Circular Motion Gravitation
Circular Motion Uniform circular motion is motion in a circle at constant speed. Centripetal force is the force that keeps an object moving in a circle. Centripetal acceleration, or radial acceleration, is the acceleration toward the center of the circular path. The period is the time it takes for one complete revolution.
Circular Motion A revolution is one trip around the path. A rotation is one turn on an axis. Circular motion can be described in linear terms or in rotational terms. Both are effective at describing the motion but the units are different.
Circular Motion Linear distance d m velocity v m/s acceleration a m/s 2 time t s Rotational distance θ rad velocity ω rad/s acceleration α rad/s 2 time t s
Conversions from linear to rotational Θ = d/r ω = v/r α = a/r
Linear Equations of Rotational Motion V = 2πr/T where T is the period. F c = ma c a c = v 2 /r Thus, F c = mv 2 /r And F c = m(4π 2 r/t 2)
Dynamics of Uniform Circular Motion Ball moving in a vertical circle. F T mg F T Ball moving in a horizontal circle. mg F c
Dynamics of Uniform Circular Motion For an object moving in a horizontal circle F T = F c. Gravity and tension provide the centripetal acceleration. For an object moving in a vertical circle At the top of the circle F T + mg = ma c At the bottom of the circle F T - mg = ma c Tension and gravity acting in opposite directions provide the centripetal acceleration. The minimum speed will be obtained when the tension is zero so (gr) 1/2 = v min
Dynamics of Uniform Circular Motion If the object is moving in a vertical circle then T = 2π(length/g) 1/2 If the object is moving in a horizontal circle then T = 2π(lcosθ/g) 1/2
A Car Rounding a Curve F N F f a F g
Car Rounding a Banked Curve y x F N F g
Nonuniform Circular Motion If the net force is not directed toward the center but is at an angle, the force has two components. The component directed toward the center gives the centripetal acceleration and keeps the object going in a circle. The component tangent to the circle acts to increase or decrease the speed and gives the tangential acceleration.
Changing Circular Motion Torque, τ, plays the role of force in circular motion. Torque is equal to the product of the force and the lever arm distance which is perpendicular to the applied force. In some respects, it is like work. The formula is τ = F x d
Newton s Law of Universal Gravitation According to the Greeks, objects have a builtin desire to fall. According to Galileo and Newton, a force called gravity exists. It is an attractive force between the Earth and other objects. Newton studied the motion of the planets and the moon. He wondered what kept the moon in its orbit around the Earth. The idea that gravity extends throughout the universe is credited to Newton who is said to thought of it when an apple fell on his head.
The Falling Moon Newton compared the falling apple to the falling moon. The moon falls in the sense that it falls below the straight line path that inertia would carry it on if no forces were acting on it. He used a cannonball example to prove his point. The cannonball eventually would have tangential velocity sufficient to carry it around the earth.
The Falling Moon Newton tested his hypothesis by reasoning that the mass of an object should not affect how it falls. How far an object falls should only relate to its distance from Earth s center. In fact, it is related to the square of the distance from Earth s center. The moon accelerates to the Earth at about 1/3600 g.
The Falling Earth Why does the earth not crash into the sun? Which attraction is greater, the sun for the earth or the earth for the sun? If there is an attraction for all objects, why do we not feel gravitated towards large buildings and other massive objects?
Newton s Law of Universal Gravitation Newton s Law states that every object attracts every other object with a force that is directly proportional to the mass of each object. He also deduced that the force decreases as the square of the distance between the objects increases. F = Gm 1 m 2 /d 2, where G is the universal gravitation constant, 6.67 x 10-11 Nm 2 /kg 2. The gravitation constant was measured by Henry Cavendish.
Gravitational Interactions The value for g of a planet can be found by g = GM/r 2, where G is the universal gravitation constant, M is the mass of the planet, and r is the planet s radius. The acceleration of objects on the surface of the moon is only 1/6 of 9.8 m/s 2. Is it correct to say that the mass of the moon is therefore 1/6 the mass of Earth? The value of v for an object can be found by v = (Gm/r) 1/2
Newton s Law of Universal Gravitation The force between you and any object is usually very small. The force of attraction between you and the earth is. Your weight depends on your distance from the center of the earth. The closer you are to the center, the smaller will be your weight. This is due to the change in the mass and radius of the planet. Cavendish went so far as to mass the earth. Its mass is 5.98 x 10 24 kg.
Newton s Law of Universal Gravitation The distance that an object is from the center of the Earth affects its acceleration due to gravity. Earth s radius is 6.38 x 10 6 m. That is the average radius. If one is on a mountain that is very high, its height must be taken into consideration.
Gravitational Interactions A force field exerts a force on objects in its vicinity. A field is represented by field lines. Where the lines are closer together, the field is stronger. A gravitational field for a planet is represented by vectors which point to the center of mass.
Weight and Weightlessness Suppose you weighed in an elevator. What would be your weight if the elevator accelerated downward? What would be your weight if the elevator accelerated upward? What would be your weight if the elevator was not accelerating? What would be your weight if the elevator cable broke and the elevator fell freely?
Weight and Weightlessness Weight then is the force that you exert against a support. Weightlessness then becomes the absence of a supporting force.
History Tycho Brahe spent his life accurately predicting astronomical events. He believed that the Earth was the center of the universe. His protege Johannes Kepler believed that the sun was the center of the universe. He formulated 3 laws based on his observations of the motions of the planets.
Kepler s Laws First Law: The paths of the planets are ellipses with the center of the sun at one focus. Second Law: An imaginary line from the sun to a planet sweeps out equal areas in equal time intervals. Thus planets move fastest when they are closest to the sun. Third Law: The ratio of the squares of the periods of any two planets revolving about the sun is equal to the ratio of the cubes of their average distances from the sun.
Types of Forces Four fundamental forces Gravitational force Electromagnetic force Strong nuclear force Weak nuclear force Grand Unification Theory So far the electromagnetic force and the weak nuclear force have been united into the electroweak force.
Simple Harmonic Motion If the restoring force varies linearly with the displacement, the motion is said to be SHM. Period and amplitude are used to describe this motion. Period is the time required for one complete cycle. Amplitude is the maximum displacement an object moves from its equilibrium position.
Springs F = kx KE = 1/2 k x 2