Center of mass 質心 and linear momentum 線性動量 (Chap. 9) Center of mass: 1.The center of mass follows a parabolic path, just like a single ball. 2.There are.

Center of mass 質心 and linear momentum 線性動量 (Chap. 9)

Center of mass:

1. The center of mass follows a parabolic path, just like a single ball.

2. There are also rotations around the center of mass

Definition of center of mass:

or

e.g. for two particles:

For a system of masses.

Extended to solid bodies:

For a body with uniform density, one can use the fact that:

Replace the summation by integration:

22

2

)2( RRm

Rm

P

S

1. One can consider only the three center of mass points: coms, comc, comP.

2. Note that because of symmetry, all three points must lay on the x-axis.

Use the definition of center of mass:

0

PS

PPSSC mm

commcommcom

Because comS=-R

R

m

Rmcom

P

SP

3

1

Newton’s second law for a system of particles

1. The internal forces (like forces in a collision) of system of particles do not change the motion of the center of mass.

2. While the external forces will govern the motion of center of mass according to the Newton’s second law.

This also applies to a solid body, like a bat, or a firework.

The forces of the explosion are internal to the system.

A ballet dance leaps across the stage in a grand jeté.

• center of mass still follow a parabolic path (no external force)

• but the dance lifts her hands and legs so that the center of mass moves up a little bit relative to her body.

• as a result, her head and torso keeps in the same height and giving an illusion of floating for a short moment.

Proof of second law on the center of mass

From the definition of center of mass:

First derivative:

second derivative:

Because of Newton’s third law, any internal force pairs will cancel with each other, so only external force, if any, left.

netF

Linear momentum

Definition:

Newton’s second law

For a system of particles:

Note that it is a vector.

Just like a particle M moving with velocity vcom.

Newton’s second law

If there is no external force, total momentum cannot be changed.

Collision 碰撞 and impulse 衝量

• When collision occurs, the objects will change shape

• note that the ball stops first and moves backwards, i.e. there is a momentum change, that gives a force.

• this applies to other collisions, like car collision.

• The collision time is short, so as the force.

Single collision

The change of momentum is:

Equal to the area under the force curve.

Average force: The longer the collision, the smaller the force.

衝量

)( ifif vvmppJ

skgmvvmJ ixfxx /910)( skgmvvmJ iyfyy /3500)(

skgmjiJ /ˆ3500ˆ910

Magnitude of impulse is: skgmJJJ yx /361622

So the average force on the driver is:

Nt

JFavg

51058.2

Or consider the driver’s average acceleration during the collision:

g

sm

kgNmFa aveave

329

/1022.3

80/10583.2/23

5

If the collision is increased by 10 times, the average force will reduced by 10 times and the driver may survive.

Conservation of linear momentum

Since we have

When there is no external force, i.e. F

net=0 0dt

dP

This applies to all situations, as long as there is no external force, even there is energy loss in the system.

Because there is no external force involves, we can apply the conservation of momentum:

)500(2.08.0

2.08.0

HSHS

MSHSi

vMMv

MvMvMv

hkmvHS /2200

Consider total py:

smM

Mv

vmvm

fB

fCCfBB

/6.940sin2.0

10sin53.0

10sin40sin

Consider total px:

10cos50cos fCCfBBfAA vmvmvm

• When a ball strikes the floor and rebounds, it appears that momentum is not conserved. Is this true? Explain why or why not.

• A sailor in a small boat has a fan and a sail available to him. Is there some way the fan can be used to propel the boat? If so, give details.

• A car collides head-on with a truck. Is the damage to the car greater if they have (a) the same kinetic energy, or (b) the same linear momentum?

• An unfortunate individual is at the center of a railway car that is at the edge of a cliff. Which way should the person walk to minimize the danger?

• (a) With the same muscular effort, why is a steel-headed hammer more effective in driving a nail than one with a rubber head. (b) Why is standing on the nail ineffective?

• A salesperson claims that an airbag absorbs the force of a collision. Explain why you agree or disagree with this statement.

• A ball is thrown at a block of wood. In which case will the ball exert the greater impulse: (a) when it sticks, or (b) when it rebounds with the same speed?

• Explain why a balloon flies off when air escapes from it.

• Cannons on ships are mounted on bases attached to springs. Exactly what purpose do the springs serve?

Nt

mvmvnFave

560

)500003.0)(2(100)(

Momentum and kinetic energy in collision

1. Energy may loss in a collision. If there is no loss in kinetic energy, it is an elastic collision 彈性碰撞 . Both energy and momentum is conserved.

2. If some kinetic energy is lost, it is an inelastic collision 非彈性碰撞 . But momentum is still conserved. The greatest loss occurs if the bodies stick together, in which case the collision is called a completely inelastic collision.

Inelastic collision

Completely inelastic collision

• when the objects stick to each other.

We know that the distance h has to do with the velocity of the block + bullet after the collision:

ghv

ghmMvmM

f

f

2

)()(2

1 2

Therefore by measuring h we can determine vf.

Using conservation of momentum:

m

ghMmv

vMmmv f

2

)(

Elastic collision

222

211

211

221111

2

1

2

1

2

1ffi

ffi

vmvmvm

vmvmvm

where 11 2ghv i

222

21

211

22111

)(

)(

ffi

ffi

vmvvm

vmvvm

ffi vvv 211

if

fifi

vmm

mmv

vvmvvm

121

211

112111 )()(