CIRCULAR MOTION

• Centripetal acceleration

• Centripetal Force

• Universal Law of Gravitation

• Keplers Laws

Uniform Circular Motionmotion in a circle of constant radius & constant speed

Instantaneous Velocity always tangent to path

Change in direction = change in velocity = acceleration

Acceleration is towards center of circle along the radius• Centripetal Acceleration• Radial Acceleration

Texas Motor Speedway

• Radius of Turns = 775ft = 230m

• Record Qualifying Speed = 233mph

• What is the centripetal Acceleration of the driver?

• How many g’s does the driver feel?

G-force Threshold Test

G-Force Simulator• Jet Pilots, Astronauts, Daredevils

• r = 7.0m

• a = 6.1gs

• V=?

480 rpm (max)

r=0.15m r=0.09m

Record player (rpm)What is the centripetal acceleration at outer edge?

Rpm m/s

Centripetal ForceFor an object to accelerate, a force must be applied.

𝐹 = 𝑚𝑎 , 𝑎𝑐 =𝑣2

𝑟, 𝐹𝑐 = 𝑚𝑎𝑐 = 𝑚𝑣2

𝑟.

Centripetal Force is a NET resultant Force

It can be provided by:

• Tension

• Gravity (moon)

• Friction (Car, motorcycles)

• Normal Force (Gravitron, Nascar)

Or any combination of these

Centrifugal Force

• DOES NOT EXIST!!!!

• Natural tendency for a body to move in a straight line (INERTIA)

• A 300 kg motor cycle and rider rounds a curve (r=60m) with a velocity of 25 m/s.

• What is the force of friction between the tires and the track?

• What is the coefficient of friction between the tires and the track?

FRICTION AS CENTRIPETAL FORCE

Normal Force as Centripetal Force Wall of Death

• Diameter of bowl = 20m

• M = 0.80

• Minimum Velocity?

Tension & gravity• m=850lbs =386 kg

• r=60m

• Vmax = 35m/s

• Max force on cable =1000lbs

• Why did the cable break?

Gravity as centripetal Force • What is the minimum speed at max height to complete the loop with a height of 5.0m?

Windseeker at Cedar Point

• Maximum Diameter = 110 ft = 33.5 m

• Velocity = 8 rpm

• What is the angle of the swing arm?

Components as Centripetal Force

Components as Centripetal Force

Horizontal Component of Normal Force is directed toward the center of turn.

Horizontal Component of Normal Force contributes to Centripetal Force

Components as Centripetal Force

Ted Ligitym=95kgV=15m/sϴ above hz = 20What is the radius of the turn?

UNIVERSAL LAW OF GRAVITATION• Isaac Newton (1686)• Observed that apple falls towards the center of the earth.

• Moon orbits the earth in nearly circular path.

• Earth orbits sun in nearly circular path.

• Common Centripetal Force that attracts all masses to each other.

• All particles attract every other particle in the universe using a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

UNIVERSAL LAW OF GRAVITATIONTHE CAVENDISH EXPERIMENT (1797-1798)• Henry Cavendish (Great Britain) Measured the Force between two objects.

• Calculated G (Universal Gravitational Consant).

• Measured the mass of the earth.

•G = 6.67 x 10-11 N*m2/kg2

UNIVERSAL LAW OF GRAVITATION

• Measure the mass of the earth.

• G = 6.67 x 10-11 N*m2/kg2

• re = 6.38*103 km

𝑚𝑔 =𝐺𝑚1𝑚2

𝑟2

UNIVERSAL LAW OF GRAVITATION• Calculate the Force between any two masses.

• m1 = 70kg

• m2=50kg

• d = 1.5m

UNIVERSAL LAW OF GRAVITATIONGravity as Centripetal Force

• What is the velocity of the International Space Station?• Altitude = 370km

• re = 6.38*103 km

• me= 5.98*1024kg

𝑚𝑠𝑣2

𝑟=𝐺𝑚𝑠𝑚𝑒

𝑟2

UNIVERSAL LAW OF GRAVITATION• Acceleration of Gravity on other Planets

• Mars• Mass = 6.39*1023kg

• Radius = 3.40*106m 𝑚𝑔 =𝐺𝑚1𝑚2

𝑟2

Keplers Laws of Planetary Motion

• 1st Law: Planets move in ellipses with the Sun at one focus.

Keplers Laws of Planetary Motion

• 2nd Law: The radius vector describes equal areas in equal times

Keplers Laws of Planetary Motion

• 3rd Law: The ratio of the squares of the revolutionary periods for two planets is equal to the ratio of the cubes of their semi-major axes.

𝑻𝟐

𝑻𝟐=𝒓𝟑

𝒓𝟑

Sample Problems

Planet Period of Orbit Average

Orbital Radius

Mercury 88 days

Earth 1 year 1 A.U.

Mars 2.125 A.U.

Saturn 29 years

Sample Problems

• The average radius of the moons orbit around the earth is 384 748 km and has a period of 27.3 days. A Direct TV satellite has a period of exactly one day.

• What is the radius of the satellites orbit?

• How far is the satellite above the Earths surface?