Transcript Slide 1
Chapter 5 Circular Motion; Gravitation Units of Chapter 5 •Kinematics of Uniform Circular Motion •Dynamics of Uniform Circular Motion •Highway Curves, Banked and Unbanked •Nonuniform Circular Motion •Centrifugation •Newton’s Law of Universal Gravitation Units of Chapter 5 •Gravity Near the Earth’s Surface; Geophysical Applications •Satellites and “Weightlessness” •Kepler’s Laws and Newton’s Synthesis •Types of Forces in Nature 5-1 Kinematics of Uniform Circular Motion Uniform circular motion: motion in a circle of constant radius at constant speed Instantaneous velocity is always tangent to circle. Centripetal Acceleration, cont. • Centripetal refers to “center-seeking” • The direction of the velocity changes • The acceleration is directed toward the center of the circle of motion Centripetal Acceleration, cont. a = Δv (eq. I) Δt By similar triangles Δv = Δs v r Therefore: Δv = Δs v r Sub into eq. I a = Δs v r Δt = v2 r Since Δs = v Δt Centripetal Acceleration and Angular Velocity • The angular velocity and the linear velocity are related (v = ωr) • The centripetal acceleration can also be related to the angular velocity a c = v2 r = (rω)2 r = rω2 aC r 2 5-1 Kinematics of Uniform Circular Motion This acceleration is called the centripetal, or radial, acceleration, and it points towards the center of the circle. 5-2 Dynamics of Uniform Circular Motion We can see that the force must be inward by thinking about a ball on a string: 5-2 Dynamics of Uniform Circular Motion For an object to be in uniform circular motion, there must be a net force acting on it. We already know the acceleration, so can immediately write the force: ΣFr = Fc = mac =mv2 r 5-2 Dynamics of Uniform Circular Motion There is no centrifugal force pointing outward; what happens is that the natural tendency of the object to move in a straight line must be overcome. If the centripetal force vanishes, the object flies off tangent to the circle. 5-3 Highway Curves, Banked and Unbanked When a car goes around a curve, there must be a net force towards the center of the circle of which the curve is an arc. If the road is flat, that force is supplied by friction. Fc=Ffriction 5-3 Highway Curves, Banked and Unbanked If the frictional force is insufficient, the car will tend to move more nearly in a straight line, as the skid marks show. 4-8 Applications Involving Friction, Inclines The static frictional force increases as the applied force increases, until it reaches its maximum. Then the object starts to move, and the kinetic frictional force takes over. 5-3 Highway Curves, Banked and Unbanked Banking the curve can help keep cars from skidding. In fact, for every banked curve, there is one speed where the entire centripetal force is supplied by the horizontal component of the normal force, and no friction is required. This occurs when: 5.2 Centripetal Acceleration Example 3: The Effect of Radius on Centripetal Acceleration The bobsled track contains turns with radii of 33 m and 24 m. Find the centripetal acceleration at each turn for a speed of 34 m/s. Express answers as multiples of g 9.8 m s 2 . 5.2 Centripetal Acceleration ac v r 2 ac 2 34 m s 35 m s 2 3.6 g ac 2 34 m s 48m s 2 4.9 g 33 m 24 m 5.3 Centripetal Force Recall Newton’s Second Law When a net external force acts on an object of mass m, the acceleration that results is directly proportional to the net force and has a magnitude that is inversely proportional to the mass. The direction of the acceleration is the same as the direction of the net force. a F m F ma 5.3 Centripetal Force Thus, in uniform circular motion there must be a net force to produce the centripetal acceleration. The centripetal force is the name given to the net force required to keep an object moving on a circular path. The direction of the centripetal force always points toward the center of the circle and continually changes direction as the object moves. 