Chapter 6: Gravity & Projectile Motion Sun at Center Orbits are Circular Tycho Brahe 1546-1601 Tycho was the greatest observational astronomer of his time. Tycho did not believe in the Copernican model because he didn’t believe that the Earth Moved. Kepler worked for Tycho as his mathematician. Kepler derived his laws of planetary motion from Tycho’s observational data. Orbits are Elliptical Kepler’s 3 Laws of Planetary Motion 1: The orbit of each planet about the sun is an ellipse with the sun at one focus. 2. Each planet moves so that it sweeps out equal areas in equal times. 22 Tr11 3. 22= Tr22 Isaac Newton (1642 -1727) UniversalUniversal LawLaw ofof GravityGravity Every particle in the Universe attracts every other particle with a force along a line joining them. The force is directly proportional to the product to their masses and inversely proportional to the square of the distance between them. M d m mM F ~ d 2 Gravitational Force is Universal The same force that makes the apple fall to Earth, causes the moon to fall around the Earth. UniversalUniversal LawLaw ofof GravityGravity Nm2 Gx= 6.67 10−11 kg 2 M d m GmM F = d 2 Measuring G: Cavendish mM GmM F ~ 2 F = d d 2 Nm2 Gx= 6.67 10−11 kg 2 G is the same everywhere in the Universe. Inverse Square Law GmM F = d 2 Gravitational Force How would the force of gravity and the acceleration due to gravity change as you fell through a hole in the Earth? Problem Calculate the acceleration of gravity acting on you at the surface of the Earth. Gmyou M E F = Fma= you d 2 GM a = E d 2 −11 2 2 24 ()6.673xNmkgxkg 10 /() 5.98 10 a = 8 2 ()3.84xm 10 ams= 9.81 / 2 Problem Calculate the force of gravity between the Earth and the Moon. Assume a circular orbit. GmM F = d 2 −11 2 2 24 22 (6.673xNmk 10 /g )( 6.00xk 10g )( 7.4xk 10 g ) F = 8 2 ()3.84xm 10 FxN= 2.01 1020 Problem Calculate the acceleration of the Earth due to the Earth-Moon gravitational interaction. F a = m 2.01x 1020 N a = 5.98x 1024 kg axms= 3.33 10−52 / Problem Calculate the acceleration of the Moon due to the Earth-Moon gravitational interaction. F a = m 2.01x 1020 N a = 7.35x 1022 kg axms= 2.73 10−32 / Problem The acceleration of gravity at the Moon due to the Earth is: axms= 2.73 10−32 / The acceleration of gravity at the Earth due to the moon is: axms= 3.33 10−52 / Why the difference? FORCE is the same. Acceleration is NOT!!! Force is not Acceleration! FFEarth on Moon= − Moon on Earth The forces are equal but the accelerations are not! Gravitational forces produce the tides and keep the same side of the Moon always facing the Earth in a Synchronous Rotation. The gravitational force between Earth and Moon results in tides Large spring tides occur when the Sun and Moon are aligned such that they BOTH pull on the Earth’s water. Smaller neap tides occur when the Sun and Moon are misaligned such that they pull on the Earth’s water in different directions. Earth’s Ocean Tides Neap Tides Spring Tides How many High and Low Tides per day? Earth- Moon System: Tidal Lock Moon pulls on Earth and causes tidal bulges in Oceans. Earth pulls on moon causing tidal bulge in Moon such that the CM is off from CG. Earth’s pull on the moon at its center of gravity produces a torque which rotates it. The moon is ‘tidally locked’ so that it rotates at the same rate at which it revolves, showing Earth only one face. Questions! Which exerts a greater gravitational force on the Earth, the moon or the Sun? Which has a greater effect on the Earth’s tides? Why? The Sun exerts a greater force on the Earth. It also causes tides, though it has about half the effect as the moon. Tides are due to differences in force on opposite sides of the Earth. The sun is so far away relative to the size of the Earth that the difference in force is not as significant as the difference due to the Moon since it is closer! Spring Spring Neap Bouncing laser beams off the Moon demonstrates that it slowly moving away from the Earth ~.25 cm/month Earth- Moon System is Changing! •Earth Rotation Slowing due to friction of ocean on bottom •.0023 s per century: 900 Million yrs ago, Earth day was 18 hrs! •Decrease of Earth’s spin angular momentum, increases the orbital angular momentum of the Moon! •Earth is slowing down and Moon is moving further away! Eventually the Earth's rotation period will be identical to the Moon's orbital period. This situation is called synchronous (1:1) rotation. In the distant future (many billions of years from