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Unformatted text preview: PHYS 172: Modern Mechanics Spring 2011 Lecture 5 – Momentum conservation, Fundamental Forces, gravity Read 3.1 – 3.2, 3.11 System consisting of two objects
system F1, surr F1,int
+ Momentum principle: p1 ∆p1 ≡ p f ,1 − pi ,1 = F1, surr + F1,int ∆t ∆p2 ≡ p f ,2 − pi ,2 = F2, surr + F2,int ∆t ( ) ( ) F2,int p f ,1 + p f ,2 − pi ,1 − pi ,2 = F1, surr + F2, surr + F1,int + F2,int ∆t ( ) p2
F2, surr ptotal , f − ptotal ,i Fnet , surr Clicker question: What is the last term in that equation? A) Zero B) Not zero, directed toward the larger object C) Not zero, directed toward the smaller object D) It is always positive E) It is always negative 1 System consisting of two objects
system F1, surr F1,int
+ Momentum principle: p1 ∆p1 ≡ p f ,1 − pi ,1 = F1, surr + F1,int ∆t ∆p2 ≡ p f ,2 − pi ,2 = F2, surr + F2,int ∆t ( ) ( ) F2,int p f ,1 + p f ,2 − pi ,1 − pi ,2 = F1, surr + F2, surr + F1,int + F2,int ∆t ( ) p2
F2, surr ptotal , f − ptotal ,i Fnet , surr 0 ∆ptotal ≡ ptotal , f − ptotal ,i = Fnet ∆t
Only from surrounding! System consisting of many objects Because all the forces inside the system come in pairs they cancel out. The only forces left over are forces from the surroundings! 2 System consisting of several objects
The Momentum Principle for a system: ∆ptotal ≡ ptotal , f − ptotal ,i = Fnet ∆t
Total momentum of the system: ptotal ≡ p1 + p2 + p3 + ...
The sum of all external forces due to surrounding system Conservation of momentum
F1,int p1 + ∆p1 ≡ p f ,1 − pi ,1 = F1,int ∆t ∆p2 ≡ p f ,2 − pi ,2 = F2,int ∆t F2 ,int p f ,1 + p f ,2 − pi ,1 − pi ,2 = F1,int + F2,int ∆t ( ) p2 ptotal , f − ptotal ,i 0 In the absence of external forces system ∆ptotal = 0 ∆p1 + ∆p2 = 0
F1 p1 Conservation of momentum F2 ∆psystem + ∆psurrounding = 0
p2 surrounding 3 Collisions: negligible external forces
p1 f p2 f
p1i
m1 1. Sticky ball Momentum conservation: p1,i + p2,i = p1, f + p2, f
Assume γ=1: m1v1i + m2 v2i = m1v f + m2 v f vf =
p2i
m2 m1v1i + m2 v2i m1 + m2 W hat if balls bounce? m1v1i + m2 v2i = m1v1 f + m2 v2 f
Two unknowns, one equation Clicker questions
A cannon ball is shot into air. note: ignore friction 1 2 Is cannon ball’s momentum conserved during flight? A)Yes B) No Is the total momentum of the cannon and the ball conserved during the flight? A)Yes B) No 3 Is the total momentum of the Earth, cannon and the ball conserved during the flight? Is the total velocity of the Earth, cannon and the ball conserved during the flight? A)Yes B) No A)Yes B) No 4 4 Clicker questions: truck and mosquito collide Which experiences a larger ∆v ? A) Mosquito B) Truck C) The same Which experiences a larger ∆p ? A) Mosquito B) Truck C) The same During such an interaction, the changes in momenta are equal and opposite, but not changes in velocity. The Four Fundamental Forces
“Composite” forces like the spring force, air drag, friction, etc. are combinations of these four fundamental forces 5 Newton’s Great Insight:
The force that attracts things toward the earth (e.g. a falling apple) is the same force that keeps planets orbiting about the sun Question: Why doesn’t the Moon fall into the Earth? The gravitational force law
m2 r2 −1 ˆ r2 −1
m1
r2
m2 Newton Cavendish
G = 6.7 ×10−11 N×m 2 kg 2 Fgrav on 2 by1 = −G m2 m1 r2−1
2 ˆ r2−1 Gravitational constant r2−1 ≡ r2 − r1 r1
m1 6 Features of gravitational force Fgrav on 2by1 = − G m2 m1 r2−1
2 ˆ r2 −1 gravity is always attractive Fgrav on 2by1 = − G m2 m1 r2−1
2 ˆ r2 −1 Fgrav on 2by1 = − G m2 m1 r2−1
2 ˆ r2 −1 gravity is an inverse square law the force depends upon the product of the masses Distance between two objects
Real objects: have size Point object: idealized object which has no size, all mass is in one point If distance between the two objects is >> than their size, can model the objects as pointmasses Special case: spherical objects (spherical symmetry) Uniformdensity spheres interact gravitationally in exactly the same way as if all their mass were concentrated at the center of the sphere. Can model as a point mass! 7 Clicker question A B Fgrav on 2 by1 = −G m2 m1 r2−1
2 ˆ r2−1 What is the distance between these two spheres to be used in gravitational law? Clicker question
The magnitude of gravitational force on a student standing on the surface of Earth is 800 N. 2R Q: what would be the magnitude of gravitational force if a student climbs on top of a ladder which has height that equals to the radius of the Earth? Fgrav on 2 by1 = −G
R A) 200 N B) 400 N C) 800 N D) 1600 N E) 3200 N m2 m1 r2−1
2 ˆ r2−1 W hat were the assumptions? 8 ...
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This note was uploaded on 04/23/2011 for the course PHYS 172 taught by Professor ? during the Spring '08 term at Purdue UniversityWest Lafayette.
 Spring '08
 ?
 Force, Gravity, Momentum

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