Circular motion and Gravitation

1. Circular motion and Gravitation Chapter 6 Physics Chapter 6

2. Dynamics of Circular Motion • Remember that when an object moves in a circle with a constant speed, its acceleration is always directed toward the center of the circle. Physics Chapter 6

3. Circular motion • The acceleration in circular motion is caused by a net force, just like any acceleration. • If this force is removed, the object will continue along a straight path with a constant velocity. Physics Chapter 6

4. Centripetal force • The net force that causes centripetal acceleration. • Not a separate force • Does not appear in the free body diagram. Physics Chapter 6

5. Newton’s second law Physics Chapter 6

6. Example • A hockey puck with mass 0.500 kg revolves in a uniform circle on the frictionless ice. It is attached to a 0.400 m long cord nailed into the ice. It makes one revolution per second. • What is the force, F, exerted by the cord on the puck? Physics Chapter 6

7. Example • You have a summer job as part of an automobile design team. You are testing new prototype tires to see whether or not the tires perform as well as predicted. In a skid test, a new BMW 530i was able to travel at a constant speed in a circle of radius 45.7 m in 15.2 s without skidding. • What was its speed, v? • What is the acceleration? • Assuming air drag and rolling friction to be negligible, what is the minimum value for the coefficient of static friction between the tires and the road? Physics Chapter 6

8. Example • A curve of radius 30 m is banked at an angle q. Find q for which a car can round the curve at 40 km/h even if the road is covered with ice so that friction is negligible. Physics Chapter 6

9. Vertical circles • Be careful about weight • Apparent weight – what you feel like you weigh • Apparent weight = normal force Physics Chapter 6

10. Normal Force • Can be equal to, less than, or greater than weight • If contact with the surface is lost, normal force is zero. Physics Chapter 6

11. Example • You swing a cup of water with mass m in a vertical circle of radius r. If its speed is vt at the top of the circle, find • The force exerted on the water by the cup at the top of the circle • The minimum value for vt for the water to remain in the cup. Physics Chapter 6

12. On your own • What is the force exerted by the cup on the water at the bottom of the circle, where the pail’s speed is vb? Physics Chapter 6

13. Universal Law of Gravitation • A gravitational force acts between every pair of particles in the universe. • Gravitational forces are always attractive. • Published by Newton in 1687. Physics Chapter 6

14. Universal Law of Gravitation • The m’s are the masses of the two objects. • r is the distance between their centers of mass. • G is a fundamental physical constant called the gravitational constant. Physics Chapter 6

15. Value of G • Newton didn’t have sensitive enough equipment to measure G. • In 1798, Henry Cavendish used a torsion balance to measure G. • In SI units, G is 6.67 x 10-11 N-m2/kg2 Physics Chapter 6

16. Spherical objects • The gravitational interaction between two objects having spherical symmetry is the same as though all the mass was concentrated at the center. • So, we can treat them as particles. Physics Chapter 6

17. Superposition of Forces • If each of two masses exerts a force on a third, the total force on the third mass is the vector sum of the individual forces from the first two. Physics Chapter 6

18. Example • Particle 1 has a mass of 6.0 kg and is located at the origin. Particle 2 has a mass of 4.0 kg and is located at (0.0 , 2.0) cm. Particle 3 has a mass of 4.0 kg and is located at (-4.0 , 0.0) cm. Find the net gravitational force on particle 1. • 4.1 x 10-6 N @ 104° Physics Chapter 6

19. Gravitational Forces • Between ordinary, household objects, they are small. • Between astronomical objects they are large. • Gravity is what keeps the universe running – orbits, energy output of stars, etc. Physics Chapter 6

20. Weight • According to the Universal law of gravitation, an object of mass m on the surface of the earth would have the following weight: Physics Chapter 6

21. Weight • Setting this equal to mg, Physics Chapter 6

22. Weight • If an object is a distance (r-RE) above the surface of the earth, then it is at a distance r above the center of the earth, and • Since r > RE, g < 9.8 m/s2 Physics Chapter 6

23. Escape speed • In order for a space shuttle to leave the earth, it must have enough speed to stay in the air long enough that the Earth curves away from it faster than it falls. • We can calculate the minimum velocity required to do this. Physics Chapter 6

24. Motion of Satellites • If a satellite is traveling in a circular orbit (which most of them do), the only force acting on it is gravity. Physics Chapter 6

25. Motion of satellites • This tells us that if you want a satellite to orbit with a certain speed, it must be at a certain radius. • Doesn’t depend on mass – apparent weightlessness of astronauts. Physics Chapter 6

26. Period of circular orbits • For a circular orbit, • If you set this equal to the velocity equation we just found, Physics Chapter 6

27. Satellites • Not always manmade • Don’t always orbit Earth • Moons • Rings of Saturn, Uranus, and Neptune Physics Chapter 6

28. You try • You want to place a communications satellite into a circular orbit 300 km above the earth’s surface. What must be its speed, its period, and its radial acceleration? The earth’s radius is 6.38 x 106 m and its mass is 5.98 x 1024 kg. Physics Chapter 6