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?

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

(1)all of the system’s mass were concentrated there and (2) all external forces were applied there.

sys A BM m m

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

1 1 2 2

1 2

1( )eff

N Nsys

sys N

R m r m r m rM

M m m m

Locating a System's Center of Mass

The components of the center of mass of a system of particles are:

1 1 2 2

1 1 2 2

1 1 2 2

1 2

1( )

1( )

1( )

COM N Nsys

COM N Nsys

COM N Nsys

sys N

x m x m x m xM

y m y m y m yM

z m z m z m zM

M m m m

Velocity of center of mass

1 1 2 2

1 2

N Ncom

N

m v m v m vv

m m m

Acceleration of center of mass

1 1 2 2

1 2

N Ncom

N

m a m a m aa

m m m

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

netsysF

netsys sys comF M a

• 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.

coma

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 FA=6 N , FB=12 N , and FC=14 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

Newton’s 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 <J> : J F t

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

Linear Momentum-Impulse Theorem

f

i

t

f itdP P P

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 impulse–momentum 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 (m=0.14 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 vf=+58 m/s. (a) Determine the impulse applied to the ball by the bat. (b) Assuming that the time of contact is Δt=1.6 × 10–3 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 v0=–15 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 (vf=0 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.

sysnet

dPF

dt

1 2( ) ( )sys sysP t P t

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 that slide across the floor. An overhead view is shown in Fig. 9-14a. Piece C, with mass 0.30M, has final speed vfc=5.0m/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 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.

If the collision occurs in a very short time or external forces can be ignored, the momentum of system is conserved.

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 m1=30 g, is pulled to the left to height h1=8.0cm, and then released from rest. After swinging down, it undergoes an elastic collision with sphere 2, whose mass m2=75 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 M=0.63 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 h=1.8m. (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 then slide with locked wheels until the frictional force from the slick road (with a low μk of 0.13) stops them, at distances dA=8.2m and dB=6.1m . 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.)