Physics 106 Lecture 9 Newton’s Law of Gravitation SJ 7th Ed.: Chap 13.1 to 2, 13.4 to 5
• Historical overview • N’Newton’s inverse-square law of gravi iitation Force Gravitational acceleration “g” • Superposition • Gravitation near the Earth’s surface • Gravitation inside the Earth (concentric shells) • Gravitational potential energy Related to the force by integration A conservative force means it is path independent Escape velocity
Example A geosynchronous satellite circles the earth once every 24 hours. If the mass of the earth is 5.98x10^24 kg; and the radius of the earth is 6.37x10^6 m., how far above the surface of the earth does a geosynchronous satellite orbit the earth? G=6.67x10-11 Nm2/kg2
The speed of a geosynchronous satellite is ______.
1 Goal Gravitational potential energy for universal gravitational force
Gravitational Potential Energy WUgravity= −Δ gravity Near surface of Earth: Gravitational force of magnitude of mg, pointing down (constant force) Æ U = mgh Generally, gravit. potential energy for a system of m1 & m2 G Gmm12 mm F = Attractive force Ur()=− G12 12 r 2 g 12 12 r12
Zero potential energy is chosen for infinite distance between m1 and m2. Urg ()012 = ∞= Æ Gravitational potential energy is always negative.
2 mm12 Urg ()12 =− G r12
r r Ug=0 1
U(r1)
Gmm U =− 12 g r
Mechanical energy 11 mM EKUrmvMVG=+ ( ) =22 + − mech 22 r m V r v
M
E_mech is conserved, if gravity is the only force that is doing work. 1 2 MV is almost unchanged. If M >>> m, 2 1 2 mM ÆWe can define EKUrmvG=+ ( ) = − mech 2 r
3 Example: A stone is thrown vertically up at certain speed from the surface of the Moon by Superman. Assuming no air resistance and no gravitational force except the one from the Moon, what would be the minimum speed to escape the Moon’s gravitational pull? Radius of the Moon is 1700 km and the mass is 7. 3x 10^ 22 kg
Stone
Moon
Note: Such velocity is called “escape velocity”.
Mechanical energy, continued
• As a particle moves from A to B, its gravitational potential energy changes by ΔU
• But the mechanical energy remains constant, as lthfitilong as no other force is acting
Emech = K + U(r)
E • The mechanical energy may be mech2 positive, negative, or zero
Emech1
4 Conservation of mechanical energy with gravitation
• Emech determines whether motion is bound, free, or at escape threshold
Emech is constant FREE r Emech = K + Ug(r) Ug=0 r1 r2
Gmem Ug(r) = − always negative E r mech 1 Turning KE For E < 0, particle is bound and point r2 mech U(r1) cannot escape. It cannot move beyond KE2 = 0 a turning point (e.g., r ) 2 BOUND For Emech > 0, particle is free. It can reach r = infinity and still have some KE
E Gm left U = − e g r Emech = 0 is t he escape condi ti on. A particle at any location r would need at
least KE = -Ug(r) to move off to the right and never return.
Escape speed formula – derivation and example
Escape condition for object of mass m from the surface of earth: 1 2 Gmem Emech ≡ K + Ug = 0 = mvesc - 2 re rradius= of earth me = mass of earth e
The mass m cancels: 1 Gm 2 e 2Gme 0 = vesc - v = = 2gr 2 r esc e e re m NtNote: e GGg2 = re
5 Example: Find the escape speed from the Earth’s surface
2 g = 9.8 m/s re = 6370 km
6 vmsesc ==≈ 2*9.8*6.370*10 11,100 / 25,000 mile/h
Example: Jupiter has 300 times the Earth’s mass and 10 times the Earth’s diameter. How does the escape velocity for Jupiter compare to that for the Earth?
6 Example: Breaking out of a stable orbit
Find the orbital speed of a satellite in a circular orbit at an altitude of 2 Earth radii Me
R re h How much additional speed (ΔV) does it need to break orbit and escape from the Earth’s gravitational pull? G = 6.67x10-11 (SI) 24 Me= 5.98x10 kg 6 re= 6.37x10 m
Gravitational Potential Energy of a system of many particles
Utotal = ∑ Uij(rij) all pairs sum over all possible pairings
Example 3 possible pairs
⎧ ⎫ ⎪Gm1m2 Gm1m3 Gm2m3 ⎪ Utotal = U12 + U13 + U23 = −⎨ + + ⎬ ⎩⎪ r12 r13 r23 ⎭⎪
7