Circular Motion ( ΣF = ma for circles ) Circular motion involves Newton’s Laws applied to objects that rotate or revolve about a fixed radius. This motion can be horizontal circles (washing machine), vertical circles (ferris wheel), partial circles (speed bump), angled circles (banked curve), or satellites about a planetary body.

Circular Motion - velocity

Circular Motion – Force & Acceleration A ball attached to string is whirled in horizontal circle by hand. What force is responsible for ball changing direction? direction What is direction of this 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 is inward or CENTER-SEEKING.

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

Centrifugal Force Centrifugal = center-fleeing

Sources of Centripetal Force You must identify the centripetal force. It may be provided by the tension in a string, the normal force, friction, among other sources.

Circular eqns

It takes a 615 kg car 14.3 s to travel at a uniform speed around a circular racetrack of 50.0 m radius. Example 1 a) What is the speed of the car? b) What is the acceleration of the car? c) What amount of inward force must the track exert on the tires to keep the car moving in the circle?

example2 A washing machine drum makes 5 rotations per second during the spin cycle. The inside drum has a radius of 25cm. a) Determine the period of rotation b) Determine speed of the drum. c) Determine the centripetal acceleration of the clothes inside d) What force provides inward force to clothes?

Example3: Rounding a Corner 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  s = 0.82, what is the maximum speed the car can have on the curve without skidding?

A toy airplane (0.040kg) is suspended by a string and flies in a circle. The diameter of the circle is 1.5m. The string makes a 20 o angle with the vertical. a)Find the tension in the string b) Find the speed of the plane Example4:

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

Example 2 a) What is the normal force exerted by seat on 80.0 kg passenger at the bottom of the dip if you are moving at 17m/s? Radius of dip is 65m. b) How fast can car go over a hill of same radius so that it doesn’t lose contact with road? (critical velocity)

example3 A stunt plane is in a vertical dive and pulls up into a vertical loop. The speed of the plane is 230m/s. What is the minimum radius of the loop so that the pilot never feels more than 3x his weight?

example4 A bucket is whirled in a vertical circle at a speed of 5m/s. Determine the tension in the rope if the mass of the bucket is 2.3kg.

Banked Curves Banked curves provide extra support towards the center. It allows moving objects to navigate a turn at a greater rate of speed. The support comes from a component of the normal force

UNIVERSAL GRAVITATION

Newton’s Law of Universal Gravitation

The acceleration of gravity decreases with altitude…why? In diagram below, it changes very gradually since altitude is minor compared to R E

Once the altitude becomes comparable to the radius of the Earth (R E ), the decrease in the acceleration of gravity is much larger: Acceleration due to gravity drops off as 1/d 2

Example1 a) Determine the force of gravity between the center of Earth (6.0x10 24 kg) and a 100kg person on the surface where the radius is 6400km. b) Compare this to F g = mg Determine the force of gravity between 2 apples that are 0.11kg each and separated by 0.50m. Example2

An 50kg astronaut climbs a ladder that is 6400km high. He stands on a scale on the top step. Example3 a) Determine his weight at that point if the mass of earth is 6.0x10 24 kg b) What would be the force of gravity on him if he stepped off the ladder? c) Determine the acceleration due to gravity (‘g’) at this point.

The planet Venus has a mass of 4.86x10 24 kg and a diameter of 12,102km. Example4 b) Determine how long it would take a 5.0kg object to fall 3.0m when dropped from rest near the surface of the planet. a) Determine the acceleration due to gravity on Venus

Force of gravity inside the Earth What would force of gravity be like if you were at exact center of planet? Your acceleration? What would force of gravity be like halfway between center and surface?

What happens to a person’s weight if the planet were to shrink? Mass of planet is still same Consider a person standing on the surface of the planet What happens to a person’s weight if the planet were to increase in mass? By what factor is a person’s weight changed if the planet were to increase its radius by a factor of 2? By what factor is a person’s weight changed if the planet were to increase its mass by a factor of 3 and its radius reduces by a factor of 2?

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?

What happens to force of gravity from the sun on the Earth if the sun were to shrink down to the size of a pea while still keeping all of its mass?

TIDES

Tides Tides are a result of differential gravity forces exerted on near side of planet vs far side of planet

Differential Forces due to the Moon Therefore, water at (A) is pulled away from earth (B). Earth is pulled away from water at (C) Near side of earth is pulled harder than center and center is pulled harder than far side by the moon.

The effect of the Sun is not as great as the Moon’s effect No tides in a glass of water…Why? 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?

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 P it is at the perihelion position or perigee. (closest) When the planet is located at point A it is at the aphelion position or apogee. (farthest)

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 perihelion, the position closest to the sun along the planet’s orbital path, the planet’s speed is maximal. At the aphelion, the position farthest from the sun along the planet’s orbital path, the planet’s speed is minimal.

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… F C = F g

Newton’s Mountain Geometric curvature of Earth

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.

Circular Orbits

Satellites Satellites are fast-moving projectiles that continually fall, but never hit the ground The centripetal force that keeps them moving in orbit is GRAVITY. Important for weather, telecommunications, military, & GPS. Satellites don’t crash because they have sufficient tangential speed + no friction.

Astronauts are not weightless! What would have to be true to be weightless? a) What is their acceleration due to gravity? b) What is the speed of the shuttle? c) What is the period of their orbit? EXAMPLE: The astronauts orbit about 200 miles (320km) above the surface of the Earth.

Example 2 The acceleration due to gravity is 1.8m/s 2 for an object in orbit about the Earth. It’s speed is 5180m/s. a)Find the period of the satellite. b) Find the weight of satellite if it has a mass of 500kg

Example 3 On July 19, 1969, Apollo 11’s orbit around the Moon was adjusted to an average altitude of 111 km. The mass of the moon is 7.3x10 22 kg and its radius is 1.7x10 6 m (a) At that altitude how many minutes did it take to orbit once? b) At what speed did it orbit the Moon?

A satellite takes 6.25 days to complete an orbit about Saturn. b) If the radius of Saturn is 6.027x10 7 m, at what altitude was the satellite orbiting? a) What is the distance of the orbit of the satellite from the center of Saturn (M= 5.69x10 26 kg)?

Assuming you are 17 Earth years old, how old would you be if you instead had lived on Pluto your whole life? Meaning, how old are you in Plutonian years? The distance from the Sun to Pluto is 5.9x10 12 m. Mass of Sun = 1.99x10 30 kg