Circular Motion

1. Circular Motion Physics chapter 7

2. Objectives: Circular Motion • Solve problems involving: • Centripetal acceleration • Centripetal force • Explain how the apparent existence of an outward force in circular motion can be explained as inertia resisting the centripetal force Definition: Centripetal – Center seeking Centrifugal – Center fleeing

3. V1 V2 Tangential Speed • The tangential speed(vt) • Used to describe the speed of an object in circular motion • is the object’s speed along an imaginary line drawn tangent to the circular path. • Tangential speed depends on the distance from the object to the center of the circular path • Which point is moving faster outside or inside of the circle? • V1 has to cover more distance, so it has a great tangential speed • When the tangential speed is constant, the motion is described as uniform circular motion.

4. Centripetal Acceleration • An object, in circular motion, with a constant speed, will there be acceleration? • Yes: acceleration is vector (both magnitude and direction) and the direction is constantly changing • An acceleration of this nature is called centripetal acceleration • Which results from change in direction • Example: car on circular track, constant speed, but changing direction = centripetal acceleration • Tangential acceleration • Results from changing the speed of rotation • Example: car on circular track, driver pushes the gas pedal down more

5. Visual Concept Centripetal Accel Centripetal Acceleration • Centripetal Acceleration is always directed toward the center of a circle • As object moves from Vi to Vf • The direction of the objects velocity vector changes • ∆ V is directed toward the central of the circle

6. Centripetal Force • Centripetal force (Fc) is a force that makes a body follow a curved path • Without a centripetal force, an object in motion continues along a straight-line path. • With a centripetal force, an object in motion will be accelerated and change its direction • Centripetal force is directed toward the center of the circle • Centripetal force is simply the name given to the net force on an object in uniform circular motion

7. Centripetal Force • Any type of force or combination of forces can provide this Centripetal force (net force) • Example:Friction between a race car’s tires and circular track is a centripetal force that keeps the car in a circular path • Example:Gravitational force is a centripetal force that keeps the moon in its orbit • Example: A mass on a string being whirled in a horizontal circular path. The string is the centripetal force

8. Centripetal Force • If the centripetal force vanishes, the object stops moving in a circular path • Inertia will cause the object to move in a direction tangent to the path • Newton’s 2nd law combined with centripetal acceleration forms an equation for centripetal force

9. Objectives: Newton’s Law of Universal Gravitation • Explainhow Newton’s law of universal gravitation accounts for various phenomena, including satellite and planetary orbits, falling objects, and the tides • Apply Newton’s law of universal gravitation to solve problems

10. Visual Gravitational Force Gravitational Force • The centripetal force that holds the planets in orbit is the same force that pulls an apple toward the ground – Gravitational Force • Gravitational force is the mutual force of attraction between particles of matter. • Gravitational force depends on the masses and on the distance between them • Newton developed the following equation to describe quantitatively the magnitude of the gravitational force if distance r separates masses m1 and m2 Constant of universal gravitation (G) = 6.67310–11 N•m2/kg

11. Gravitational Force • The gravitational forces that two masses exert on each other are always equal in magnitude and opposite in direction • This is an example of Newton’s third law of motion • Example: Earth-moon system • The Moon and Earth each orbit the center of mass of the Earth-moon system. • Because Earth has a much greater mass than the moon, this center of mass lies within Earth

12. Gravitational Force • Orbiting objects are in free fall • 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

13. Applying the Law of Gravitation • Newton’s Law of gravitation accounts for ocean tides • High and low tides are partly due to the gravitational force exerted on Earth by its moon • The tides result from the difference between the gravitational force at Earth’s surface and at Earth’s center

14. Objectives: Torque and Simple Machines • Distinguish between torque and force • Calculate the magnitude of a torque on an object • Indentify the six types of simple machines • Calculate the mechanical advantage of a simple machine

15. Rotational Motion • Rotational and Translational motion can be analyzed separately. • Translational motion is movement that changes the position of an object, as opposed to rotation • Example: when a bowling ball strikes the pins, the pins may spin in the air as they fly backward. • These pins have both rotational & translational motion. • We are going to isolate • Rotational motion. • Explore how to measurethe ability of a force to rotate an object

16. Visual Concept Torque • Torque is a quantity that measures the ability of a force to rotate an object around some axis • How easily an object rotates on both: • how much force is applied and on • where the force is applied. • The perpendicular distance from the axis of rotation to a line drawn along the direction of the force is equal to d sin q and is called the lever arm • The applied force may act at an angle • The direction of the lever arm (d sin q) is always perpendicular to the direction of the applied force t = Fd sin q torque = force  lever arm

17. Visual Concept Right Hand Rule for Torque • Torque is a vector quantity. • Part of the torque calculation is the determination of direction. • The direction is perpendicular to both the radius from the axis and to the force. It is conventional to choose it in the right hand rule direction along the axis of rotation. The torque is in the direction of the angular velocity which would be produced by it in the absence of other influences. In general, the change in angular velocity is in the direction of the torque.

18. Torque Example Problem • A basketball is being pushed by two players during tip-off. One player exerts an upward force of 15 N at a perpendicular distance of 14 cm from the axis of rotation. The second player applies a downward force of 11 N at a distance of 7.0 cm from the axis of rotation. Find the net torque acting on the ball about its center of mass Given: F1 = 15 N F2 = 11 N d1 = 0.14 m d2 = 0.070 m tnet = ? Eqns: t = Fd tnet = t1 + t2 = F1d1+ F2d2 Tip: The factor sin q is not included in the torque equation because each given distance is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force. In other words, each given distance is the lever arm.

19. Torque Example Problem Soln: t = Fd tnet = t1 + t2 = F1d1+ F2d2 t1 = F1d1= (15 N)(–0.14 m) = –2.1 N•m t2 = F2d2 = (–11 N)(0.070 m) = –0.77 N•m tnet = t1 + t2 = –2.1 N•m – 0.77 N•m tnet = –2.9 N•m The net torque is negative, so the ball rotates in a clockwise direction.

20. Simple Machines • A machine is any device that transmits or modifies force, usually by changing the force applied to an object • All machines are combinations or modifications of six fundamental types of machines, called simple machines. • These six simple machines are the lever, pulley, inclined plane, wheel and axle, wedge,andscrew

21. Simple Machines • The purpose of a simple machine is to change the direction or magnitude of an input force • A useful way of characterizing a simple machine is to compare the output and input force • This ratio is called mechanical advantage • If friction is disregarded, mechanical advantage can also be expressed in terms of input and output distance.

22. Simple Machines • The diagrams show two examples of a trunk being loaded onto a truck. • In the first example, a force (F1) of 360 N moves the trunk through a distance (d1) of 1.0 m. This requires 360 N•m of work. • In the second example, a lesser force (F2) of only 120 N would be needed (ignoring friction), but the trunk must be pushed a greater distance (d2) of 3.0 m. This also requires 360 N•m of work.

23. Visual Concept- Efficiency Simple Machines • The simple machines we have considered so far are ideal, frictionless machines • Real machines, however, are not frictionless • Some of the input energy is dissipated as sound or heat • The efficiencyof a machine is the ratio of useful work output to work input. • The efficiency of an ideal (frictionless) machine is 1, or 100 percent. • The efficiency of real machines is always less than 1.

24. Objectives: Motion in Space • Describe Kepler’s law of planetary motion • Solve problems involving orbital speed and period