Linear Momentum. Lecture 15. Chapter 9. Physics I Department of Physics and Applied Physics

Transcription

1 Lecture 15 Chapter 9 Physics I Linear Momentum Course website: Lecture Capture:

2 Outline Chapter 9 Momentum Conservation of momentum Impulse Collisions (Elastic & Inelastic)

3 Exam II Info Exam II Mon Nov 4, 9:00-9:50am, OH 150. Exam II covers Chapters 5-8 Same format as Exam I Prior Examples of Exam II posted Ch. 5: Using Newton s Laws, Friction, Uniform Circular motion Ch. 6: Universal Law of Gravitation, Kepler s Laws Ch. 7: Work & Energy Ch. 8: Conservation of Energy Exam Review Session Thursday 6-8pm, Ball 210

4 Linear Momentum m Linear momentum is defined as the product of an object s mass and velocity: p v mv Momentum is a VECTOR! Units of momentum: kg m s The greater the linear momentum of a body, the greater its tendency to continue in motion. n iron shot (m larger) is harder to stop than a baseball (m small) of the same velocity.

5 Force and Momentum (Newton s 2 nd law) Let s rewrite Newton s 2 nd law in terms of momentum: F ma m d v d dt (m v) dt F dp dt d p dt The rate of change of momentum is equal to the net force So, a force is required to change momentum of an object.

6 How to find an average force? This can be used to get average force from momentum change: p m i v i F t net p f mv f F So, let s rewrite N. 2 nd law for an average force: dp dt F average p t p f p i t Let s look at an example:

7 Example: finding an average force The speed of a fastball is about 40 m/s, and the speed of the ball coming off of player s bat for a home run is about 54 m/s. The ball (0.145kg) is in contact with the bat for 1ms. What is the average Force exerted by the player? x F average p f t p i m( v f ( v t i )) Pay attention to directions!!!!!!!! ( 0.145kg)[54 F average s F average 136, 300N 40] m / s in the direction of v f or x

8 ConcepTest 1 Two Boxes/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 1 second. Which box has more momentum after the force acts? ) the heavier one B) the lighter one C) both the same F We know: p f av p t p f pi t In this case F and t are the same for both boxes! Both boxes will have the same final momentum. F av t p i 0 F light F heavy

9 ConcepTest 2 Two Boxes/velocity In the previous question, which box has the larger velocity after the force acts? ) the heavier one B) the lighter one C) both the same Mv h p f mv l Since M m, then v l v h

10 From Newton s 2 nd law: Conservation of Momentum dp F dt dp If Fexternal 0, then 0 thus p dt If no net external force acts on a system, its momentum is conserved. m v m v B B P initial m v m BvB P initial P final m, const v Pfinal m v ' m v m B vb m v ' m m B vb ' B v B m B vb '

11 Why is Momentum conserved? 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

12 Internal forces of an isolated system Internal forces of an isolated system. acts on B: B acts on : F F B B F B F B Forces equal and opposite (Newton s 3rd Law). F B F B During collision Thus, they all cancel each other. That s why they cannot ruin conservation of momentum. (It is not a proof)

13 Impulse During a collision, objects are deformed because of the large forces involved. How to relate those forces with a change in momentum? From Newton s 2 nd law: Integrate it: Before collision p i J F d p dt t f Fdt p dp f t i p f Force exerted on one of the balls fter collision p i Define Impulse as: J Fdt p f p i p t t f i Fdt dp J So, Impulse= area under F-vs-t curve p Impulse= change in momentum

14 Impulse/verage force The exact variation of F with time is very often not known. So, it is easier to find an average force. J the same J J F avg t t f Fdt t i p Having a certain p, a cat by bending its lags tries to increase t (impact time), so that an impact force would be reduced. F avg t p (intuitive knowledge of Physics ) How to avoid broken legs

15 Example: Tennis ball/impulse The force exerted by a tennis racket on the ball (mass 56 g) during a serve ( v i 0 )can be approximated by the F vs time plot below. What is the impulse on the ball? t f What is the speed of the serve? Force (kn) 2 0 Time (ms) 10 Fdt J p f p 0 i t i rea under force-time curve is an impulse: 2kN 2ms J rea 2( ) 4N s 2 J J p f mv f v f m 4 N s v f 71m/s kg

16 Different types of collisions Momentum is conserved in any two-body collision (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

17 1-D Elastic Collisions

18 m v Elastic Collision Math (1D) m v B B m v m B v B Conservation of momentum Conservation of mech. energy m v m B v B m v m B v 1 B 2 m v 2 1 m 2 Bv 2 B 1 m v m 2 2 B m v m v m B v B m B v B m v 2 m v 2 m B v 2 2 B m B v B m (v v ) m B ( v B v B ) Relative velocities switch signs in the collision v v B m (v 2 v 2 ) m B ( v 2 B v 2 B ) m (v v )(v v ) m B ( v B v B )( v B v B ) ( v v ) B v v v B v B v B

19 Elastic Collision Math (1D) 1 st m v m B v B m v m B v B 1 m 2 v 2 1 m 2 Bv 2 B 1 m v m 2 B v 2 B Conservation of momentum Conservation of mechanical energy 2 nd m v m B v B m v m B v v ( v v ) B B v B 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)

20 Example: Ballistic Pendulum device used to measure the speed of a bullet. m v o M M+m v 1 h

21 Example: Ballistic Pendulum (cont.) For Bullet mass 10 g Block mass is 3 kg Block swings up to a height of 5 cm the speed of a bullet v o 298 m s

22 Different Ballistic Pendulum device used to measure the speed of a bullet. Bullet mass 30 g Block mass is 5 kg Spring compresses by 12 cm Spring constant k = 300 N/m Bullet velocity before collision? mv 0 (M m)v 1 v 0 M m m v 1 m 1 2 (M m)v kx2 v 1 x M k M m v o m s v m s

23 Thank you See you on Monday