Physics Chapter 7 What To Expect In this chapter, you will learn how to describe circular motion and the forces associated with it, including the force due to gravity. Circular Motion Newton’s Law of Universal Gravitation Motion in Space Torque and Simple Machines Circular Motion and Gravitation Centripetal Force Describing a Rotating System Centripetal Acceleration Any object that revolves about a single axis undergoes circular motion Vocabulary The line about which the rotation occurs is called the axis of rotation. the acceleration directed toward the center of a circular path Centripetal acceleration is due to a change in direction Tangential speed depends on distance Tangential speed (vt) can be used to describe the speed of an object in circular motion. When the tangential speed is constant, the motion is described as uniform circular motion Remember! Problems A test car moves at a constant speed around a circular track. If the car is 48.2 m from the track’s center and has a centripetal acceleration of 8.05 m/s2,what is the car’s tangential speed? 1. A rope attaches a tire to an overhanging tree limb. A girl swinging on the tire has a centripetal acceleration of 3.0 m/s2. If the length of the rope is 2.1 m, what is the girl’s tangential speed? 2. As a young boy swings a yo-yo parallel to the ground and above his head, the yo-yo has a centripetal acceleration of 250 m/s2. If the yo-yo’s string is 0.50 m long, what is the yo-yo’s tangential speed? 3. A dog sits 1.5 m from the center of a merry-go-round. The merry-goround

is set in motion, and the dog’s tangential speed is 1.5 m/s.What is the dog’s centripetal acceleration? 4. A race car moving along a circular track has a centripetal acceleration of 15.4 m/s2. If the car has a tangential speed of 30.0 m/s, what is the distance between the car and the center of the track? Tangential acceleration is due to a change in speed an acceleration due to a change in speed is called tangential acceleration One More Thing... Problems A pilot is flying a small plane at 56.6 m/s in a circular path with a radius of 188.5 m. The centripetal force needed to maintain the plane’s circular motion is 1.89 × 10^4 N.What is the plane’s mass? 1. A 2.10 m rope attaches a tire to an overhanging tree limb. A girl swinging on the tire has a tangential speed of 2.50 m/s. If the magnitude of the centripetal force is 88.0 N, what is the girl’s mass? 2. A bicyclist is riding at a tangential speed of 13.2 m/s around a circular track. The magnitude of the centripetal force is 377 N, and the combined mass of the bicycle and rider is 86.5 kg.What is the track’s radius? 3. A dog sits 1.50 m from the center of a merry-go-round and revolves at a tangential speed of 1.80 m/s. If the dog’s mass is 18.5 kg, what is the magnitude of the centripetal force on the dog? 4. A 905 kg car travels around a circular track with a circumference of 3.25 km. If the magnitude of the centripetal force is 2140 N, what is the car’s tangential speed? Centripetal force is necessary for circular motion Because centripetal force acts at right angles to an object’s circular motion, the force changes the direction of the object’s velocity centrifugal force centripetal force Inertia is often misinterpreted as a force Gravitational force Applying the law of gravitational force the mutual force of attraction

between particles of matter Orbiting objects are in free fall Planets Satellites People Apparent Weightlessness Gravitational force depends on the masses and the distance G is called the constant of universal gravitation Outside = Center Gravitational force acts between all masses Gravitational force always attracts objects to one another Gravitational force exists between any two masses, regardless of size. Even those masses that are really small Problems: Find the distance between a 0.300 kg billiard ball and a 0.400 kg billiard ball if the magnitude of the gravitational force between them is 8.92 10^-11 N. 1. What must be the distance between two 0.800 kg balls if the magnitude of the gravitational force between them is equal to that in the previous problem? 2. Mars has a mass of about 6.4 × 10^23 kg, and its moon Phobos has a mass of about 9.6 × 10^15 kg. If the magnitude of the gravitational force between the two bodies is 4.6 × 10^15 N, how far apart are Mars and Phobos? 3. Find the magnitude of the gravitational force a 66.5 kg person would experience while standing on the surface of each of the following planets: Newton’s law of gravitation accounts for ocean tides Cavendish finds the value of G and Earth’s mass Gravity is a field force Gravitational field strength equals free-fall acceleration Weight changes with location Gravitational mass equals inertial mass The tides result from the difference between the gravitational force at Earth’s surface and at Earth’s center On the side of Earth that is nearest to the moon, the moon’s gravitational force is greater than it is at Earth’s center (because gravitational force decreases with distance) On this side, all mass is still pulled toward the moon, but the water is pulled least In and out 4X a day 1798 Cavendish Experiment masses create a gravitational field in space.

