CIRCULAR MOTION & GRAVITATION

Circular Motion - velocity Direction of velocity Velocity is not constant. Why? T =

Circular Motion – Force & Acceleration A ball attached to string is whirled in horizontal circle by hand. What force is responsible for ball changing direction? What is direction of force on ball? What would occur to ball if force vanished?

Circular Motion – Force & Acceleration If force on ball is directed INWARD to prevent ball from flying outward (inertia), then net force This force A net force produces an acceleration.

Centripetal Force causes objects to navigate a circle Centripetal Force causes objects to navigate a circle. Objects do not move towards center because why? Objects in circular motion at constant speed are not balanced. Why?

Centrifugal Force Centrifugal = center-fleeing

Centripetal Force eqns

Example1 A washing machine drum makes 5 rotations per second during the spin cycle. The inside drum has a radius of 25cm. a) Determine speed of the drum. b) What acts as centripetal force on clothes? c) Determine the normal force on a 1.0kg pair of jeans in the washing machine during this rotation.

Example 2: A 1,200 kg car rounds a corner of radius r = 45.0 m. If the coefficient of friction between the tires and the road is ms = 0.82, what is the maximum speed the car can have on the curve without skidding?

Example3 An amusement park ride consists of a rotating circular platform 8.00m in diameter from which 10.0kg seats are suspended at the end of 2.50m light chains.  a) When the system rotates, the chains make an angle of 28.0o with the vertical. What is the speed of each seat? b) Find the tension in the chain when a 40.0 kg child is riding in a seat.

Example4 An early major objection to the idea that the Earth is spinning on its axis was that Earth would turn so fast at the equator that people would be thrown off into space.  a) Show the error in this logic by solving for the NET inward force necessary to keep a 97.0 kg person standing on the equator with speed 444m/s and Rearth = 6400km.  b) What force(s) are responsible for composing the centripetal force on person?  c) Determine the length of day in hours so that a person would just be ‘weightless’ (scale can’t push on you).

Vertical Circles A rollercoaster executes a loop moving at 30m/s at the bottom and 20m/s at the top. The radius of loop is 10m. Negligible friction. a) What does a scale read on 50kg passenger at bottom of loop? b) What is the slowest speed coaster can go at top of loop so as not to fall?

Example 2: Tarzan, whose mass is 85 kg, tries to cross a river by swinging from a vine. The vine is 10.0 m long, and his speed at the bottom of the swing (as he just clears the water) is 8.00m/s. Tarzan doesn't know that the vine has a breaking strength of 1000 N. Does he make it safely across the water? Water & board demo

Example 3 A fighter jet is in a vertical dive and pulls up into a vertical loop. The speed of the plane is 230m/s. a) What provides inward force on pilot? On plane? b) What is the minimum radius of the loop so that the pilot never feels more than 3x his weight?

A car rounds a curve at angle θ. The radius of the curve is R A car rounds a curve at angle θ. The radius of the curve is R. Assuming negligible friction, determine the expression for the speed it could negotiate the curve without sliding.

Newton’s Law of Universal Gravitation Newton realized that ALL objects Figure 6-1. Caption: Anywhere on Earth, whether in Alaska, Australia, or Peru, the force of gravity acts downward toward the center of the Earth.

Example1 Two bowling balls each have a mass of 6.8 kg.  They are located next to one another with their centers 21.8 cm apart.  What amount of gravitational force does one exert on the other?  Example2 The Earth exerts a pull of gravity on the moon as does the less massive moon on the Earth. Which planetary body pulls harder?

The acceleration due to gravity doesn’t change that much for altitudes that are much less than radius of Earth.

Acceleration due to gravity (g) drops off as 1/r2 Once the altitude becomes comparable to the radius of the Earth (RE) , the decrease in the acceleration of gravity is much larger: Acceleration due to gravity (g) drops off as 1/r2

Example3: A 50kg astronaut climbs a ladder that is 6400km high Example3: A 50kg astronaut climbs a ladder that is 6400km high. He stands on a scale on the top step. a) Determine scale reading on her at that point if the mass of earth is 6.0x1024kg and RE = 6.4x106m. Ignore rotation of Earth b) What would be the force of gravity on her if she stepped off the ladder? c) Determine the acceleration due to gravity (‘g’) at this point.

