Transcript ISNS3371_012307_bw

ISNS 3371 - Phenomena of Nature
A Rifle and a Bullet
When a bullet is fired from a rifle, the rifle recoils due to the interaction
between the bullet and the rifle.
The force the rifle exerts on the bullet is equal and opposite to the force
the bullet exerts on the rifle.
But the acceleration of the bullet is much larger that the acceleration of
the rifle - due to Newton’s 2nd law: a = F/m
The acceleration due to a force is inversely proportional to the mass.
The force on the rifle and the bullet is the same but the mass of the rifle
is much larger than the the mass of the bullet so the acceleration of the
rifle is much less than the acceleration of the bullet.
ISNS 3371 - Phenomena of Nature
Tension
Consider a block being pulled by a rope. The person doing the pulling at one
end of the rope is not in contact with the block, and cannot exert a direct
force on the block. Rather a force is exerted on the rope, which transmits
that force to the block. The force experienced by the block from the rope is
called the stretching force, commonly referred to as tension.
Tension is actually not a force - tension transmits the stretching force. A
force always has a direction - the tension in a string or rope must follow the
rope! The tension may have to extend around corners, over and under
pulleys, etc. So, tension transmits a force through a string or rope, but
tension is not a force. Tension doesn't work exactly the way force does.
ISNS 3371 - Phenomena of Nature
Suppose you hang a 5 Newton weight from a string, and hold the other
end of the string in your hand. If the weight (and the string and your
hand) is at rest, then the weight exerts a 5 N downward force on the
lower end of the string, and you exert a 5 N upward force on the upper
end of the string. What is the stretching force/tension in the string? It is
possible to build very plausible arguments that the tension in the string is
10 N, or that it is 0 N, or that it is 5 N - but what is it, really, and why?
Remember - tension transmits the force. It would be the same as if you
were holding the weight in your hand - the force on your hand would be 5
N. Therefore the stretching force/tension is 5 N.
In a tug-of-war, the tension in the rope is produced by the people pulling
on opposite ends of the rope. The forces at either end of the rope are
equal and opposite. What is the tension in the rope?
What happens if a 200 lb man wearing socks and a 100 lb girl wearing
rubber-soled shoes have a tug-of-war? Who wins?
ISNS 3371 - Phenomena of Nature
Momentum
Momentum is mass times velocity, a vector quantity:
Mom=mv
The more massive an object, the greater its momentum.
The greater the velocity of an object, the larger its momentum.
The momentum of an object is changed by applying a force:
- the larger the applied force, the greater the change in
momentum.
- the longer the force is applied, the greater the change in
momentum
ISNS 3371 - Phenomena of Nature
Impulse
Impulse of a force is the force times the time over which the
force acts on a body.
I = F x ∆T
∆ means a change in a quantity - ∆T is the time over which the
force is acting.
From Newton’s second law:
v
F  ma  m
t
v
I  Ft  m t
t
I  mv
Therefore, an Impulse produces a change in momentum of a
body.
ISNS 3371 - Phenomena of Nature
Process of minimizing an impact force - approached from the definition of
the impulse of force: If an impact stops a moving object, then the change
in momentum is a fixed quantity, and extending the time of the collision
will decrease the impact force by the same factor.
This principle is applied in many common-sense situations:
• If you jump to the ground from any height, you bend your knees
upon impact, extending the time of collision and lessening the impact
force.
• A boxer moves away from a punch, extending the time of impact
and lessening the force.
• Automobiles are made to collapse upon impact, extending the time
of collision and lessening the impact force.
• If you drop a glass on hard floor - it breaks. If you drop it on a soft
carpet, the impact time is extended as the glass sinks into the carpet
- impact force reduced - glass doesn’t break.
ISNS 3371 - Phenomena of Nature
Conservation of Momentum
Law of Conservation of Momentum
The total momentum of an isolated system is conserved,
I.e., it remains constant.
An outside or external force is required to change the momentum of an
isolated system.
The Law of Conservation of Momentum is an alternate way of stating
Newton’s laws:
1. An object’s momentum will not change if left alone
2. A force can change an object’s momentum, but…
3. Another equal and opposite force simultaneously changes
some other object’s momentum by same amount
ISNS 3371 - Phenomena of Nature
Collisions
In a collision, momentum is conserved because the forces acting are
internal forces - momentum is simply redistributed.
net momentum before collision = net momentum after collision
ISNS 3371 - Phenomena of Nature
Elastic Collisions
An elastic collision is one in which the objects collide without generating heat
or being permanently deformed. The objects do not stick together - they
“bounce”.
Given two masses, m1 and m2 at initial velocities v1 and v2
After they collide, they have velocities V1 and V2
Conservation of momentum says that
m1v1  m2v2  m1V1  m2V2
Solving for V1 and V2 (and using conservation of energy) gives

m1  m2
2m2
V1 
v1 
v2
m1  m2
m1  m2
2m1
m2  m1
V2 
v1 
v2
m1  m2
m1  m2
ISNS 3371 - Phenomena of Nature
m1  m2
2m2
V1 
v1 
v2
m1  m2
m1  m2
V2 
2m1
m  m1
v1  2
v2
m1  m2
m1  m2
Let v2 = 0 and m1 = 475 gr and m2 = 266 gr