(Similarly, charged objects generate an electric field.) A gravitational force is an interaction between a mass and the gravitational field created by other masses. What about potential energy? At any point, Earth’s gravitational field can be described by the gravitational-field strength, abbreviated g. The value of g at any given point is equal to the acceleration due to gravity a=F/m g=Fg/m when you hang an object from a spring scale, you are measuring gravitational field strength Note: field strength and free-fall acceleration are equivalent but not the same New definition: weight is mass times gravitational field strength Thus, as your distance from Earth’s center increases, the value of g decreases, so your weight also decreases Because gravitational field strength equals free-fall acceleration free-fall acceleration on the surface of Earth likewise depends only on Earth’s mass and radius. it does not depend on an object's mass!!! inertial mass vs. gravitational mass inertial mass = gravitational mass Rotational Motion rotation of a rigid object center of mass the point about which an object rotates not always at the center of the object! Rotational and translational motion can be separated The Magnitude of a Torque Torque a quantity that measures the ability of a force to rotate an object around some axis Lever arm the perpendicular distance from the axis of rotation to a line drawn along the direction of the force Torque depends on the force and the lever arm The lever arm depends on the angle The quantity d sin (theta) is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force. note: d sin (theta) is the lever arm Therefore The Sign of Torque Torque, like displacement and force, is a vector quantity Torque (+ or -)? positive if the rotation is counterclockwise negative if the rotation is clockwise Problems A basketball is being pushed by two players during tip-off. One player exerts an upward force of 15 N at a perpendicular distance of 14 cm from the axis of rotation. The second player applies a downward force of 11 N at a perpendicular distance of 7.0 cm from the axis of rotation. Find the net torque acting on the ball about its center of mass. 1. Find the magnitude of the torque produced by a 3.0 N force applied to a door at a perpendicular distance of 0.25 m from the hinge. 2. A simple pendulum consists of a 3.0 kg point mass hanging at the end of a 2.0 m long light string that is connected to a pivot point.

a. Calculate the magnitude of the torque (due to gravitational force) around this pivot point when the string makes a 5.0° angle with the vertical.

b. Repeat this calculation for an angle of 15.0°. 3. If the torque required to loosen a nut on the wheel of a car has a magnitude of 40.0 N•m, what minimum force must be exerted by a mechanic at the end of a 30.0 cm wrench to loosen the nut? Types of Simple Machines A machine is any device that transmits or modifies force, usually by changing the force applied to an object All machines are combinations or modifications

of six fundamental types of machines, called simple machines. lever, pulley, inclined plane, wheel and axle, wedge, and screw Simple Machines Using simple machines purpose change the direction or magnitude of an input force Compare how large the output force is relative to the input force, in a ration called: mechanical advantage IF Then Machines can alter the force and the distance moved Efficiency is a measure of how well a machine works Real machines dissipate energy! The efficiency of a machine is the ratio of useful work output to work input the mechanical efficiency of an ideal machine is 1, or 100 percent Because of friction, the efficiency of real machines is always less than 1. Problems 2. At what distance above Earth would a satellite have a period of 125 min? Newton used Kepler's laws to spport his law of gravitation. The color-enhanced image of Venus shown here was compiled

from data taken by Magellan, the first planetary

spacecraft to be launched from a space shuttle.During the

spacecraft’s fifth orbit around Venus,Magellan traveled at

a mean altitude of 361 km. If the orbit had been circular,

what would Magellan’s period and speed have been? Kepler's laws are consistent with Newton's law of gravitation Kepler's Laws 1. Find the orbital speed and period that the Magellan satellite from Sample Problem D would have at the same mean altitude above Earth, Jupiter, and Earth’s moon. Newton proved that if force is inveresly proportional to distance swared, the resulting orbit must be an ellipse or a circle. Nicolaus Copernicus propsed that the earth and other planets orbited the sun in perfect circles Kepler's three laws describe the motion of the planets Note: the poprtionality constant depends on the mass of the central object Kepler's third law desribes orbital period The constant of proportionality between these two variables turns out to be 4pi^2/Gm, where m is the mass of the object being orbited. The speed of an object that is in a circular orbit depends on the same factors that the period does. Assumption circular orbit provides a close approximation for real orbits in our solar system because all planets except Mercury and Pluto have orbits that are nearly circular. Note: For an artificial satellite orbiting Earth, r is equal to Earth’s mean

radius plus the satellite’s distance from Earth’s surface (its “altitude”).

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