Lecture 16. Conservation of Linear Momentum

Transcription

1 Lecture 16 Chapter 11 Conservation of Linear Momentum nother conservation? I like conservations! Course website: Department of Physics and pplied Physics

2 IN THIS CHPTER, you will learn to use the concepts of linear momentum. Today we are going to discuss: Chapter 11: Conservation of Linear Momentum: Section D Collisions (Elastic/Inelastic): Section 11.3 Department of Physics and pplied Physics

3 Example How to avoid broken legs for a cat? F avg t F, that is what can break cat s bones and the cat feels that and tries to reduce F as much as it can. p Since the cat falls from a certain height, p(initial)=δp is given and the cat cannot do anything about that during the collision. y bending legs and increasing an impact time, Δt. Having a certain p, a cat by bending its lags tries to increase t (impact time), so that an impact force would be reduced. (intuitive knowledge of Physics ) Department of Physics and pplied Physics Initial linear momentum Final linear momentum 0 F avg Collision with a floor

4 ConcepTest Two oxes/momentum Two boxes, one heavier than the other, are initially at rest on a horizontal frictionless surface. The same constant force F acts on each one for exactly 1sec. Which box has more linear momentum after the force acts? ) the heavier one ) the lighter one C) both the same We know: p i 0 F av p t p f p f pi t F av t In this case F and t are the same for both boxes! oth boxes will have the same final momentum. F light F heavy

5 ConcepTest Two oxes/velocity (In the previous situation) Which box has the larger velocity after the force acts? ) the heavier one ) the lighter one C) both the same Mv h p f mv l Since M m, then v l v h

6 Conservation of Linear Momentum Department of Physics and pplied Physics

7 Conservation of linear momentum (derivation) Let s consider some cruelty Consider two colliding particles (a ball and a bird). ssume: there are external and internal forces Internal forces obey N.3 rd law: F F From modified Newton s 2 nd law during the collision: dp ext F F dt dd these two eq-ns dp ext ext F F F dt F d( p p) ext ext F F F ( F) dt Thus, the internal forces cancel each other!!! Let s introduce a total linear momentum of two objects Thus, we don t need to know the internal forces to solve collision problems. Department of Physics and pplied Physics ext ext F F dp dt P F F p p ext ext F ext F Cont. collision ext F F

8 Conservation of Linear Momentum Let s look at a special case of zero net external forces. If F ext 0 dp F dt dp, then 0, dt P const thus If no net external force acts on a system, its momentum is conserved. gain, the total momentum remains constant, regardless of whatever interactions (internal forces) are going on inside the system ext Department of Physics and pplied Physics

9 Why is momentum conserved during collision? F N1 F N 2 m g m g 1 2 mg s are canceled by normal forces, so net external force is zero, and the momentum is conserved m g 1 F N 2 m g 2 The net external force is m 1 g, and the momentum is NOT conserved Isolated system is a system on which no external forces act. There are only internal forces acting between objects. this system (two balls) is isolated this system (two balls) is NOT isolated The total momentum of an isolated system of objects is conserved Department of Physics and pplied Physics

10 Example Rolling away girl running with speed of 4.0 m/s jumps on a stationary cart. The girl has a mass of 30 kg and the cart s mass is 10 kg. What is the cart s speed just after the girl jumps on? m 30kg v v 0 m 10kg v v? Initial (before collision) pply Conservation of Momentum: P initial P final Final (after collision) (Since this system is isolated) p initial mv m P initial P final v m v m v p final ' ( m m ) v m v ' m v ' ' ( m m ) v Solve for v : v mv m m 30kg 4.0m / s 30kg 10kg 3m / s Department of Physics and pplied Physics

11 Different types of collisions Linear momentum is conserved in both of these two-body collisions (since there is no net external force) Is mechanical energy conserved in these collisions? Metal balls get deformed and restored Mech. energy is conserved Elastic collision Cars get deformed and not restored. Some Mech. Energy is spent on deformation. Mech. energy is NOT conserved Inelastic collision Department of Physics and pplied Physics

12 1-D Elastic Collisions Department of Physics and pplied Physics

13 m v Elastic Head-on Collision Math (1D) m v m v m v Conservation of momentum Conservation of mech. energy m v m v m v m v 1 2 m v 2 1 m 2 v 2 1 m v m 2 2 m v m v m v m v m v 2 m v 2 m v 2 2 m v m (v v ) m ( v v ) m (v 2 v 2 ) m ( v 2 v 2 ) m (v v )(v v ) m ( v v )( v v ) v v v v v m v m v m v m Department of Physics and pplied Physics v v v ( v v ) Relative velocities switch signs in the collision

14 Elastic Collision Math (1D) 1 st m v m v m v m v 1 m v m v m v m 2 v 2 Conservation of momentum Conservation of mechanical energy 2 nd m v m v m v m v v ( v v ) v Conservation of momentum Conservation of mechanical energy So, instead of the 1 st set of crazy equations, we can use the 2 nd one which is easier (both are linear). It is only true for an elastic head-on collision Department of Physics and pplied Physics

15 Example allistic Pendulum device used to measure the speed of a bullet. the speed of a bullet Department of Physics and pplied Physics For ullet mass 10 g lock mass is 3 kg lock swings up to a height of 5 cm v o 298 m s

16 Department of Physics and pplied Physics Thank you