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Chapter 7 Circular Motion and Gravitation 7.1 Angular Measure The position of an object can be described using polar coordinates—r and θ—rather than x and y. 7.1 Angular Measure The distance r extends from the origin, and the angle Θ is commonly measured counterclockwise from the positive xaxis 7.1 Angular Measure The length of the arc between r and the x-axis is known as the ―arclength‖ (see what they did there?) and is represented by the variable ‗s‘ An object ―subtends‖ an angle. In other words… the angle that the object takes up. 7.1 Angular Measure It is most convenient to measure the angle θ in radians: 7.1 Angular Measure It is extremely handy to know that… 1 rad = 57.3° 360° = 2π rad 7.1 Angular Measure 1. What is the radius of the Earth if walking around it twice is a distance of 8.02E4 km? 2. The moon is 4.08E8 m away from Earth. If it moves an angle of 1 rad through the sky, what distance has it actually moved? 7.2 Angular Speed and Velocity In analogy to the linear case, we define the average and instantaneous angular velocity: Δd v= Δt d v= t 7.2 Angular Speed and Velocity The direction of the angular velocity is along the axis of rotation, and is given by a right-hand rule. 7.2 Angular Speed and Velocity Relationship between tangential and angular speeds: This means that parts of a rotating object farther from the axis of rotation move faster. 7.2 Angular Speed and Velocity The period is the time it takes for one rotation; the frequency is the number of rotations per second. f = frequency T = period The relation of the frequency to the angular velocity: Warm-Up Questions 1. Fred runs four and a half laps around a circular track. If this took him seven minutes at a constant speed, what is his angular displacement and angular velocity? 2. Bob is swinging his 0.30 meter long ID lanyard around at 2 rev/s when he lets go, what is the tangential velocity of the ID right when it is let go? 3. The moon is 3.84x105 km from the Earth. If the distance from the space station to the Earth‘s center is is 6.4x103 km, how many orbits must they complete to have travelled a distance equal to that of a journey to the moon? 4. From #3, if it takes them 90 minutes to orbit the earth what is their angular velocity? Recap from Yesterday! r = meters Θ = radians Θ = radians t = seconds r = meters ω = rad/s s = … meters?! ω = rad/s v = m/s (why?) 7.3 Buckets of Fun What is happening? Does the water fall? What direction is the overall acceleration? What velocity or velocities does the bucket have? 7.3 Uniform Circular Motion and Centripetal Acceleration A careful look at the change in the velocity vector of an object moving in a circle at constant speed shows that the acceleration is toward the center of the circle. 7.3 Uniform Circular Motion and Centripetal Acceleration The same analysis shows that the centripetal acceleration is given by: 7.3 Uniform Circular Motion and Centripetal Acceleration The centripetal force is the mass multiplied by the centripetal acceleration. This force is the net force on the object. As the force is always perpendicular to the velocity, it does no work. Centrifuge Centripetal? Centrifugal? • The centripetal force constantly accelerates the object towards the center of rotation. • The centrifugal force has two potential definitions (one real, one not so real). • The misunderstood idea of centrifugal force is that it is a force pulling the rotating object outward. – There is no such force! • In reality, the centrifugal force is the reactive force of the object pushing against whatever is making it rotate. – Water against the bucket – The object wants to keep going with the tangential velocity, but its spinning container prevents this. • 1. The largest salami in the world, made in Norway, was more than 20 m long. If a hungry mouse ran around the salami‘s circumference with a tangential speed of 0.17 m/s, the centripetal acceleration of the mouse was 0.29 m/s2.What was the radius of the salami? • 2. The largest steerable single-dish radio telescope is located at the branch of the Max Planck Institute in Bonn, Germany. Suppose this telescope rotates about its axis with the same angular speed as Earth. The centripetal acceleration of the points at the edge of the telescope is 2.65 x 10-7m/s2. What is the radius of the telescope dish? • 3. In 1994, Susan Williams of California blew a bubble-gum bubble with a diameter of 58.4 cm. If this bubble was rigid and the centripetal acceleration of the equatorial points of the bubble were 8.50 x 102 m/s2, what would the