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Notes

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CHAPTER 5: Circular Motion; Gravitation

 

Uniform Circular Motion Kinematics

 

†  Uniform circular motion describes the movement of an object in a circular path at constant speed.

†  The velocity vector, however, is constantly changing in order for the object to stay on the circular path.

†  At any point in the path, the velocity vector is tangent to the circular path in the direction of motion.

†  The acceleration at any point is directed toward the center of the circle.

†  This radial acceleration is given by ar = v2/r

†  At any point, the velocity and radial acceleration vectors are perpendicular

†  The number of revolutions per second, the frequency f, is given in seconds.

†  The period T is the reciprocal of the frequency, and is the time in seconds for each revolution.

†  From the above, it can be seen that the speed is given by 2pr/T

 

Uniform Circular Motion Dynamics

 

†  åFR = mv2/r    This force, directed toward the circle’s center, is that required to keep the object in circular motion

†  Remove the force, and the object continues in the direction of the velocity vector at that point.

†  For cars, road friction provides the inward force to move in a circle.

†  Satellites use gravity to provide the force for circular orbits

 

Non Uniform Circular Motion (not covered)

 

†  This is the situation when for net force on an object is not directed toward the center of the circle.

†  In this case, the force vector can be broken into 2 perpendicular components

     ]  Inward directed radial force: FR

    ]  Tangential force: FTan

†  The acceleration can also be broken into two components

     ]  Inward radial acceleration: aR

     ]  Tangential acceleration: aTan

†  Note the speed is changing in this case, as a result of the tangential force resulting in a tangential acceleration

†  The magnitude of the acceleration at any point can be found from the Pythagorean theorem, using the above 2 components.

 

Newton’s Law of Universal Acceleration

 

†  The magnitude of the attractive force between two objects is given by

 

     F = Gm1m2/r2

 

     Where G is a proportionality constant, m1 is the mass of one object, and m2 is the mass of the other object, and r is the distance between the

     centers of the two objects.

 

†  The value for G was first determined experimentally by Cavendish, about 100 years after Newton proposed his law.