velocity of the CM and also
show it on the diagram.
d) What is the total linear momentum of the
system ?
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Example 3
A rocket is fired vertically upward. At the
instant it reaches an altitude of 1000 m and a
speed of 300 m/s, it explodes into three
fragments having equal mass. One fragment
continues to move upward with a speed of 450 m/s
following the explosion. The second fragment has
a speed of 240 m/s and is moving east right after
the explosion. What is the velocity of the third
fragment right after the explosion ?