m1  m2
2m2
V1 
v1 
v2
m1  m2
m1  m2
2m1
m2  m1
V2 
v1 
v2
m1  m2
m1  m2
475  266
209
v1 
v1
475  266
741
2(475)
950
V2 
v1 
v1
475  266
741
V1 

m1v1  m1V1  m2V2
209
950
475v1  475(
)v1  266(
)v1  475v1

741
741
ISNS 3371 - Phenomena of Nature
V1 
m1  m2
2m2
v1 
v2
m1  m2
m1  m2
V2 
2m1
m  m1
v1  2
v2
m1  m2
m1  m2
A heavy car collides with a
stationary lighter car
Let v2 = 0 and m1 = 475 gr and m2 = 266 gr


V1 
m1  m2
2m2
v1 
v2
m1  m2
m1  m2
V2 
2m1
m  m1
v1  2
v2
m1  m2
m1  m2
m1(the heavier car) is still moving
475  266
209
V1 
v1 
v1 after the collision, but slower.
475  266
741 m2(the lighter car) is moving after
2(475)
950 the collision with a velocity greater
V2 
v1 
v1 than the velocity of m before the
1
475  266
741
collision.
m1v1  m1V1  m2V2
209
950
)v1  266(
)v1  475v1 Momentum is conserved
475v1  475(
741
741
ISNS 3371 - Phenomena
Nature
m1 ofm
2
2m2
V1 
v1 
v2
m1  m2
m1  m2
V2 
2m1
m  m1
v1  2
v2
m1  m2
m1  m2
A light car collides with a
stationary heavier car
Let v2 = 0 and m1 = 266 gr and m2 = 475 gr


V1 
m1  m2
2m2
v1 
v2
m1  m2
m1  m2
V2 
2m1
m  m1
v1  2
v2
m1  m2
m1  m2
m1(the lighter car) is still moving
266  475
209 after the collision, but in the
V1 
v1 
v1
475  266
741 opposite direction.
m2(the heavier car) is moving after
2(266)
532
V2 
v1 
v1 the collision with a velocity smaller
475  266
741
than the velocity of m1 before the
collision.
m1v1  m1V1  m2V2
209
532

266v1  266(
)v1  475(
)v1  266v1 Momentum is conserved
741
741
ISNS 3371 - Phenomena of Nature
m1  m2
2m2
V1 
v1 
v2
m1  m2
m1  m2
V2 
Two moving cars with the
same mass collide
2m1
m  m1
v1  2
v2
m1  m2
m1  m2
m1 = m2

V1 
m1  m2
2m2
v1 
v2
m1  m2
m1  m2
V2 
2m1
m  m1
v1  2
v2
m1  m2
m1  m2
V1 
2m2
v2  v2
m1  m2
V2 
2m1
v1  v1
m1  m2

m1 and m2 simply switch velocities it doesn’t matter whether they are
going in the same or opposite
directions.
m1v1  m2v2  m1V1  m2V2  m1v2  m2v1

Momentum is conserved
ISNS 3371 - Phenomena of Nature
Elastic Collisions in 2 Dimensions - Pool
Remember: Momentum is a vector quantity - so the vector sum of the two balls’
momentum must equal the momentum of the que ball (the red ball) before
collision.
Note: the angle that the que ball and the object ball make after collision is always
a right angle (we will show this later).
ISNS 3371 - Phenomena of Nature
q  que ball
o  object ball
mq v q1  mq v q 2 cos( 2 )  mov o cos(1 )
mq  mo
so
v q1  v q 2 cos( 2 )  v o cos(1 )
ISNS 3371 - Phenomena of Nature
Inelestic Collisions
In an inelastic collision, the objects stick together after collision. Again,
momentum is conserved:
m1v1  m2v2  m1V1  m2V2
V1 = V2 because the objects are moving together and:

m1v1  m2v2  (m1  m2 )V

m1v1  m2v 2
V
m1  m2
ISNS 3371 - Phenomena of Nature
m1v1  m2v 2
V
m1  m2
Two cars of the same mass, one moving and the other stationary: v2 = 0
m1v1  m2v 2
m1
1

V

v1  v1
m1  m2
m1  m2
2

Velocity after
collision is
1/2 velocity of
m1 before
collision
Two cars of the same mass and velocities equal but in the opposite
direction: v2 = -v1
m1v1  m2v1
V
0
m1  m2
Velocity after the
collision is 0
ISNS 3371 - Phenomena of Nature
Angular Momentum
Momentum associated with rotational or orbital motion
angular momentum = mass x velocity x radius
ISNS 3371 - Phenomena of Nature
Torque and Conservation of Angular Momentum
Conservation of angular momentum - like conservation of momentum in the absence of a net torque (twisting force), the total angular
momentum of a system remains constant
Torque - twisting force
ISNS 3371 - Phenomena of Nature
A spinning skater speeds up as she brings her arms in and slows down
as she spreads her arms because of conservation of angular momentum