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

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

2. Definition of center of mass: For a system of masses. or e.g. for two particles:

3. Extended to solid bodies: Replace the summation by integration: For a body with uniform density, one can use the fact that:

4. One can consider only the three center of mass points: coms, comc, comP. • Note that because of symmetry, all three points must lay on the x-axis. Use the definition of center of mass: Because comS=-R

5. Newton’s second law for a system of particles • The internal forces (like forces in a collision) of system of particles do not change the motion of the center of mass. • 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.

6. 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.

7. 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.

12. Linear momentum Note that it is a vector. Definition: Newton’s second law For a system of particles: Just like a particle M moving with velocity vcom. Newton’s second law If there is no external force, total momentum cannot be changed.

13. 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.

14. Single collision • The collision time is short, so as the force. The change of momentum is: Equal to the area under the force curve. 衝量 Average force: The longer the collision, the smaller the force.

15. Or consider the driver’s average acceleration during the collision: Magnitude of impulse is: If the collision is increased by 10 times, the average force will reduced by 10 times and the driver may survive. So the average force on the driver is:

16. Conservation of linear momentum Since we have When there is no external force, i.e. Fnet=0 This applies to all situations, as long as there is no external force, even there is energy loss in the system.

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

18. Consider total py: Consider total px:

19. 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?

20. (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?

22. Momentum and kinetic energy in collision • 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. • 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

23. Completely inelastic collision • when the objects stick to each other.

24. We know that the distance h has to do with the velocity of the block + bullet after the collision: Using conservation of momentum: Therefore by measuring h we can determine vf.

25. Elastic collision where