Uniform Circular Motion and Gravitation Rotational Motion in

Uniform Circular Motion and Gravitation Rotational Motion: in close analogy with linear motion (distance/displacement, velocity, acceleration) Angular measure in “natural units” Angles and Rotation in radians r q s Angle = arc length / radius r from one complete rotation = 360 o = 2 p rad 45 o = p/4 rad 90 o = p/2 rad 180 o = p rad 1 rad ≈ 57. 30 o Phys 250 Ch 5 p 1

Example: The ancient Greek Eratosthenes new that when the sun was directly overhead in Syene, the sun was about 7 degrees from overhead in Alexandria. (see astro 03 f 14. jpg) Using the known distance between the cities, he was able to determine the radius of the Earth. Using the distance of 770 km between these cities, calculate the radius of the earth. Phys 250 Ch 5 p 2

A curved path requires an “inward” force “Center seeking” = Centripetal force is the force perpendicular to the velocity of an object moving along a curved path. The centripetal force is directed toward the center of curvature of the path. v Dv Dv Dv = a. Dt = (F/m)dt examples: ball on a string, car rounding a corner. Centrifugal Effect: the “fictitious force” felt by an object when the frame of reference moves along (and therefore accelerates) along a curved path. This effect is simply inertia. Stop the force and the object will undergo straight line motion. Phys 250 Ch 5 p 3

Uniform Circular Motion motion in a circle at constant speed centripetal force Fc and centripetal acceleration ac is always directed towards the center centripetal force and acceleration have constant magnitudes the period T of the motion is the time to make one orbit the frequency f is the number of complete revolutions per unit time f=1/T Phys 250 Ch 5 p 4

Example: A bicycle racer rides with a constant speed around a circular track 25 m in diameter. What is the centripetal acceleration of the bicycle if its speed is 6. 0 m/s? Example: A grinding wheel with a 25. 4 cm diameter spins at a rate of 1910 revolutions per minute. What is the linear speed of a poihnt on the rim? What is the acceleration of a point on the rim? Phys 250 Ch 5 p 5

Example: What is the centripetal acceleration of the moon as it circles the earth? Its orbital period is 27. 3 days and its orbital radius is 3. 84 E 8 m. Phys 250 Ch 5 p 6

Angular velocity an object which rotates an angle q in a time t has an average angular velocity w : usually rad/s but sometime rpm, rps For a particle traveling in uniform circular motion r q Phys 250 Ch 5 p 7 s

Example: An amusement park ride carries passengers in a circular path 7. 70 m in radius. The ride makes a complete rotation every 4. 00 s. What is the angular velocity of the passengers? What is the angular acceleration of the passengers? Example: A student ties a 0, 060 kg lead fishing weight to the end of a string and whirls it around his head in a horizontal circle. if the radius of the circle is 0. 30 m and the object moves with a speed of 2. 0 m/s, what is the horizontal component of the force that keeps the string in circular motion? What is the tension in the string? Phys 250 Ch 5 p 8

Example: A space station is to consist of a torus with an outside diameter of 1. 5 km. What period of rotation must the space station have in order to simulate earth’s gravity? Example: A centrifuge separates blood cells from blood plasma by rotating a tube at 55 rotations per second. What is the acceleration at the center of a centrifuge tube 8. 0 cm from the axis of rotation? Phys 250 Ch 5 p 9

Example: A race track designed for average speeds of 240 km/hr (66. 7 m/s) is to have a turn with a radius of 975 m. To what angle must the track be banked so that cars traveling at the design speed have no tendency to slip sideways? Phys 250 Ch 5 p 10

Gravitation History: Kepler’s Laws ellipses, equal areas and period-distance pick units for T: earth years pick units for R: Astronomical Units (AU) = Earth’s orbit radius about the sun k = 1 Gravitation as a force A fundamental force of nature (electromagnetism, weak nuclear force, strong nuclear force) Newton’s law of universal gravitation All objects interact by virtue of having mass Force is proportional to each mass Force is inversely proportional to the square of the distance The force between two 1 kg masses separated by 1 m is 6. 67 x 10 -11 N ~ 1. 5 x 10 -11 lb Phys 250 Ch 5 p 11

Example: What is the force of gravity exerted by a 1 kg object on the surface of the earth? Earth’s mass: 5. 98 E 24 kg Earth’s radius: 6. 38 E 6 m Example: Show that Kepler’s third law follows from Universal Gravitation. C Phys 250 Ch 5 p 12

Example omitted: We can use our knowledge about g and G and Universal Gravitation to determine the mass of the earth, and from there its average density Example: What would be the period of an artificial satellite orbiting just above the earth’s surface? Example: What would be the radius of a satellite in geosynchronous orbit? Phys 250 Ch 5 p 13