tangential speed of those points be? • 4. An ostrich lays the largest bird egg. A typical diameter for an ostrich egg at its widest part is 12 cm. Suppose an egg of this size rolls down a slope so that the centripetal acceleration of the shell at its widest part is 0.28 m/s2.What is the tangential speed of that part of the shell? • 5. A waterwheel built in Hamah, Syria, rotates continuously. The wheel‘s radius is 20.0 m. If the wheel rotates once in 16.0 s, what is the magnitude of the centripetal acceleration of the wheel‘s edge? • 6. In 1995, Cathy Marsal of France cycled 47.112 km in 1.000 hour. Calculate the magnitude of the centripetal acceleration of Marsal with respect to Earth‘s center. Neglect Earth‘s rotation, and use 6.37 x 103 km as Earth‘s radius. 7.4 Angular Acceleration The average angular acceleration is the rate at which the angular speed changes: If the angular speed is changing, the linear speed must be changing as well. Linear Units Recap Linear... Variable Symbol Unit Displacement Velocity Acceleration Mass Force Momentum x v a m F p m m/s 2 m/s kg N kg·m/s Angular Units Recap Angular... Variable Symbol Unit Displacement Velocity Acceleration Mass Force Momentum θ ω α ?? ?? ?? rad rad/s 2 rad/s ?? ?? ?? 7.4 Angular Acceleration Ch. 7 Equations 1. Peter Rosendahl of Sweden rode a unicycle with a wheel diameter of 2.5 cm. If the wheel‘s average angular acceleration was 2.0 rad/s2, how long would it take for the wheel‘s angular speed to increase from 0 rad/s to 9.4 rad/s? 2. Jupiter has the shortest day of all of the solar system‘s planets. One rotation of Jupiter occurs in 9.83 h. If an average angular acceleration of –3.0 × 10–8 rad/s2 slows Jupiter‘s rotation, how long does it take for Jupiter to stop rotating? 3. In 1989, Dave Moore of California built the Frankencycle, a bicycle with a wheel diameter of more than 3 m. If you ride this bike so that the wheels‘ angular speed increases from 2.00 rad/s to 3.15 rad/s in 3.6 s, what is the average angular acceleration of the wheels? 4. In 1990, David Robilliard rode a bicycle on the back wheel for more than 5 h. If the wheel‘s initial angular speed was 8.0 rad/s and Robilliard tripled this speed in 25 s, what was the average angular acceleration? 5. Earth takes about 365 days to orbit once around the sun. Mercury, the innermost planet, takes less than a fourth of this time to complete one revolution. Suppose some mysterious force causes Earth to experience an average angular acceleration of 6.05 × 10–13 rad/s2, so that after 12.0 days its angular orbital speed is the same as that of Mercury. Calculate this angular speed and the period of one orbit. 6. The smallest ridable tandem bicycle was built in France and has a length of less than 40 cm. Suppose this bicycle accelerates from rest so that the average angular acceleration of the wheels is 0.800 rad/s2.What is the angular speed of the wheels after 8.40 s? Newton‘s Epiphany 7.5 Newton’s Law of Gravitation Newton’s law of universal gravitation describes the force between any two point masses: G is called the universal gravitational constant: 7.5 Newton’s Genius It was through logic, reasoning, and luck that Newton’s work led to this equation and the value for his constant, G. Using this equation he was able to calculate the mass of the Earth, which in turn allowed the mass of the moon, sun and other planets to be determined! But… how did he do it? 7.5 Newton’s Law of Gravitation Gravity provides the centripetal force that keeps planets, moons, and satellites in their orbits. We can relate the universal gravitational force to the local acceleration of gravity: 7.5 Newton’s Law of Gravitation The gravitational potential energy is given by the general expression: Einstein / Newton Higgs Particle? • Last week there was a press conference where it was announced that they have some additional evidence of a ―Higgs‖ Particle – Predicted to exist mathematically – May be responsible for giving other particles mass – huge step towards understanding how gravity actually works (not just understanding how to calculate it) 1. A giant asteroid flying towards us has an average radius of 4.3 km. If the gravitational force between the asteroid and a 5.0 kg rock at its surface is 4.5 ×10–2 N, what is the mass of the asteroid? 