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## Chapter 5 Circular Motion, the Planets, and Gravity

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**Does the circular motion of the moon around the Earth ...**... have anything in common with circular motion on Earth?**Circular Motion**An object is moving with a constant speed in a circular track. Is the object’s velocity changing? If so, how we define the change? In what direction? How can we calculate the acceleration? .**Centripetal Acceleration**• Centripetal acceleration is the rate of change in velocity of an object that is associated with the change in direction of the velocity. • Centripetal acceleration is always perpendicular to the velocity and points toward the center of the circle. • Centripetal acceleration always relates to the velocity and radius of the circle.**A ball is whirled on the end of a string with constant speed**when the string breaks. Which path will the ball take?**Centripetal Force**• Because of Centripetal acceleration is involved in all the curved motions, according to Newton’s Second Law of motion, there must be a total net force involved to produce such as acceleration. • The centripetal force may be due to one or more individual forces, such as a normal force and/or a force due to friction. • The centripetal force refers to any force or combination of forcesthat produces a centripetal acceleration.**The horizontal component of T produces the centripetal**acceleration. • The vertical component of T is equal to the weight of the ball. • At higher speeds, the string is closer to horizontal because a large horizontal component of T is needed to provide the required centripetal force.**What force helps a car negotiate a flat curve?**• The Static force of friction is the frictional force acting when there is no motion along the surfaces (no skidding or sliding) • The Kinetic force of friction is the frictional force acting when there is motion along the surfaces. • The friction between the tires and road produces the centripetal acceleration on a level curve.**What happens if the curve is banked?**• On a banked curve, the horizontal component of the normal force also contributes to the centripetal acceleration. The bank slope is designed according to the speed and the radius of the curve.**What forces are involved in riding a Ferris wheel?**Depending on the position: • Weight of the rider • Normal force from seat • Seat-belt force**Example (Box 5.1):Circular motion of a ball on a string**• A ball has a mass of 50g (0.05kg) and is revolving at the end of a string in a circle with a radius of 40 cm (0.40m). The ball moves with a speed of 2.5 m/s, or one revolution per second. a. What is the centripetal acceleration? b. What is the horizontal component of the tension needed to produce this circulation?**Examples**• E2. A car rounds a curve with a radius of 25 m at a speed of 20m/s. What is the centripetal acceleration of the car? • E4. How much larger is the required centripetal acceleration fro a car rounding a curve at 60 mph than for one rounding the same curve at 30mph • E6. A car with a mass of 1200 kg is moving around a curve with a radius of 40m at a constant speed of 20m/s (about 45mph). a. What is the centripetal acceleration of the car? b. What is the magnitude of the horizontal component of the normal force that would be required to produce this centripetal acceleration in the absence of any friction?**Example**• E8. A Ferris wheel at a carnival has a radius of 12 m and turns so that the speed of the rider is 8m/s. a. A What is the magnitude of the centripetal acceleration of the riders? b. What is the magnitude of the net force required to produce this centripetal acceleration for a rider with a mass of 70 kg?**Planetary Motion**• The ancient Greeks believed the sun, moon, stars and planets all revolved around the Earth: geocentric view (Earth-centered). • A time-lapse photograph showing the apparent motions of stars in the northern sky. (Polaris (the “North star)” lies near the center of the pattern and does not appear to move very much. The entire pattern appears to rotate during the night about a point near Polaris.**Planetary Motion**• To explain the apparent retrograde motion of the planets, Ptolemy (second century, A.D) invented the idea of epicycles.**Copernicus (1472-1543) developed a model of the universe in**which the planets (including Earth!) orbit the sun: heliocentric view (sun-centered) of the universe. • Galileo was an early advocate the Copernican model • Careful astronomical observations were needed to determine which view of the universe was more accurate. • Tycho Brahe spent several years collecting data on the precise positions of the planets with naked eyes (before the invention of the telescope)**Tycho Brahe’s large quadrant permitted accurate**measurement of the positions of the planets and other heavenly bodies (to an accuracy of 1/60 of a degree) . These data fell to his assistant, Johannes Kepler (1571-1630).**Kepler’s First Law of Planetary Motion**• The planets all move in elliptical orbits about the sun, with the sun located at one focus of the ellipse.**Kepler’s Second and Third Laws of Planetary Motion**• An imaginary line drawn from the sun to any planet moves through equal areas in equal intervals of time • If Tis the amount of time taken for the planet to complete one full orbit around the sun (period) and ris the average radius of the distance of the orbit around the sun for each planet, then**Newton’s Law of Universal Gravitation**• Newton realized the similarity between the motion of a projectile on Earth and the orbit of the moon. • Newton’s law of universal • Gravitation: • G is the Universal • gravitational constant.**The gravitational force is attractive and acts along the**line joining the center of the two masses. It obeys Newton’s third law of motion.**Universal Gravitation Constant**• Universal gravitational constant G was measured by Henry Cavendish (1731-1810) . • The measurement of the universal gravitational constant gives the mass of the earth. • Example: Earth’s radius is 6,370 km, the gravitational acceleration at the surface is g=9.8 m/s2. What is the mass of the Earth?**Examples**• Box 5.3 (p 93) The mass of the Earth is 5.98x1024 kg, and its average radius is 6370km (6.37x106 m). Find the gravitational force (the weight) of a 50-kg person standing on the surface of the earth (a) by using Newton’s Law of Gravitation (b) by using the gravitational acceleration. • Suppose that an astronaut is in a space that is 6370km (the same as the earth’s radius) above the surface of the Earth. If the Astronaut’s weight on Earth is 200 pound, What will be his weight in that point of the space? The gravitational acceleration at the surface of the Moon is only about 1/6 of that on the surface on the Earth, decided by the Moon’s mass and radius. For the same reason, other planets also have different gravitational accelerations.**Examples**Q22. Three equal masses are located as shown. What is the direction of the total force acting on m2? E13. Two masses are attracted by a gravitational force of 0.14 N. What will the force of attraction be if the distance between these two masses is halved? E14. The acceleration of gravity at the surface of the moon is approximately 1/6 of that at the surface of the Earth. What is the weight of an astronaut standing on the moon whose weight on Earth is 180 pound?**The Moon and Other Satellites**Phases of the moon result from the changes in the positions of the moon, Earth, and sun.**An artist depicts a portion of the night sky as shown. Is**this view possible?