AP Physics 1: Chapter 5 Circular Motion, Universal Gravitation In-class notes Uniform Circular Motion (UCM) occurs when a mass orbits a fixed central point due to the action of a centripetal force. Know the basics: - Centripetal = “towards the center” - Tangential = tangent to circular path (“skims” the surface) - Speed is constant, but the mass is accelerating…why? - Acceleration is constant in magnitude - Acceleration (centripetal) is perpendicular to velocity (tangential) - Centripetal force is always in the same direction as acceleration (towards the center) - Period (T) = time required for object in UCM to make one complete revolution or cycle Quantities in UCM: 𝑣𝑡𝑎𝑛𝑔𝑒𝑛𝑡𝑖𝑎𝑙 = 𝑣 𝑇 = 𝑎𝑐𝑒𝑛𝑡𝑟𝑖𝑝𝑒𝑡𝑎𝑙 = 𝑎𝑐 = 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑝𝑒𝑟𝑖𝑜𝑑 = 2𝜋𝑟 𝑇 𝑣2 𝑟 𝐹𝑐𝑒𝑛𝑡𝑟𝑖𝑝𝑒𝑡𝑎𝑙 = 𝐹𝑐 = 𝑚𝑎𝑐 = 𝑚𝑣 2 𝑟 = 𝑚( 4𝜋 2 𝑟 𝑇2 ) Note: “Centripetal Force” is not a new type of force or a “real” force…it is a designation given to whatever force keeps an object moving in UCM (see Table 1) Table 1: Forces acting as a Centripetal Force Scenario: Centripetal Force: Racecar moving around a circular track* Rock being twirled on a string Satellite orbiting a planet *On a banked track, a component of the normal force also acts centripetally - Can you think of another instance where the normal force acts as a centripetal force? Often the key to solving centripetal force problems is to realize what real force (or what component of a real force) is causing the constant acceleration towards the center. Fly Me a River: A plane is traveling a circular path with a radius of 200 m, as shown in the diagram to the right. At the location in the diagram, what is the direction of the plane’s a. velocity, and b. acceleration? c. If the plane completes this circular route in 12 s, find the magnitude of the plane’s acceleration. Little Tykes Get Tired: A clump of mud gets stuck on the edge of a .60-m-diameter tire on a car. A kid sitting in the next car over counts how many times the clump completes a circle in one minute: 92 times. This genius kid then calculates the speed of the car. What is it? Doesn’t apply to Bond cars: A car travels around a circular curve with a radius of 18 m. If the coefficient of friction between the car tires and the road is .95, calculate the maximum speed with which this car can round the curve without slipping. Bank on it: A different car is traveling around a circular curve of radius 15 m, but the road is so icy that it can be considered frictionless. Luckily, the road is banked at an angle of 20°. Find the necessary speed of the car so that it does not slip either up or down the inclined road as it travels along it. Coni-cademy: A conical pendulum is set up with a .75-kg mass on the end. The mass revolves with a velocity of 1.19 m/s as it travels through a circle of radius .30 m. Calculate the tension in the rope 𝐹𝑇 and the angle 𝜃 made by the string. 2 3 Newton's Law of Universal Gravitation = every mass in the universe attracts every other mass with a force of gravity given by: - magnitude: |Fg| = Fg = where: Fg (N) = mA, mB (kg) = R (m) = G= - direction: (attracts) B A Calculating the (magnitude of the) Local Gravitational Field Constant, g (aka the acceleration due to gravity) consider: force of gravity by M on m: m R Fg = and M Fg = Gravity decreases as you move away from the earth… What happens to the strength of gravity as you go deeper INTO the Earth? 4 Circular Orbits Kinematics of bodies in orbit: apply UCM equations UCM What provides the centripetal force on an object in orbit? 𝑣= Radius (R) is now orbital radius (distance from center of orbiting object to center of object being orbited); Period (T) is now time for one complete revolution around orbited object (a “year”); Velocity (v) is orbital velocity of the orbiting object through space; Centripetal Acceleration (ac) is now acceleration due to the gravity (g) of the orbited object. 2𝜋𝑅 𝑇 𝑎𝑐 = 𝑣2 𝑅 𝐹𝑐 = 𝑚𝑎𝑐 Note on orbital radius being measured from “center to center:” - Altitude (height above the surface of a planet) by itself means nothing…add to radius of planet to get orbital radius (R) This is the “R” used in UCM calculations Alt. + Rplanet = R Be aware of situations where the radius of a planet is insignificant…for example, if the circle below represents the Sun and the period represents a planet, the radius of either is insignificant for the orbital radius R. PLANET SUN . R For an object in orbit… - Gravity is the only force acting - No fuel required to maintain orbital speed - Must orbit above any atmosphere present Brew Me Some Gravi-tea: Calculate the magnitude of the Earth’s local gravitational field constant (g) at a distance from the Earth equal to the Moon’s orbital radius (2 possible methods). GIVEN Lunar orbital period 2.3 x 106 s Lunar orbital radius 3.8 x 108 m 5 G Whiz: Consider a satellite in a circular orbit around the Earth which is gradually moved to an orbit with greater and greater radius. As the satellite’s orbital radius is increased, will the following increase, decrease, or stay the same? Justify with equations, starting with Fc = Fg. Remember, g decreases as you move away from Earth. - Centripetal acceleration - Orbital Period - Orbital Velocity When Life Hands You LEM: During the Apollo 10 lunar mission, the Lunar Excursion Module (L.E.M.) orbited the moon at a lunar altitude of approximately 1.5 x 104 m (about 10 miles). a. Calculate the orbital period of the LEM when in this close lunar orbit. GIVEN Radius of moon 1.74 x 106 m Near moon’s surface… |g| = 1.6 m/s2 b. Calculate the orbital speed of the LEM. Planet’s Best Friend: To find the mass of ANYTHING given the period and orbital radius of one of its satellites (in this case Proteus, a moon orbiting Neptune, is used to find Neptune’s mass): GIVEN Proteus orbital radius 1.18 x 108 m Proteus orbital period 9.69 x 104 s - Allowed earlier scientists to calculate the mass of planets with moons - Later scientists could put an artificial satellite in orbit around a planet to find its mass Weight a Minute: At the altitude at which the International Space Station orbits, the acceleration due to Earth’s gravity is only equal to about 8.8 m/s2. Consider a 62-kg astronaut on the space station crew. a. What is the mass of the astronaut? b. What is the actual weight of the astronaut? c. What is the astronaut’s apparent weight?