Chapter 9 (Cont.) Linear Momentum and Collisionsnsmn1.uh.edu/hpeng5/Peng09b_LectureOutline.pdfQuestion 9.20 Center of Mass (1) X CM (2) a) higher b) lower c) at the same place d) there

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Chapter 9 (Cont.)

Linear Momentum and Collisions


Units of Chapter 9

• Linear Momentum

• Momentum and Newton’s Second Law

• Impulse

• Conservation of Linear Momentum

• Inelastic Collisions

• Elastic Collisions• Center of Mass


Review: 9-1 Linear Momentum

Momentum is a vector; its direction is the same as the direction of the velocity.


Review: 9-2 Momentum and Newton’s Second Law

Newton’s second law, Form I:

Form II:


Review: 9-3 Impulse

Impulse is a vector, in the same direction as the average force.


Review: 9-3 Impulse

We can rewrite

as

So we see that

The impulse is equal to the change in momentum.

(from Newton’s 2nd law)


Review: 9-4 Conservation of Linear Momentum

Internal Versus External Forces:

Internal forces act between objects within the system.

As with all forces, they occur in action-reaction pairs. As all pairs act between objects in the system, the internal forces always sum to zero:

Therefore, the net force acting on a system is the sum of the external forces acting on it.

net external force=0 conservation of momentum


Review: 9-5 Inelastic Collisions

Time of collision is short enough that external forces may be ignored

Inelastic collision: momentum is conserved but kinetic energy is not

Completely inelastic collision: objects stick together afterwards


9-6 Elastic CollisionsIn elastic collisions, both kinetic energy and momentum are conserved.

One-dimensional elastic collision:


9-6 Elastic CollisionsWe have two equations (conservation of momentum and conservation of kinetic energy) and two unknowns (the final speeds). Solving for the final speeds:

Q: what if m1>>m2? Or m2>>m1?

Question 9.10aQuestion 9.10a Elastic Collisions IElastic Collisions I

v 2v

1at rest

at rest

a) situation 1

b) situation 2

c) both the same

Consider two elastic collisions: 1) a golf ball with speed v hits a

stationary bowling ball head-on. 2) a bowling ball with speed v

hits a stationary golf ball head-on. In which case does the golf ball have the greater speed after the collision?

Remember that the magnitude of the relative velocity has to be equal before and after the collision!

Question 9.10aQuestion 9.10a Elastic Collisions IElastic Collisions I

v1

In case 1 the bowling ball will almost remain at rest, and the golf ball will bounce back with speed close to v.

v 22v

In case 2 the bowling ball will keep going with speed close to v, hence the golf ballwill rebound with speed close to 2v.

a) situation 1

b) situation 2

c) both the same

Consider two elastic collisions: 1) a golf ball with speed v hits a

stationary bowling ball head-on. 2) a bowling ball with speed v

hits a stationary golf ball head-on. In which case does the golf ball have the greater speed after the collision?


9-6 Elastic CollisionsTwo-dimensional collisions can only be solved if some of the final information is known, such as the final velocity of one object:



(Ki=Kf)

(Py=0)


9-7 Center of Mass

The center of mass of a system is the point where the system can be balanced in a uniform gravitational field.

Q: Where do you draw the gravitational force for an object in a free-body diagram?


9-7 Center of Mass

For two objects:

The center of mass is closer to the more massive object.

Question 9.20 Center of Mass

(1)

XCM

(2)

a) higherb) lowerc) at the same placed) there is no definable

CM in this case

The disk shown below in (1) clearly has its center of mass at the center.Suppose the disk is cut in half and the pieces arranged as shown in (2).Where is the center of mass of (2) ascompared to (1) ?

Question 9.20 Center of Mass

(1)

XCM

(2)The CM of each half is closer to the top of the semicirclethan the bottom. The CM of the whole system is located at the midpoint of the two semicircle CMs, which ishigher than the yellow line.

a) higherb) lowerc) at the same placed) there is no definable

CM in this case

CM

The disk shown below in (1) clearly has its center of mass at the center.Suppose the disk is cut in half and the pieces arranged as shown in (2).Where is the center of mass of (2) ascompared to (1) ?


9-7 Center of Mass

The center of mass need not be within the object:


Center of mass for a system of many objects:



(b)

(a)


9-7 Center of MassMotion of the center of mass:


9-7 Center of Mass

The total mass multiplied by the acceleration of the center of mass is equal to the net external force:

The center of mass accelerates just as though it were a point particle of mass Macted on by

Two equal-mass particles (A and B) are located at some distance from each other. Particle A is held stationary while B is moved away at speed v. What happens to the center of mass of the two-particle system?

a) it does not move

b) it moves away from A with speed v

c) it moves toward A with speed v

d) it moves away from A with speed ½v

e) it moves toward A with speed ½v

Question 9.19Question 9.19 Motion of CMMotion of CM

Two equal-mass particles (A and B) are located at some distance from each other. Particle A is held stationary while B is moved away at speed v. What happens to the center of mass of the two-particle system?

a) it does not move

b) it moves away from A with speed v

c) it moves toward A with speed v

d) it moves away from A with speed ½v

e) it moves toward A with speed ½v

Let’s say that A is at the origin (x = 0) and B is at some position x. Then the center of mass is at x/2 because A and B have the same mass. If v = Δx/Δttells us how fast the position of B is changing, then the position of the center of mass must be changing like Δ(x/2)/Δt, which is simply v.

Question 9.19Question 9.19 Motion of CMMotion of CM

12



(a)

(b)(conservation of momentum)


Summary of Chapter 9

•Elastic collision: kinetic energy is conserved

• Center of mass:


Summary of Chapter 9

• Center of mass:

• Motion of center of mass:

Documents

Chapter 9 (Cont.) Linear Momentum and Collisionsnsmn1.uh.edu/hpeng5/Peng09b_LectureOutline.pdfQuestion 9.20 Center of Mass (1) X CM (2) a) higher b) lower c) at the same place d) there