Example 4: The planet Saturn has a mass of 5 Example 4: The planet Saturn has a mass of 5.68x1026 kg and a radius of 5.85x107m. Determine the time it takes a 1.0kg object to fall 10.0m from rest near the surface of the planet.

EXAMPLE 5 Imagine an astronaut in between the Earth and moon. Would a 70kg astronaut have to be placed closer to the earth or the moon (7.3x1022kg) so that the net gravitational force on them is zero? Determine this distance if the avg distance between moon and earth is 384,403km (center to center).

Force of gravity INSIDE earth What happens to force of gravity when you enter the earth on way towards center? What would force of gravity be like halfway between center and surface?

Tall buildings, but not in a single bound If you walked into the lobby of a skyscraper, what would happen to your weight or force of gravity, technically speaking?

Describe what would occur to orbit of Earth if our Sun were to collapse into a very tiny volume (while still maintaining same mass).

Tides Tides are a result of

Differential Forces due to the Moon

Full cycle of 2 highs & 2 lows is about 25hrs, not 24hrs It takes more than 24hours for us to rotate through 2 tidal bulges of 2 high and 2 lows. The reason is because by the time we make one full rotation on Earth, the moon has moved from its position it was at 24hrs prior. As a result, in order to get back to the high tide bulge, we have to rotate a little bit further than just one rotation. (see figure below)

We need about our hour's worth of extra rotation to accomplish this We need about our hour's worth of extra rotation to accomplish this. Therefore, there are typically two high tides every 25 hours, rather than every 24 hours. This is why the times of high tides are about an hour later each day.

Tides due to both Sun and Moon – Spring Tides and Neap Tides Spring tides are much larger…this is when both sun and moon are lined up making attractive pull greater Neap tides are smaller…this is when sun and moon are pulling earth at 90o with respect to one another

The effect of the Sun is not as great as the Moon’s effect Sun’s pulls on earth 180x more than moon does but only has ½ the effect. So, why does moon have more effect on tides? No tides in a glass of water…Why? 31

Kepler's 1st Law: The Law of Elliptical Orbits Each planet travels in an elliptical orbit with the sun at one focus. When the planet is located at point A it is at the aphelion position or apogee. (farthest) When the planet is located at point P it is at the perihelion position or perigee. (closest)

Kepler’s 2nd Law: The Law of Equal Areas A line from the planet to the sun sweeps out equal areas of space in equal intervals of time. At the aphelion, the position farthest from the sun along the planet’s orbital path, the planet’s speed is minimal. At the perihelion, the position closest to the sun along the planet’s orbital path, the planet’s speed is maximum.

Kepler’s 3rd Law: The Law of Periods The square of a planet’s orbital period is directly proportional to the cube if its average distance from the sun. We make an assumption that the orbits are circular since they are only SLIGHTLY elliptical… FC= Fg

Newton’s Mountain Geometric curvature of Earth 35

Astronauts are not weightless! Newton reasoned that if you fired a projectile fast enough horizontally, it would continually fall from its straight-line path but never hit the earth…”falling around” or orbiting the earth. Astronauts are not weightless!

Circular Orbits

Satellite Example1 The Magellan space probe was placed into orbit around the planet Venus in 1992.  The probe was to orbit at an altitude of 4370km.  The mass of Venus is 4.87x1024 kg and its radius is 6100 km.  a) What speed is required to maintain this orbit?  b) What was the orbital period in hours?

Example 2 On July 19, 1969, Apollo 11’s orbit around the Moon (7.36x1022kg) was at an altitude of 111km.  a) How many minutes did it take to orbit once? (Rmoon=1.74x106m)   b) At what speed did it orbit the Moon?

Example 3: Determine the height above the earth of a geosynchronous satellite.