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Circular Motion and Gravitation: Angular Velocity, Polar Coordinates, Centripetal Force & Newton's Law of Gravitation

This chapter covers angular velocity, polar coordinates, centripetal acceleration, angular acceleration, and Newton's Law of Gravitation. Learn about the motion of objects in circular paths and the forces involved.

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Circular Motion and Gravitation: Angular Velocity, Polar Coordinates, Centripetal Force & Newton's Law of Gravitation

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  1. Chapter 7 Circular motion and gravitation October 1 Angular velocity 7.1 Angular measure Polar coordinates: The position of a point on a plane can be measured by polar coordinates r(radial position) and q (angular position). The relations between the Cartesian coordinates and the polar coordinates are Angular displacement is the difference between two angular positions: Radian: The angle in radians is given by the ratio of the arc length sand the radius r:

  2. Small-angle approximation: When q is small, we have Example7.2: How Far Away? A Useful Approximation

  3. Average angular speed is the magnitude of the angular displacement divided by the total time to travel the angular distance: If the angular speed is constant, then Angular velocity is a vectorwhose magnitude is the angular speed, and whose direction is determined by the right-hand-rule: When the fingers of the right hand are curled in the direction of the rotation, the extended thumb points in the direction of the angular velocity. Practice: What is the angular velocity direction of 1) the spin of the earth, and 2) a clock placed vertically?

  4. Tangential speed (vt) and the angular speed: All the particles of a rigid object rotating with constant angular velocity have the same angular speed, but the parts farther from the axis of rotation move faster. Example 7.3: Merry-Go-Rounds: Do Some Go Faster than Others? Period and frequency: Periodis the time it takes for one rotation.Frequencyis the number of rotations per second. The unit of frequency is hertz (Hz, 1 Hz=1 1/s). Example 7.4: Frequency and Period: An Inverse Relationship

  5. Read: Ch7: 1-2 Homework: Ch7: E4,6,23 Due: October 9

  6. October 2 Circular motion 7.3 Uniform Circular Motion and Centripetal Acceleration Uniform circular motion occurs when an object moves at a constant speed in a circular path. Examples: amusement park riders, a car on the exit of a highway, the moon around the earth. Centripetal acceleration: As shown in the figure, as Dt approaches zero, the instantaneous change in the velocity points directly toward the center of the circle. The acceleration in uniform circular motion is therefore called centripetal acceleration. Centripetal acceleration is directed radially inward, with no component in the direction of the tangential velocity, or else the magnitude of that velocity would change.

  7. Centripetal acceleration is given by: Example7.5: A Centrifuge: Centripetal Acceleration Centripetal force: The centripetal force is the force needed to provide the centripetal acceleration. According to Newton’s second law, The centripetal force is directed radially toward the center of the circular path. Centripetal force does no work and does not change the kinetic energy of the object. The centripetal force is not a new kind of force. It is supplied by either a real force or the vector sum of several forces that we have studied.

  8. Example7.6: Breaking Away Example 7.7: Where the Rubber Meets the Road: Friction and Centripetal Force Example7.9: Center-Seeking Force: One More Time

  9. Read: Ch7: 3 Homework: Ch7: E30,36,43 Due: October 9

  10. October 7 Angular acceleration 7.4 Angular Acceleration In circular motion, if there is an angular acceleration, the motion is non-uniform. The magnitude of average angular acceleration is given by If the angular acceleration is constant, then The SI unit of angular acceleration is rad/s2. If the angular acceleration increases the angular velocity, both quantities have the same sign. If the angular acceleration decreases the angular velocity, then the two quantities have opposite signs. Example7.10: A Rotating CD: Angular Acceleration

  11. Tangential acceleration is associated with changes in tangential speed in a non-uniform circular motion. Centripetal acceleration is necessary for circular motion, but tangential acceleration is not. When there are both tangential and centripetal accelerations, the instantaneous acceleration is their vector sum. Equations for angular kinematics are analogous to their counterparts in linear kinematics. Example7.11: Even Cooking: Rotational Kinematics

  12. Read: Ch7: 4 Homework: Ch7: E44,48 Due: October 18

  13. October 8 Gravitation 7.5 Newton’s Law of Gravitation Newton’s law of universal gravitation: Every particle in the universe has an attractive gravitational interaction with every other particle. The gravitational interaction between two particles with m1 and m2 separated by a distance ris The universal gravitational constant is It was first measured by Henry Cavendish, from which the mass of the earth was evaluated. Newton derived the inverse-square relationshipfrom Johannes Kepler’s laws of planetary motion. For calculating the gravitational forces between spherical homogeneous objects, they can be thought as point masses located at their respective centers. Example7.12: Greater Gravitational Attraction?

  14. Acceleration due to gravity revisited: The acceleration due to gravityagat a distance r from the center of a spherical mass Mis Let ME be the mass of the earth, and RE be its radius. The acceleration due to gravity at the surface of the earth is The acceleration due to gravity varies with altitude. At a distance h above the surface of the earth, the acceleration due to gravity is It also slightly depends on the latitudes because of the oblate shape of the earth. Discussion: What happens to the acceleration due to gravity if we dig a deep hole? Example7.13: Geosynchronous Satellite Orbit Note this example effectively proved r3/T2=constant, one of the Kepler’s laws of planetary motion.

  15. Gravitational potential energy away from the earth: The gravitational potential energy is calculated from the work done to place an object into a certain location. Choosing the zero reference point to be at the infinity, for a mass m at an altitude h above the surface of the earth, we have To prove this we unfortunately need calculus. On the earth, we can visualize ourselves as being in a negative gravitational potential energy well. The total mechanical energy of a mass m1 moving at a distance r from mass m2 is Example7.14: Different Orbits: Change in Gravitational Potential Energy

  16. Read: Ch7: 5 Homework: Ch7: E53,57 Due: October 18

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