Chapter 5

5.1 Uniform Circular Motion  Uniform circular motion is the motion of an object traveling at a constant (uniform) speed on a circular path.  Since the path is in a circle, period (T)is often used to express time.  The distance around the circle is equal to the circumference (2πr) If the magnitude of velocity if constant, what causes acceleration?

5.2 Centripetal Acceleration  Since the direction of the velocity vector is constantly changing, the object must be accelerating.  Centripetal acceleration is a vector quantity.  The direction of acceleration is always toward the center of the circle or arc.

Centripetal Acceleration Defined  Magnitude: The centripetal acceleration of an object moving with a speed v on a circular path of radius r has a magnitude a c given by: Direction: The centripetal acceleration always points toward the center of the circle and continually changes direction as the object moves.

5.3 Centripetal Force  Newton’s 2 nd Law of Motion holds true for objects moving with uniform circular motion  Remember F=ma  Applying our equation for centripetal acceleration to Newton’s 2 nd Law gives the equation

Points to Ponder  Centripetal means “directed toward the center”  Centripetal force is caused by other forces such as friction, tension, normal force, applied force, etc.  You may need to use equations from previous chapters to determine the centripetal force  Remember: NET FORCE!

5.4 Banked Curves  Roadways are banked as a means of eliminating friction as the cause of centripetal force.  On a banked turn, the normal force is perpendicular to the surface of the road. The vertical component of the normal force (F N sinθ) provides the centripetal force.  This will only work for a certain velocity at a certain banking angle.

Useful Equations for Banked Curves The vertical component of the normal force is F n cosθ and since the car doesn’t accelerate in the vertical direction, this component must balance the weight (mg) of the car. Therefore this component of the normal force will equal the weight. Using this relationship we derive the equation….

5.5 Satellites in Circular Orbits  Gravitational force will act as the centripetal force for satellites in orbit about the earth.  There is only one speed that a satellite can have if the satellite is to remain in an orbit with a fixed radius.  For a given orbit, a satellite with a large mass has exactly the same orbital speed as a satellite with a small mass.

Equations for Satellites in Circular Orbit Remember: Centripetal force comes from any number of forces. As long as the force is directed toward the center of an arc or circle, it is considered a centripetal force. These equations apply to man- made earth satellites or to natural satellites like the moon. It also applies to circular orbits about any astronomical object. Replace the mass of the earth with the mass of the object on which the orbit is centered.

Relating Period to Centripetal Force  The period of a satellite is the time required to complete one orbital revolution.  Shown on the next slide is an equation that can be used to find the period of planets in nearly circular orbits.  Replace the mass of the earth with the mass of the sun.  The period is proportional to three-halves power of the orbital radius. This is Kepler’s third law of planetary motion.

More Equations! The first equation sets velocity of a satellite in orbit at a fixed radius equal to the velocity of an object with uniform circular motion. If we rearrange this equation to solve for T, the equation on the right will be useful to determine the period of a satellite with fixed orbital radius. This equation is useful for communications experts launching “synchronous satellites” where the period is equal to one day. Check out Example 11 on page 147!

5.6 Apparent Weightlessness and Artificial Gravity  When a person is in an orbiting satellite, both the person and the scale are experiencing uniform circular motion.  Both will continually accelerate or “fall” toward the center of the circle.  Since both fall at the same rate, the person cannot push on the scale  The apparent weight in the satellite is zero!  Read through Example 13 on page 149 of your text!

5.7 Vertical Circular Motion  Consider a motorcycle stunt driver driving around a vertical loop.  The magnitude of the normal force changes because the speed changes and because the weight does not have the same effect at every point.  In the “east and west” positions, the weight is tangent to the circle and has no effect on the centripetal force.

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