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© 2010 Pearson Education, Inc. Lecture Outline Chapter 7 College Physics, 7 th Edition Wilson / Buffa / Lou

Chapter 7 Circular Motion and Gravitation © 2010 Pearson Education, Inc.

Units of Chapter 7 Angular Measure Angular Speed and Velocity Uniform Circular Motion and Centripetal Acceleration Angular Acceleration Newton’s Law of Gravitation Kepler’s Laws and Earth Satellites © 2010 Pearson Education, Inc.

7.1 Angular Measure The position of an object can be described using polar coordinates— r and θ— rather than x and y. The figure at left gives the conversion between the two descriptions. © 2010 Pearson Education, Inc.

7.1 Angular Measure r is a distance that extends from the origin. r is the same for any point on a given circle. (like the radius!) Θ is an angle, and it changes with time. Linear Displacement…how do we calculate? Angular Displacement is VERY similar

7.1 Angular Measure Δθ = θ - θ i The unit for angular displacement is the degree. There are 360 degrees in one complete circle.

7.1 Angular Measure The arc length, s, is the distance that is traveled along the circular path. The θ is said to define the arc length. It is most convenient to measure the angle θ in radians.

7.1 Angular Measure Relationship between arc length, the radius, and the angle: © 2010 Pearson Education, Inc. For one full circle, with s = 2πr (this is the circumference of the circle)

7.1 Angular Measure A spectator standing at the center of a circular running track observes a runner start a practice race 256m due east of her own position. The runner runs on the track to the finish line, which is located due north of the observer’s position. What is the distance of the run?

7.1 Angular Measure A sailor sights a distance tanker ship and finds that it subtends an angle of 1.15 degrees. He knows from the shipping charts that the tanker is 150m in length. Approximately how far away is the tanker?

7.2 Angular Speed and Velocity How do we calculate speed? What’s the difference between average speed and instantaneous speed?

7.2 Angular Speed and Velocity In analogy to the linear case, we define the average and instantaneous angular speed: © 2010 Pearson Education, Inc. Units?? Angular Velocity??

7.2 Angular Speed and Velocity The direction of the angular velocity is along the axis of rotation, and is given by a right-hand rule. How does this work? Counterclockwise is positive © 2010 Pearson Education, Inc.

7.2 Angular Speed and Velocity A particle moving in a circle has an instantaneous velocity tangential to its circular path. What is a tangent? Tangential speed (the particle’s orbital speed)

7.2 Angular Speed and Velocity Relationship between tangential and angular speeds: This means that parts of a rotating object farther from the axis of rotation move faster. © 2010 Pearson Education, Inc.

7.2 Angular Speed and Velocity An amusement park merry go round at its constant operational speed makes one complete rotation in 45 seconds. Two children are on horses, one at 3.0 m from the center of the ride and the other farther out at 6.0 m from the center. –What are the angular speeds of each? –What are the tangential speeds of each?

7.2 Angular Speed and Velocity The period is the time it takes for one complete revolution (rotation) For example: The period of revolution of the Earth around the Sun is one year. Or the period of the Earth’s axial rotation is 24 hours. Units: seconds or sometimes seconds/cycle © 2010 Pearson Education, Inc.

7.2 Angular Speed and Velocity the frequency is the number of revolutions (rotations) per second. [Units: Hertz] The relation of the frequency to the angular speed:

7.2 Angular Speed and Velocity A CD rotates in a player at a constant speed of 200 rpm. What are the CD’s –Frequency? –Period?

7.3 Uniform Circular Motion and Centripetal Acceleration The acceleration in uniform circular motion is called centripetal acceleration. Centripetal means “center-seeking.” Centripetal acceleration is directed inward or “into” the circle. The tangential velocity is perpendicular to the centripetal acceleration.

7.3 Uniform Circular Motion and Centripetal Acceleration © 2010 Pearson Education, Inc. Instantaneous centripetal acceleration Can also be written as…

7.3 Uniform Circular Motion and Centripetal Acceleration A laboratory centrifuge operates at a rotational speed of 12,000 rpm. –What is the magnitude of the centripetal acceleration of a red blood cell at a radial distance of 8.00 cm from the centrifuge’s axis of rotation? –How does this acceleration compare with g?

7.3 Uniform Circular Motion and Centripetal Acceleration The centripetal force (net inward force) is the mass multiplied by the centripetal acceleration. This force is the net force on the object. As the force is always perpendicular to the velocity, it does no work. © 2010 Pearson Education, Inc.

7.3 Uniform Circular Motion and Centripetal Acceleration A ball is attached to a string is swung with uniform motion in a horizontal circle above a person’s head. If the string breaks, which of the trajectories shown on the following slide would the ball follow.

7.3 Uniform Circular Motion and Centripetal Acceleration Centripetal force is not a new individual force, but rather the cause of the centripetal acceleration. –What do I mean?? Examples?

