Circular Motion

1. Circular Motion Any object that revolves about a single axis undergoes circular motion. The line about which the rotation occurs is called the axis of rotation. Ex. Spinning a ferris wheel

2. Tangential Speed, vt • Tangential speed can be used to describe the speed of an object in circular motion. When the tangential speed is constant, the motion is called uniform circular motion. • The tangential speed depends on the distance from the object to the center of the circular path. E.g. pair of horses side by side on a carousel. The outside horse has a greater tangential speed.

3. Centripetal acceleration, ac • Centripetal acceleration is due to a change in direction. • The acceleration of a ferris wheel car moving in a circular path and at constant speed is due to a change in direction, and is termed centripetal acceleration. • Centripetal acceleration-the acceleration of an object in uniform circular motion.

4. ac=vt2/r • Sample Problem A: A test car moves at a constant speed around a circular track. If the car is 48.2m from the track’s center and has an ac of 8.05m/s2, what is the car’s tangential speed?

5. ac • Given: • r=48.2m ac=8.05m/s2 • vt-=? • 8.05m/s2=vt2/48.2m=19.7m/s

6. Tangential Acceleration • Tangential acceleration is acceleration due to a change in speed. • A car moving in a circle has ac. If the speed of the car changes, it also has tangential acceleration.

7. Fc, centripetal force • Centripetal force is the net force directed toward the center of an object’s circular path. Newton’s Second Law applies. • Fc=mvt2/r • Sample Problem B: • A pilot is flying a small plane at 56.6m/s in a circular path with a radius of 188.5m. The centripetal force needed to maintain the plane’s circular motion is 1.89x104N. What is the plane’s mass?

8. Solution: • Given: r=188.5m Fc=1.89x104N • Vt=56.5m/s m=? • 1.89x104N=m(56.6m/s)/188.5m=1110kg

9. More on Fc • It acts at right angles to an object’s circular motion, so the force changes the direction of the object’s velocity. Without centripetal force, the object stops moving in a circular path and leads to a straight path that is tangent to the circle.

10. Newton’s Law of Universal Gravitation • Gravitational force is the mutual force of attraction between particles of matter. • Orbiting objects are in free fall- Newton observed that if an object were projected at just the right speed, the object would fall down toward Earth in just the same way that Earth curved out from under it. So, it would orbit the Earth. A gravitational attraction between Earth and our sun keeps Earth in its orbit around the sun.

11. What does Fgrav depend on? • It depends on the masses and the distance. • Newton’s Law of Universal Gravitation: Fg=G m1m2/r2; G=constant of universal gravitation • What is the value of G? 6.673x10-11Nxm2/kg2

12. Newton demonstrated that the gravitational force that a spherical mass exerts on a particle outside the sphere would be the same if the entire mass of the sphere were concentrated at the sphere’s center. Gravitational force acts between all masses. It always attracts objects to one another. E.g. the force that the moon exerts on the Earth is equal and opposite to the force that Earth exerts on the moon. (Example of Newton’s Third Law). Universal Gravitation con’t…

13. Universal Gravitation… • Earth’s acceleration is so small that it cannot be detected for its mass is so large and acceleration is inversely proportional to mass, the Earth’s acceleration is negligible. • Sample Problem C: 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 8.92x10-11N.

14. Sample Problem C con’t… • Given: m1=0.300-kg m2=0.400-kg Fg=8.92x10-11N G=6.673x10-11Nxm2/kg2 8.92x10-11N=6.673x10-11Nxm2/kg2 (0.300-kg)(0.400-kg)/r2 r=0.30m

15. Newton’s Law of Universal Gravitation accounts for ocean tides. • High and low tides are partly due to the gravitational force exerted on Earth by its moon.

16. Henry Cavendish • In 1798, determined the value of G via experimentation. He took two small spheres that were fixed to the ends of a suspended light rod, and attracted to two large spheres by gravitational force. Once you have the value of G, it can be used to determine Earth’s mass.

17. Gravitational Field • A gravitational field is an interaction between a mass and the gravitational field created by other masses. Earth’s gravitational field can be explained by gravitational field strength, g. The value of g is equal to the magnitude of the gravitational force exerted on a unit mass at that point, or g=Fg/m. • Gravitational filed strength is equal to free fall acceleration, however, they are not the same thing. E.g. object hanging from a spring scale.

18. Weight changes with location. • Here, weight is mass times gravitational field strength. • Fg=GmmE/r2 • g=Fg/m=GmE/r2 • What does gravitational field strength depend on? Mass and distance • On the surface of any planet, the value of g will depend on the planet’s m and r, and so will your weight.

19. Motion in Space • Claudius Ptolemy’s view on motion- Planets travel in small circles called epicycles while simultaneously traveling in larger circular orbits. • Nicolaus Copernicus published a book and proposed that Earth and other planets orbit the sun in perfect circles.

20. Kepler and planetary motion • Tycho Brahe-an astronomer who made precise observations about the planets and stars. Some of his data did not have face validity with the model of Copernicus. • Johannes Kepler (astronomer): did work to reconcile Copernican theory with the data of Brahe. He developed three laws of planetary motion:

21. Kepler’s Laws • First Law: Each planet travels in an elliptical orbit around the sun, and the sun is at one of the focal points. • Second Law: An imaginary line drawn from the sun to any planet sweeps out equal areas in equal time intervals. • Third Law: The square of a planet’s orbital period (T2) is proportional to the cube of the average distance between the planet and the sun or T2 is roughly r3.

22. What did Kepler find? • He concluded the 1st law while examining Mars and found that the planet’s orbits are ellipses and not circles. • The 3rd law relates to the orbital period and mean distance for two orbiting planets as follows: • T=Period, T12/T22=r13/r23

23. Kepler continued… • Kepler’s Third law applies to satellites orbiting Earth, including our moon. (r is the distance between the orbiting satellite and the Earth in this situation). • Newton utilized Kepler’s laws to support and validate his law of gravitation. Newton proved that if force is inversely proportional to the distance squared, the resulting orbit must be an ellipse or circle. He also portrayed that his law could be utilized to validate Kepler’s third law.

24. Rotational Motion and Torque • The motion of a rotating rigid object (e.g. a football spinning as it flies through the air). If the only force acting on the football is gravity, the football spins around a point called its center of mass. As it moves through the air, its center of mass follows a parabolic path.

25. Torque is the quantity that measures the ability of a force to rotate an object around some axis. • Example: a cat flap door • The perpendicular distance from the axis of rotation to a line drawn along the direction of force is called the lever arm. The lever arm depends on the angle. • A force applied to an extended object can produce torque. This torque, in turn, causes the object to rotate.

26. Torque calculation: • Torque=Fd sin theta • Net torque = the sum of all torques

27. Machines and Efficiency • Machine-a device used to multiply forces or simply change the direction of forces. The law of conservation of energy underlies every machine. • Lever-a simple machine. • Fulcrum-pivot point of a lever. • Mechanical advantage-the ratio of output force to input force for a machine. • Pulley-a kind of lever that can be used to change the direction of a force. • No machine can put out more energy than is put into it.