Momentum - Physics - Lecture Slides, Slides of Physics

Download Momentum - Physics - Lecture Slides and more Slides Physics in PDF only on Docsity!

MOMENTUM!

Momentum

Impulse

Conservation of Momentum in

1 Dimension

Conservation of Momentum in

2 Dimensions

Angular Momentum

Torque

Moment of Inertia

Momentum Defined

p = m v

p = momentum vector

m = mass

v = velocity vector

Momentum Examples

10 kg

3 m /s 10 kg

30 kg · m /s

Note: The momentum vector does not have to be drawn 10 times longer than the velocity vector, since only vectors of the same quantity can be compared in this way.

5 g

p = 45 kg · m /s

at 26º N of E

Equivalent Momenta

Bus: m = 9000 kg; v = 16 m /s

p = 1.44 ·10^5 kg · m /s

Train: m = 3.6 ·10^4 kg; v = 4 m /s p = 1.44 ·10^5 kg · m /s

Car: m = 1800 kg; v = 80 m /s p = 1.44 ·10^5 kg · m /s

continued on next slide

Impulse Defined

Impulse is defined as the product force acting on an

object and the time during which the force acts. The

symbol for impulse is J. So, by definition:

J = F t

Example: A 50 N force is applied to a 100 kg boulder

for 3 s. The impulse of this force is J = (50 N) (3 s)

= 150 N · s.

Note that we didn’t need to know the mass of the

object in the above example.

Impulse Units

J = F t shows why the SI unit for impulse is the Newton · second. There is no special name for this unit, but it is equivalent to a kg · m /s.

proof: 1 N · s = 1 (kg · m /s^2 ) (s) = 1 kg · m /s

Fnet = m a shows this is

equivalent to a newton.

Therefore, impulse and momentum have the same units, which leads to a useful theorem.

Imagine a car hitting a wall and coming to rest. The force on the car due to the wall is large (big F ), but that force only acts for a small amount of time (little t ). Now imagine the same car moving at the same speed but this time hitting a giant haystack and coming to rest. The force on the car is much smaller now (little F ), but it acts for a much longer time (big t ). In each case the impulse involved is the same since the change in momentum of the car is the same. Any net force, no matter how small, can bring an object to rest if it has enough time. A pole vaulter can fall from a great height without getting hurt because the mat applies a smaller force over a longer period of time than the ground alone would.

Stopping Time

F t = F t

Impulse - Momentum Example

A 1.3 kg ball is coming straight at a 75 kg soccer player at 13 m/s who kicks it in the exact opposite direction at 22 m/s with an average force of 1200 N. How long are his foot and the ball in contact?

answer: We’ll use F net t = p. Since the ball changes direction, p = m v = m ( v f - v 0 ) = 1.3 [22 - (-13)] = (1.3 kg) (35 m/s) = 45.5 kg · m /s. Thus, t = 45.5 / 1200 = 0.0379 s, which is just under 40 ms.

During this contact time the ball compresses substantially and then decompresses. This happens too quickly for us to see, though. This compression occurs in many cases, such as hitting a baseball or golf ball.

Conservation of Momentum in 1-D

Whenever two objects collide (or when they exert forces on each other without colliding, such as gravity) momentum of the system (both objects together) is conserved. This mean the total momentum of the objects is the same before and after the collision.

before: p = m 1 v 1 - m 2 v 2

after: p = - m 1 va + m 2 v b

m 1 m 2

v 1 v 2

(Choosing right as the +

direction, m 2 has - momentum.)

m 1 m 2

va v b

m 1 v 1 - m 2 v 2 = - m 1 va + m 2 v b

Directions after a collision

On the last slide the boxes were drawn going in the opposite direction after colliding. This isn’t always the case. For example, when a bat hits a ball, the ball changes direction, but the bat doesn’t. It doesn’t really matter, though, which way we draw the velocity vectors in “after” picture. If we solved the conservation of momentum equation (red box) for vb and got a negative answer, it would mean that m 2 was still moving to the left after the collision. As long as we interpret our answers correctly, it matters not how the velocity vectors are drawn.

m 1 v 1 - m 2 v 2 = - m 1 va + m 2 v b

m 1 m 2

v 1 v 2

m 1 m 2

va v b

Sample Problem 1 (cont.)

7 kg

v = 0

700 m/s

35 g

7 kg v =?

35 g 4 cm/s

p (^) before = 7 (0) + (0.035) (700)

= 24.5 kg · m /s

Let’s choose left to be the + direction & use conservation of momentum, converting all units to meters and kilograms.

p (^) after = 7 (0.04) + 0.035 v = 0.28 + 0.035 v

p (^) before = p (^) after 24.5 = 0.28 + 0.035 v v = 692 m/s

v came out positive. This means we chose the correct

direction of the bullet in the “after” picture.

Sample Problem 2

7 kg

v = 0

700 m/s

35 g

Same as the last problem except this time it’s a block of wood rather than butter, and the bullet does not pass all the way through it. How fast do they move together after impact?

v

7. 035 kg

(0.035) (700) = 7.035 v v = 3.48 m/s

Note: Once again we’re assuming a frictionless surface, otherwise there would be a frictional force on the wood in addition to that of the bullet, and the “system” would have to include the table as well. Docsity.com

Conservation of Momentum applies only in the absence of external forces!

In the first two sample problems, we dealt with a frictionless surface. We couldn’t simply conserve momentum if friction had been present because, as the proof on the last slide shows, there would be another force (friction) in addition to the contact forces. Friction wouldn’t cancel out, and it would be a net force on the system.

The only way to conserve momentum with an external force like friction is to make it internal by including the tabletop, floor, or the entire Earth as part of the system. For example, if a rubber ball hits a

brick wall, p for the ball is not conserved, neither is p for the ball-

wall system, since the wall is connected to the ground and subject to

force by it. However, p for the ball-Earth system is conserved!

Sample Problem 3

Earth M

apple

An apple is originally at rest and then dropped. After falling a short time, it’s moving pretty fast, say at a speed V. Obviously, momentum is not conserved for the apple, since it didn’t have any at first. How can this be?

m

F

F

answer: Gravity is an external force on the apple, so momentum for it alone is not conserved. To make gravity “internal,” we must define a system that includes the other object responsible for the gravitational force--Earth. The net force on the apple-Earth system is zero, and momentum is conserved for it. During the fall the Earth attains a very small speed v. So, by conservation of momentum:

V

v

m V = M v