7.3 Uniform Circular Motion and Centripetal Acceleration Suppose two masses, m 1 = 2.5 kg and m 2 = 3.5 kg are connected by two light strings and are in uniform circular motion on a horizontal frictionless surface where r 1 = 1.0 m and r 2 = 1.3 m. The tension forces acting on the masses at T 1 = 4.5 N and T 2 = 2.9 N. Find the magnitude of the centripetal acceleration.

7.3 Uniform Circular Motion and Centripetal Acceleration A 1.0 m cord is used to suspend a tetherball from the top of a pole. After being hit several times, the ball goes around the pole in uniform circular motion with a tangential speed of 1.1 m/s at an angle of 20 degrees. The force that supplies the centripetal acceleration is: –A.) the weight of the ball? –B.) a component of the tension force in the string? –C.) the total tension in the string?

7.4 Angular Acceleration The average angular acceleration is the rate at which the angular speed changes: In analogy to constant linear acceleration: © 2010 Pearson Education, Inc.

7.4 Angular Acceleration Just like we had angular speed/velocity and tangential speed, we also have the same for acceleration: © 2010 Pearson Education, Inc.

7.4 Angular Acceleration A CD accelerates uniformly from rest to its operational speed of 500 rpm in 3.50 s. –What is the angular acceleration of the CD during this time? –What is the angular acceleration of the CD as its playing the song at a constant speed? –If the CD comes to a stop in 4.50 seconds, what is the angular acceleration?

7.4 Angular Acceleration © 2010 Pearson Education, Inc.

7.4 Angular Acceleration A microwave oven has a 30 cm diameter rotating plate for even cooking. The plate accelerates from rest at a uniform rate of 0.87 rad/s 2 for 0.50 s before reaching its constant operational speed. –A.) How many revolutions does the plate make before reaching its operational speed? –B.) What are the operational angular speed of the plate and the operational tangential speed?

7.5 Newton’s Law of Gravitation What is Newton’s Law of Gravitation? Do you remember the video you watched last year that deals with this? –Had to do with 2 lead balls over time coming together.

7.5 Newton’s Law of Gravitation Newton’s law of universal gravitation describes the force between any two point masses: G is called the universal gravitational constant: © 2010 Pearson Education, Inc.

7.5 Newton’s Law of Gravitation The gravitational attractions of the Sun and the moon give rise to ocean tides. It is sometimes said that since the Moon is closer to the Earth than the Sun, the Moon’s gravitational attraction is much stronger, and therefore has a greater influence on ocean tides. Is this true? –m E = 6.0 x 10 24 kg –m M = 7.4 x 10 22 kg –m S = 2.0 x 10 30 kg –r EM = 3.8 x 10 8 m –r ES = 1.5 x 10 8 km

7.5 Newton’s Law of Gravitation Gravity provides the centripetal force that keeps planets, moons, and satellites in their orbits. We can relate the universal gravitational force to the local acceleration of gravity: © 2010 Pearson Education, Inc.

7.5 Newton’s Law of Gravitation The acceleration due to gravity does vary with altitude. This means we need to account for any height changes. a g = GM E (R E + h) 2

7.6 Kepler’s Laws and Earth Satellites Kepler’s laws were the result of his many years of observations. They were later found to be consequences of Newton’s laws. Kepler’s first law: Planets move in elliptical orbits, with the Sun at one of the focal points. © 2010 Pearson Education, Inc.

7.6 Kepler’s Laws and Earth Satellites Kepler’s second law: A line from the Sun to a planet sweeps out equal areas in equal lengths of time. © 2010 Pearson Education, Inc.

7.6 Kepler’s Laws and Earth Satellites Kepler’s third law: The square of the orbital period of a planet is directly proportional to the cube of the average distance of the planet from the Sun; that is,. This can be derived from Newton’s law of gravitation, using a circular orbit. © 2010 Pearson Education, Inc.

Summary of Chapter 7 Angles may be measured in radians; the angle is the arc length divided by the radius. Angular kinematic equations for constant acceleration: © 2010 Pearson Education, Inc.

Summary of Chapter 7 Tangential speed is proportional to angular speed. Frequency is inversely proportional to period. Angular speed: Centripetal acceleration: © 2010 Pearson Education, Inc.

Summary of Chapter 7 Centripetal force: Angular acceleration is the rate at which the angular speed changes. It is related to the tangential acceleration. Newton’s law of gravitation: © 2010 Pearson Education, Inc.

Summary of Chapter 7 Gravitational potential energy: Kepler’s laws: 1. Planetary orbits are ellipses with Sun at one focus 2. Equal areas are swept out in equal times. 3. The square of the period is proportional to the cube of the radius. © 2010 Pearson Education, Inc.

Summary of Chapter 7 Escape speed from Earth: Energy of a satellite orbiting Earth: © 2010 Pearson Education, Inc.

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