now), the Earth will have a day which is 47 current days long, and the Moon will only be visible from one side of the Earth!! Projectile Motion Ignore Air Resistance! Most Important: X and Y components are INDEPENDENT of each other! Zero at the Top! Y component of velocity is zero at the top of the path in both cases! You know they have the y component of velocity is the same in both cases because they reached the same height! Projectile Motion Horizontal and vertical components are independent of each other! Gravity acts in the vertical direction but not in the horizontal direction!! Speed in vertical direction speeds up! Speed in horizontal direction stays the same! First the SIMPLE Case: Horizontal Launch (Ignore Air Resistance) The x-component doesn’t change (no acceleration in x-direction.) The y-component changes (a = -g.) Projectile Motion No Change ax = 0 Actual path is a vector sum of horizontal and vertical motions. change agy = Dropping From Moving Frame Plane and Package An airplane traveling at a constant speed and height drops a care package. Ignoring air resistance, at the moment the package hits the ground, where is it relative to the plane? a) Behind the plane. b) Under the plane. c) In front of the plane. Any object dropped from a plane has the same initial velocity as the plane! Projectile Motion Which hits the ground first? SAME! Both falling the same height! Horizontal speed doesn’t effect the time to hit the ground! If it takes 1 second to hit the ground, how tall is the tower? Projectile Motion dt==55(1)522 = m If it takes 1 second to hit the ground, how tall is the tower? Same rock, same speed, same angle. Which rock hits the water first? a) Rock 1 b) Rock 2 c) same Which rock hits the water with the greatest speed? a) Rock 1 b) Rock 2 c) same Same rock, same speed, same angle. Which rock hits the water first? a) Rock 1 b) Rock 2 c) same Which rock hits the water with the greatest speed? a) Rock 1 b) Rock 2 c) same Spatial Symmetry In G Field! Projectile Motion Launched at an Angle (Ignore Air Resistance) The x-component doesn’t change (no acceleration in x-direction.) The y-component changes (a = -g.) Shooting UP From Moving Frame Horizontal and vertical components are independent of each other! Horizontal component remains unchanged without air resistance. Only the vertical component changes! Projectiles Launched at an Angle: The simple case: Δy=0 Question gms=10 / 2 The ball is thrown horizontally at 20 m/s. About how long does it take to hit the ground? 0 1 2 2Δy Δ=y vt + gt ts==1 yi 2 g d How far does it travel in the horizontal direction? m Δ=x vt =20 1 s = 20 m xi s Question gms=10 / 2 The ball is thrown horizontally at 30 m/s. About how long does it take to hit the ground? 1 2 2Δy Δ=y vt + gt ts==1 yi 2 g Only Δy determines time! d How far does it travel in the horizontal direction? m Δ=x vt =30 1 s = 30 m xi s Question gms=10 / 2 The ball is thrown horizontally at 100 m/s. How long does it take to hit the ground? 1 Second!! How far does it travel in the horizontal direction? m Δ=x vt =100 1 s = 100 m xi s Curvature of Earth If you threw the ball at 8000 m/s off the surface of the Earth (and there were no buildings or mountains in the way) how far would it travel in the vertical and horizontal directions in 1 second? Curvature of Earth If you threw the ball at 8000 m/s off the surface of the Earth (and there were no buildings or mountains in the way) how far would it travel in the vertical and horizontal directions in 1 second? horizontal: Δ= x vx t =(8000 m / s)( 1 s) = 8000 m 1 2 vertical : Δ=ygtt22 =5 = 5() 1 s = 5 m 2 Curvature of Earth If you threw the ball at 8000 m/s off the surface of the Earth (and there were no buildings or mountains in the way) how far would it travel in the vertical and horizontal directions in 1 second? horizontal: Δ= x vx t =(8000 m / s)( 1 s) = 8000 m 1 2 vertical : Δ=ygtt22 =5 = 5() 1 s = 5 m 2 Curvature of Earth If you threw the ball at 8000 m/s off the surface of the Earth (and there were no buildings or mountains in the way) how far would it travel in the vertical and horizontal directions in 1 second? Does the ball ever hit the Earth???? Curvature of Earth Curvature of the Earth: Every 8000 m, the Earth curves by 5 meters! Orbital Velocity If you can throw a ball at 8000m/s, the Earth curves away from it so that the ball continually falls in free fall around the Earth – it is in orbit around the Earth! Ignoring air resistance.