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Chapter 9. Center of Mass and Linear Momentum

- 9.1. What is Physics?
- 9.2. The Center of Mass
- 9.3. Newton's Second Law for a System of

Particles - 9.4. Linear Momentum
- 9.5. The Linear Momentum of a System of

Particles - 9.6. Collision and Impulse
- 9.7. Conservation of Linear Momentum
- 9.8. Momentum and Kinetic Energy in

Collisions - 9.9. Inelastic Collisions in One Dimension
- 9.10. Elastic Collisions in One Dimension
- 9.11. Collisions in Two Dimensions

What is physics?

central axis.)

Defining the Position of a Complex Object

The effective position of the system is

- The effective position of a system of particles

is the point that moves as though - all of the systems mass were concentrated there

and - all external forces were applied there.

N particles system

- The effective position is also called as the

center of mass of a system. It represents the

average location for the total mass of a system

Locating a System's Center of Mass

The components of the center of mass of a system

of particles are

Velocity of center of mass

Acceleration of center of mass

EXAMPLE 1 Three Masses

- Three particles of masses mA 1.2 kg, mB 2.5

kg, and mC 3.4 kg form an equilateral triangle

of edge length a 140 cm. Where is the center of

mass of this three-particle system?

Solid Bodies

If objects have uniform density,

- For objects such as a golf club, the mass is

distributed symmetrically and the center-of-mass

point is located at the geometric center of the

objects.

Question

- Where would you expect the center of mass of a

doughnut to be located? Why?

Checkpoint 1

- The figure shows a uniform square plate from

which four identical squares at the corners will

be removed. (a) Where is the center of mass of

the plate originally? Where is it after the

removal of (b) square 1 (c) squares 1 and 2 (d)

squares 1 and 3 (e) squares 1, 2, and 3 (f) all

four squares? Answer in terms of quadrants, axes,

or points (without calculation, of course).

EXAMPLE 2 U-Shaped Object

- The U-shaped object pictured in Fig. has

outside dimensions of 100 mm on each side, and

each of its three sides is 20 mm wide. It was cut

from a uniform sheet of plastic 6.0 mm thick.

Locate the center of mass of this object.

Problem 3 Build your skill

- Figure 9-4a shows a uniform metal plate P of

radius 2R from which a disk of radius R has been

stamped out (removed) in an assembly line. Using

the x-y coordinate system shown, locate the

center of mass comP of the plate.

Newton's Laws for a System of Particles

- is the net force of all external

forces that act on the system. - Msys is the total mass of the system. We assume

that no mass enters or leaves the system as it

moves, so that M remains constant. The system is

said to be closed. - is the acceleration of the center of

mass of the system. Equation 9-14 gives no

information about the acceleration of any other

point of the system.

EXAMPLE 4 Center-of-Mass Acceleration

- The three particles in Fig. a are initially at

rest. Each experiences an external force due to

bodies outside the three-particle system. The

directions are indicated, and the magnitudes are

FA6 N , FB12 N , and FC14 N. What is the

magnitude of the acceleration of the center of

mass of the system, and in what direction does it

move?

Collisions and Explosions

- A COLLISION or EXPLOSION is an isolated event

in which two or more bodies exert relatively

strong forces on each other over a short time

compared to the period over which their motions

take place.

What is Properties of Collision?

- When objects collide or a large object explodes

into smaller fragments, the event can happen so

rapidly that it is impossible to keep track of

the interaction forces

Linear Momentum of a particle

- m is the mass of the particle
- is its instantaneous velocity

Newtons second law

- The rate of change of the momentum of a

particle is proportional to the net force acting

on the particle and is in the direction of that

force.

The Linear Momentum of a System of Particles

M is the mass of the system

Newton's Laws

- The sum of all external forces acting on all

the particles in the system is equal to the time

rate of change of the total momentum of the

system. That leaves us with the general

statement

Collision and Impulse

- Impulse

The average impulse ltJgt

- Impulse is a vector quantity
- It has the same direction as the force

Linear Momentum-Impulse Theorem

Check Your Understanding 1

- Suppose you are standing on the edge of a dock

and jump straight down. If you land on sand your

stopping time is much shorter than if you land on

water. Using the impulsemomentum theorem as a

guide, determine which one of the following

statements is correct. - a.In bringing you to a halt, the sand exerts a

greater impulse on you than does the water. - b.In bringing you to a halt, the sand and the

water exert the same impulse on you, but the sand

exerts a greater average force. - c.In bringing you to a halt, the sand and the

water exert the same impulse on you, but the sand

exerts a smaller average force.

Example 1 A Well-Hit Ball

- A baseball (m0.14 kg) has an initial velocity

of v0 38 m/s as it approaches a bat. We have

chosen the direction of approach as the negative

direction. The bat applies an average force that

is much larger than the weight of the ball, and

the ball departs from the bat with a final

velocity of vf58 m/s. (a) Determine the impulse

applied to the ball by the bat. (b) Assuming that

the time of contact is ?t1.6 103 s, find the

average force exerted on the ball by the bat.