2 v Fc m ac m r Centripetal force can be caused by, tension, friction, or gravitational attraction. In which case: Fc = T Fc = Ffr Fc = F g 5.3 Centripetal Force Example 5: The Effect of Speed on Centripetal Force The model airplane has a mass of 0.90 kg and moves at constant speed on a circle that is parallel to the ground. The path of the airplane and the guideline lie in the same horizontal plane because the weight of the plane is balanced by the lift generated by its wings. Find the tension in the 17 m guideline for a speed of 19 m/s. 2 v Fc T m r 19 m s T 0.90 kg 2 17 m 19 N 5.4 Banked Curves On an unbanked curve, the static frictional force provides the centripetal force. A car rounds a curve having a 100m radius Ffr Travelling at 20m/s. What is the minimum Coefficient of friction between the tires and the road required? Fc = Ffr =Fn mv2 = mg r = v2 = (20m/s)2 gr (9.8m/s2)(100m) = 0.41 Fn W = mg 5.4 Banked Curves On a frictionless banked curve, the centripetal force is the horizontal component of the normal force. The vertical component of the normal force balances the car’s weight. 5.4 Banked Curves 2 v Fc FN sin m r FN cos mg 5.4 Banked Curves 2 v FN sin m r FN cos mg 2 v tan rg 5.4 Banked Curves Example 8: The Daytona 500 The turns at the Daytona International Speedway have a maximum radius of 316 m and are steeply banked at 31 degrees. Suppose these turns were frictionless. At what speed would the cars have to travel around them? 2 v tan rg v 316 m 9.8 m v rg tan s 2 tan 31 43 m s 96 mph 5-6 Newton’s Law of Universal Gravitation If the force of gravity is being exerted on objects on Earth, what is the origin of that force? Newton’s realization was that the force must come from the Earth. He further realized that this force must be what keeps the Moon in its orbit. 5-6 Newton’s Law of Universal Gravitation The gravitational force on you is one-half of a Third Law pair: the Earth exerts a downward force on you, and you exert an upward force on the Earth. When there is such a disparity in masses, the reaction force is undetectable, but for bodies more equal in mass it can be significant. 5-6 Newton’s Law of Universal Gravitation Therefore, the gravitational force must be proportional to both masses. By observing planetary orbits, Newton also concluded that the gravitational force must decrease as the inverse of the square of the distance between the masses. In its final form, the Law of Universal Gravitation reads: (5-4) where 5-6 Newton’s Law of Universal Gravitation The magnitude of the gravitational constant G can be measured in the laboratory. This is the Cavendish experiment. 5-7 Gravity Near the Earth’s Surface; Geophysical Applications Now we can relate the gravitational constant to the local acceleration of gravity. We know that, on the surface of the Earth: Solving for g gives: (5-5) Now, knowing g and the radius of the Earth, the mass of the Earth can be calculated: Example A 10kg mass and a 15kg mass are separated by 1.5m. Find the force of attraction between the two masses. Gravitational Force and Satellites • Orbiting objects are in free fall. • To see how this idea is true, we can use a thought experiment that Newton developed. Consider a cannon sitting on a high mountaintop. Each successive cannonball has a greater initial speed, so the horizontal distance that the ball travels increases. If the initial speed is great enough, the curvature of Earth will cause the cannonball to continue falling without ever landing. 5-8 Satellites and “Weightlessness” Satellites are routinely put into orbit around the Earth. The tangential speed must be high enough so that the satellite does not return to Earth, but not so high that it escapes Earth’s gravity altogether. 5-8 Satellites and “Weightlessness” The satellite is kept in orbit by its speed – it is continually falling, but the Earth curves from underneath it. 5.5 Satellites in Circular Orbits There is only one speed that a satellite can have if the satellite is to remain in an orbit with a fixed radius. 5.5 Satellites in Circular Orbits Fg = Fc 2 m ME v G 2 m r r GM E v r 5.5 Satellites in Circular Orbits Example 9: Orbital Speed of the Hubble Space Telescope Determine the speed of the Hubble Space Telescope orbiting at a height of 598 km above the earth’s surface. v 6.6710 11 N m kg 5.9810 kg 6.38 106 m 598103 m 7.56 103 m s 2 2 16900mi h 24 5.5 Satellites in Circular Orbits GM E 2 r v r T 2 r T GM E 32 = 42r3 GM 5-9 Kepler’s Laws and Newton's Synthesis Kepler’s laws describe planetary motion. 1. The orbit of each planet is an ellipse, with the Sun at one focus. 