2. The largest turtle ever found has a mass of 621 kg. If the force of gravitational attraction between this turtle and a person with a mass of 65.0 kg is 1.0 × 10–12 N, what is the distance between them? 3. A group of ants descended on a picnic basket. The group‘s mass was estimated to be 3.2 × 109 kg. If this group was split in half and the halves separated by 1.0 × 102 m, what would the magnitude of the gravitational force between the halves be? 1. In the year 34928 Earth has won the War of the Universe by obliterating all of the other planetary bodies in the entire universe, becoming the only mass. A 5.8x104 kg starship is sent away from Earth, travelling a distance of 150 light-years away. What is the gravitational potential energy of the starship? 2. Why can‘t the potential energy ever be zero or positive when considered this way? (answer is not specific to the math) 1. The radius of the Earth is 6378 km and its mass is 5.97x1024 kg. Compare the acceleration of gravity for you on Earth to the acceleration experienced by astronauts in the International Space Station 322 km above the Earth’s surface. 2. What centripetal acceleration does the ISS experience? 3. What is the angular velocity of the ISS? 4. What is the tangential velocity of the ISS? 5. What is the period of one orbit for the ISS? “Zero Gravity” Astronauts in Earth orbit report the sensation of weightlessness. The gravitational force on them is not zero; what’s happening? “Zero Gravity” What’s missing is not the weight, but the normal force. We call this apparent weightlessness. “Artificial” gravity could be produced in orbit by rotating the satellite; the centripetal force would mimic the effects of gravity. Johannes Kepler (1571-1630) Mathematician Astronomer Astrologer Supernova of 1604 Revelation 7.6 Kepler’s Laws and Earth Satellites Kepler‘s laws were the result of his many years of observations. They were later found to be consequences of Newton‘s laws. Kepler’s first law: Planets move in elliptical orbits, with the Sun at one of the focal points. Earth‘s Orbit Earth‘s Orbit • Aphelion: • Perihelion: 152,091,221 km 147,095,260 km • Very close to circular orbit • Also changes on a 100,000 year cycle due to gravitational influences from neighboring masses (primarily Jupiter) 7.6 Kepler’s Laws and Earth Satellites Kepler’s second law: A line from the Sun to a planet sweeps out equal areas in equal lengths of time. 7.6 Kepler’s Laws and Earth Satellites Kepler’s third law: The square of the orbital period of a planet is directly proportional to the cube of the average distance of the planet from the Sun; that is, . This can be derived from Newton‘s law of gravitation, using a circular orbit. Basically: there exists a relationship between period and radius of the circular path. 7.6 Kepler’s Laws and Earth Satellites This minimum initial speed to leave orbit around a planetary body is called the escape speed. 7.6 Kepler’s Laws and Earth Satellites Any satellite in orbit around the Earth has a speed given by 7.6 Kepler’s Laws and Earth Satellites Escape Velocity Practice 1. The mass of the Earth is 5.97 1024 kg and its radius is 6371 km. Find the escape velocity from the earth. 2. Suppose the Earth shrunk suddenly to one-fourth its radius without any change in its mass. What would be the escape velocity then? 3. An imaginary planet X has a mass eight times that of the earth and radius twice that of the earth. What would be the escape velocity from this planet? 4. Your Physics textbook has a mass of _____, to what radius would you need to compress it to make a black hole? (vesc of black hole = speed of light) Summary of Chapter 7 Angles may be measured in radians; the angle is the arc length divided by the radius. Angular kinematic equations for constant acceleration: Summary of Chapter 7 Tangential speed is proportional to angular speed. Frequency is inversely proportional to period. Angular speed: Centripetal acceleration: Summary of Chapter 7 Centripetal force: Angular acceleration is the rate at which the angular speed changes. It is related to the tangential acceleration. Newton’s law of gravitation: Summary of Chapter 7 Gravitational potential energy: Kepler’s laws: 1. Planetary orbits are ellipses with Sun at one focus 2. Equal areas are swept out in equal times. 3. The square of the period is proportional to the cube of the radius. Summary of Chapter 7 Escape speed from Earth: Chapter 7 Review Pages 249-254: Questions: • • • • • • 11-15 27-32 44, 46-49, 53 66-67 77-78, 80-82, 84 91-93 7.1: Angular Measure 7.2: Angular Velocity 7.3: Centripetal Acceleration 7.4: Angular Acceleration 7.5: Newton‘s Law of Grav. 7.6: Kepler‘s Laws Mearth = 5.97x1024 kg Rearth = 6371 km Mearth = 5.97x1024 kg Rearth = 6371 km Mearth = 5.97x1024 kg Rearth = 6371 km Mearth = 5.97x1024 kg Rearth = 6371 km Mearth = 5.97x1024 kg Rearth = 6371 km Mearth = 5.97x1024 kg Rearth = 6371 km