Chapter 5 Circular Motion; Gravitation

Centripetal Acceleration Centripetal means “Center Seeking” and the centripetal force on an object moving in a circle always acts towards the center of the circle. By definition: a = V 2 /R where V = linear or tangential speed, and R is the radius of the circle.

Centripetal Force  Centripetal Acceleration results from a Centripetal Force since accelerations are always caused by outside forces Since F = ma and centripetal acceleration is a c = V 2 /R then it follows that Centripetal Force, F c = mV 2 /R

Centripetal Force  According to Newton’s 1st Law. An object in motion tends to stay in motion. It has been shown that this is straight line motion. Things want to move in straight lines, so when something is moving in a circle there must be a force causing it to stay in the circle and constantly changing the direction of its motion. Examples  When car is rounding a curve, friction between the tires and the road which is providing (not opposing) the centripetal force.  A satellite in orbit is held their by gravity (i.e. gravity is providing the centripetal force. The object actually wants to travel in a straight line, tangent to the circular path, but the provided centripetal force keeps it in a circle.

A Ball Revolving in a Vertical Circle  At the top of the circle the tension on the string is in the same direction as the weight together they equal the Net Force which provides the centripetal force F ta + mg = mV 2 /R Minimum velocity to stay in a circle, minimum tension  At the bottom of the circle the tension on the string is in the opposite direction as the weight F ta - mg = mV 2 /R Maximum Velocity, maximum tension mg F tb F ta mg

Special Case Rounding a Curve  When rounding a curve it is the friction between the tires and the road which provides the centripetal force F N = F w = mg F f = μF N = μmg F f =F c = mV 2 /R  μmg = mV 2 /R V = SQRT(μgR) FNFN F c = F f mg

F n sin θ F c = F f Special Case Rounding a Banked Curve  Special note on road problems: The appropriate banking angle is one where the car would stay on the road without friction  mg = F N cos   F c = F N sin .  F N = mg/cos  F c = mg sin /cos  = mg tan .  mv 2 /r = mg tan   tan  = v 2 /gr  θ = banking angle without friction The red arrows are the Vertical and Horizontal components of F N. The vertical component is F N cos  and the horizontal component is F N sin . FwFw FnFn F n cos θ θ θ

Universal Gravitation  Every object is gravitationally attracted to every other object in the Universe F = Gm 1 m 2 /R 2  G = universal gravitation constant 6.67 x Nm 2 /kg 2  M 1, M 2 = mass of each object  R – distance between the objects

Universal Gravitation  So F = Gm 1 m 2 /R 2 and F = ma  Attraction between earth and another object m p g = Gm p m e /R 2 Since m p cancels on both sides you get: g = Gm e /R 2 So this tells us that gravity caused by any object is the ratio of the mass to the distance between the centers of the objects multiplied by the universal gravitation constant

Special Case: Satellites  Since the weight of the satellite is given by the gravitational force: m sat g = Gm sat me/R 2  And since the weight of the satellite provides the centripetal force we get: m sat V 2 /R = Gm sat me/R 2  Since m sat cancels you can rearrange the equation to get: V = SQRT(Gme/R)  Which is the minimum velocity required to keep a satellite in an orbit R (measured from the center of the earth to the satellite

Kepler’s Laws  Kepler studied planetary motion. He found: All planetary bodies move in elliptical orbits Planetary objects sweep equal areas in equal amounts of time If two satellites revolve around the same object they are related by the equation: (t 1 /t 2 ) 2 = (r 1 /r 2 ) 3

Sample Problems Centripetal Acceleration  Centripetal Force  Vertical Circles  Rounding a Curve  Banking Curves  Universal Gravitation  Satellites  Kepler’s Laws