Example 2 A Rain Storm

- During a storm, rain comes straight down with

a velocity of v015 m/s and hits the roof of a

car perpendicularly (see Figure ). The mass of

rain per second that strikes the car roof is

0.060 kg/s. Assuming that the rain comes to rest

upon striking the car (vf0 m/s), find the

average force exerted by the rain on the roof.

Conservation of Momentum

- If no net external force acts on a system of

particles, the total translational momentum of

the system cannot change.

Note If the component of the net external

force on a closed system is zero along an axis,

then the component of the linear momentum of the

system along that axis cannot change.

Conceptual Example 4 Is the Total Momentum

Conserved?

- Imagine two balls colliding on a billiard

table that is friction-free. Use the momentum

conservation principle in answering the following

questions. (a) Is the total momentum of the

two-ball system the same before and after the

collision? (b) Answer part (a) for a system that

contains only one of the two colliding balls.

Example 5

- Bullet and Two Blocks In Fig. a, a 3.40 g

bullet is fired horizontally at two blocks at

rest on a frictionless tabletop. The bullet

passes through the first block, with mass 1.20

kg, and embeds itself in the second, with mass

1.80 kg. Speeds of 0.630 m/s and 1.40 m/s,

respectively, are thereby given to the blocks

(Fig.b). Neglecting the mass removed from the

first block by the bullet, find (a) the speed of

the bullet immediately after it emerges from the

first block and (b) the bullet's original speed.

Example 7

- The drawing shows a collision between two

pucks on an air-hockey table. Puck A has a mass

of 0.025 kg and is moving along the x axis with a

velocity of 5.5 m/s. It makes a collision with

puck B, which has a mass of 0.050 kg and is

initially at rest. The collision is not head-on.

After the collision, the two pucks fly apart with

the angles shown in the drawing. Find the final

speed of (a) puck A and (b) puck B.

Sample Problem 9

- Two-dimensional explosion A firecracker placed

inside a coconut of mass M, initially at rest on

a frictionless floor, blows the coconut into

three pieces thatacross the floor. An

overhead view is shown in Fig. 9-14a. Piece C,

with mass 0.30M, has final speed vfc5.0m/s. (a)

What is the speed of piece B, with mass 0.20M?

(b) What is the speed of piece A?

Momentum and Kinetic Energy in Collisions

- If the collision occurs in a very short time

or external forces can be ignored, the momentum

of system is conserved.

- If the kinetic energy of the system is conserved,

such a collision is called an elastic collision. - If the kinetic energy of the system is not

conserved, such a collision is called an

inelastic collision. - The inelastic collision of two bodies always

involves a loss in the kinetic energy of the

system. The greatest loss occurs if the bodies

stick together, in which case the collision is

called a completely inelastic collision.

Velocity of the Center of Mass

- In a closed, isolated system, the velocity of

the center of mass of the system cannot be

changed by a collision because, with the system

isolated, there is no net external force to

change it.

Example of elastic collision

- Two metal spheres, suspended by vertical cords,

initially just touch, as shown in Fig. 9-22.

Sphere 1, with mass m130 g, is pulled to the

left to height h18.0cm, and then released from

rest. After swinging down, it undergoes an

elastic collision with sphere 2, whose mass m275

g. What is the velocity v1f of sphere 1 just

after the collision?

Example of elastic collision

- A small ball of mass m is aligned above a

larger ball of mass M0.63 kg (with a slight

separation, as with the baseball and basketball

of Fig. 9-70a), and the two are dropped

simultaneously from a height of h1.8m. (Assume

the radius of each ball is negligible relative to

h.) (a) If the larger ball rebounds elastically

from the floor and then the small ball rebounds

elastically from the larger ball, what value of m

results in the larger ball stopping when it

collides with the small ball? (b) What height

does the small ball then reach (Fig. 9-70b)?

Example of inelastic collision

- In the before part of Fig. 9-60, car A (mass

1100 kg) is stopped at a traffic light when it is

rear-ended by car B (mass 1400 kg). Both cars

thenwith locked wheels until the

frictional force from the slick road (with a low

µk of 0.13) stops them, at distances dA8.2m and

dB6.1m . What are the speeds of (a) car A and

(b) car B at the start of the sliding, just after

the collision? (c) Assuming that linear momentum

is conserved during the collision, find the speed

of car B just before the collision. (d) Explain

why this assumption may be invalid.

Example of completely inelastic collision

- A completely inelastic collision occurs

between two balls of wet putty that move directly

toward each other along a vertical axis. Just

before the collision, one ball, of mass 3.0 kg,

is moving upward at 20 m/s and the other ball, of

mass 2.0 kg, is moving downward at 12 m/s. How

high do the combined two balls of putty rise

above the collision point? (Neglect air drag.)

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