CIRCULAR MOTION & GRAVITATION

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.

Uniform Circular Motion - velocity

Circular Motion: Force & Acceleration What force is responsible for car changing direction? What is the direction of THIS force on the car? What would car do if this force vanished?

Circular Motion – Force & 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 = Name given to Net Force is always

Centripetal Force equations

Example 1: A washing machine drum makes 5 rotations per second during the spin cycle. The inside drum has a radius of 25cm. a) Determine the speed of the drum. b) What acts as the centripetal force on the 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 45.0 m. If the coefficient of friction between the tires and the road is µs = 0.82, what is the maximum speed the car can have on the curve without skidding?

Example 3: 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) What would happen if the speed were to increase? c) Would the chair eventually become horizontal?

Example 4: 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 (Fc) 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 Example 1: A rollercoaster executes a loop moving at 20m/s at the bottom and 12m/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, assuming no underneath locking wheels?

EXAMPLE 2: In an automatic clothes drier, a hollow cylinder moves the clothes on a vertical circle (radius r = 0.39 m). It is designed so that the clothes tumble gently as they dry where a piece of clothing reaches an angle of θ=69o above the horizontal, it loses contact with the wall of the cylinder and falls onto the clothes below. How many revolutions per second should the cylinder make?

Example 3: A fighter jet is in a vertical dive when it 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?

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

Example: A car rounds a curve at angle θ. The radius of the curve is R Example: 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.

UNIVERSAL GRAVITATION

Newton’s Law of Universal Gravitation

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.

Once the altitude becomes comparable to the radius of the Earth (RE) , the decrease in the acceleration of gravity is much larger:

Example3: A 50kg astronaut climbs a ladder that is 6400km high Example3: A 50kg astronaut climbs a ladder that is 6400km high. She 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) Determine force of gravity on her if she steps off 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.

Force of gravity INSIDE the earth (assume uniform density) What happens to force of gravity when you enter the earth on way towards center? Explain. What would force of gravity be like halfway between center and surface?

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 Near side of earth is pulled harder than center and center is pulled harder than far side by the moon.

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? 29

Kepler's 1st Law: The Law of Elliptical Orbits As we approach the winter solstice, where is the position of the earth in the diagram? 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) Each planet travels in an elliptical orbit with the sun at one focus.

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.

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.

Newton’s Mountain Geometric curvature of Earth 33

Astronauts are not weightless in terms of gravity! 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 in terms of gravity!

Circular Orbits In circular Earth orbit, gravity is perpendicular to the velocity, responsible for changing the direction only.

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?

Satellite Example 2 On July 19, 1969, Apollo 11’s orbit around the Moon (7.36x1022kg) was at an altitude of 111km.  a) Determine the centripetal acceleration of the spacecraft. (Rmoon=1.74x106m)   b) Determine the value of ‘g’ on the surface of the moon.

Example 3: Determine the height IN MILES above the earth of a geostationary (form of a geosynchronous) satellite (stays over same location on Earth).