5-9 Kepler’s Laws and Newton's Synthesis 2. An imaginary line drawn from each planet to the Sun sweeps out equal areas in equal times. Kepler’s Third Law • The square of the orbital period of any planet is proportional to cube of the average distance from the Sun to the planet. T Kr 2 3 – T = circumference of orbit orbital speed – For orbit around the Sun, KS = 2.97x10-19 s2/m3 – K is independent of the mass of the planet • K = 42 GM Example: A planet is in orbit 109 meters from the center of the sun. Calculate its orbital period and velocity. Ms=1.991 x 1030 kg 5-9 Kepler’s Laws and Newton's Synthesis The ratio of the square of a planet’s orbital period is proportional to the cube of its mean distance from the Sun. 5.5 Satellites in Circular Orbits 5.5 Satellites in Circular Orbits Global Positioning System 2 r = T GM E 32 42r3 GM T 24 hours T = (24 hours)(3600s/hour) = 86400s r= 3 T2GMe 42 r = (86400s)2(6.67 x 10-11 Nm2/kg2)(5.98 x 1024kg) 42 3 r = 42250474m = distance from center of the earth to GPS r = Re + h h = r – Re = 42250474m – 6380000m = 35870474m = 22,300 mi 5.6 Apparent Weightlessness and Artificial Gravity Example 13: Artificial Gravity At what speed must the surface of the space station move so that the astronaut experiences a push on his feet equal to his weight on earth? The radius is 1700 m. 2 v Fc m m g r v rg 1700m9.80 m s = 130 m/s 2 5.7 Vertical Circular Motion 2 1 v FN 1 m g m r FN 2 FN 4 2 2 v m r 2 4 v m r 2 3 v FN 3 m g m r Example A satellite orbits the earth at an altitude of 1000km. What must the velocity of the satellite be in order for it to maintain a circular orbit. Once in circular orbit, what happens if something causes the satellite to speed up or slow down. 5-8 Satellites and “Weightlessness” Objects in orbit are said to experience weightlessness. They do have a gravitational force acting on them, though! The satellite and all its contents are in free fall, so there is no normal force. This is what leads to the experience of weightlessness. 5-10 Types of Forces in Nature Modern physics now recognizes four fundamental forces: 1. Gravity 2. Electromagnetism 3. Weak nuclear force (responsible for some types of radioactive decay) 4. Strong nuclear force (binds protons and neutrons together in the nucleus) 5-10 Types of Forces in Nature So, what about friction, the normal force, tension, and so on? Except for gravity, the forces we experience every day are due to electromagnetic forces acting at the atomic level. Summary of Chapter 5 • An object moving in a circle at constant speed is in uniform circular motion. • It has a centripetal acceleration • There is a centripetal force given by •The centripetal force may be provided by friction, gravity, tension, the normal force, or others. Summary of Chapter 5 • Newton’s law of universal gravitation: •Satellites are able to stay in Earth orbit because of their large tangential speed. 5-4 Nonuniform Circular Motion If an object is moving in a circular path but at varying speeds, it must have a tangential component to its acceleration as well as the radial one. 5-4 Nonuniform Circular Motion This concept can be used for an object moving along any curved path, as a small segment of the path will be approximately circular. 5-5 Centrifugation A centrifuge works by spinning very fast. This means there must be a very large centripetal force. The object at A would go in a straight line but for this force; as it is, it winds up at B. 5-7 Gravity Near the Earth’s Surface; Geophysical Applications The acceleration due to gravity varies over the Earth’s surface due to altitude, local geology, and the shape of the Earth, which is not quite spherical. 5-8 Satellites and “Weightlessness” More properly, this effect is called apparent weightlessness, because the gravitational force still exists. It can be experienced on Earth as well, but only briefly: 5-9 Kepler’s Laws and Newton's Synthesis Kepler’s laws can be derived from Newton’s laws. Irregularities in planetary motion led to the discovery of Neptune, and irregularities in stellar motion have led to the discovery of many planets outside our Solar System. 5.6 Apparent Weightlessness and Artificial Gravity Conceptual Example 12: Apparent Weightlessness and Free Fall In each case, what is the weight recorded by the scale?