Chapter 5 Circular Motion © 2014 Pearson Education, Inc.

Kinematics of Uniform Circular Motion -motion in a circle of constant radius at constant speed -instantaneous velocity is always tangent to circle -circumference and period (time a revolution takes) help establish velocity On equation sheet T = 1/f Different on equation sheet Period (s) = 1 ÷ rev/s V = C/T not on equation sheet

Velocity is constantly changing, therefore the object is accelerating Centripetal (radial) acceleration is directed toward the center of the circle © 2014 Pearson Education, Inc. Different on equation sheet a c = v 2 /r

Dynamics of Uniform Circular Motion For an object to be in uniform circular motion, there must be a net force acting on it. © 2014 Pearson Education, Inc. Not on equation sheet – just substitute a c for a

Fictitious force – centrifugal force If the real force, centripetal force vanishes, the object flies off tangent to the circle © 2014 Pearson Education, Inc.

Highway Curves, Banked and Unbanked When a car goes around a curve, there must be a net force towards the center of the circle. If the road is flat, that force is supplied by friction. © 2014 Pearson Education, Inc. Insufficient friction = moving tangent to circle

As long as the tires do not slip, the friction is static. If the tires do start to slip, the friction is kinetic, which is bad in two ways: 1.kinetic friction < static friction. 2.static friction force can point towards the center of the circle, but the kinetic frictional force opposes the direction of motion, making it very difficult to regain control of the car and continue around the curve. Antilock brakes function to keep the car is static friction not kinetic, so control is maintained © 2014 Pearson Education, Inc.

Banking the curve can help keep cars from skidding. -there is one speed where F c is supplied by the horizontal component of the F N, and no friction is required. © 2014 Pearson Education, Inc. Ideal speed for the curve Angle of the bank Radius of the curve Mass of car Not on equation sheet Tanθ = v 2 /rg Banking angle independent of mass

Practice Problems 5-4. the game of tetherball is played with a ball tied to a pole on a string. When the ball is struck it whirls around the pole. In what direction is the acceleration of the ball and what causes this? Draw of free body diagram to support your answer. © 2014 Pearson Education, Inc. 7 th edition 5-5.

Practice Problems 5-6. A rider on a Ferris wheel moves in a vertical circle at a constant speed. Is the normal force that the seat exerts on the rider at the top of the circle a) less than, b) more than, or c) the same as the force exerted at the bottom of the circle. 7 th edition 5-4c.

Practice Problems 5-1. A 150g ball at the end of a string is revolving uniformly in a horizontal circle of radius 0.600m. The ball makes 2.00 revolutions a second. What is the centripetal acceleration? 5-3. Estimate the force a person must exert on a string attached to a 0.150kg ball to make the ball revolve in a horizontal circle of radius 0.600m. The ball makes 2.00 revolutions per second.

Practice Problems 5-2. The Moon’s nearly circular orbit about the Earth has a radius of about 384,000km and period of 27.3 days. Determine the acceleration of the Moon toward Earth. © 2014 Pearson Education, Inc.

Practice Problems 5-5. A 0.150kg ball on the end of a 1.10m long cord is swung in a vertical circle. A) Determine the minimum speed the ball must have at the top to continue moving in a circle (Hint: if F T =0, the ball won’t continue in a circle). B) Calculate the tension in the cord at the bottom of the circle assuming the ball is moving twice as fast as in part A. 7 th edition 5-4.

Practice Problems 5-8. What is the angle for an expressway off-ramp curve of radius 50m at a design speed of 50km/h (14m/s)? 7 th edition 5-7.

Practice Problems 5-7. A 1000kg car rounds a curve on a flat road of radius 50m at a speed of 14m/s (50km/h). Will the car make the turn or will it skid off if A)the pavement is dry and µ s =0.60 B)the pavement is icy and µ s =0.25? 7 th edition 5-6.

Chapter 5 Gravitation © 2014 Pearson Education, Inc.

Newton’s Law of Universal Gravitation © 2014 Pearson Education, Inc. ALL masses exert a force of gravity The gravitational force on an object is one-half of a 3rd Law pair: Earth on object, object on Earth With such a disparity in masses, the reaction force is undetectable, but for bodies more equal in mass it can be significant.

Gravitational force is proportional to both masses. Gravitational force decreases as the distance increases The Law of Universal Gravitation G = 6.67 × 10 −11 N·m 2 /kg 2 © 2014 Pearson Education, Inc. BOTH on equation sheet – also look at Coulomb’s Law of F E

Gravity Near the Earth’s Surface Next 2 slides = trivia, no need to copy it 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: Now, knowing g and the radius of the Earth, the mass of the Earth can be calculated: © 2014 Pearson Education, Inc. (5-5)

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. © 2014 Pearson Education, Inc.

Satellites and “Weightlessness” Satellite 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. -it is continually falling, but the Earth curves from underneath it. © 2014 Pearson Education, Inc.

The free fall means no normal force, so experience of weightlessness. Fg=mg always exists though! © 2014 Pearson Education, Inc. E83phk Zero Gravity Flight – Weightlessness Video (5min)

Briefly experiencing weightlessness © 2014 Pearson Education, Inc.

Practice Problems 5-9. A 50kg person and a 70kg person are sitting on a bench so that their centers are 50cm apart. Estimate the magnitude of the gravitational force each exerts on the other. © 2014 Pearson Education, Inc.

Practice Problems What is the force of gravity acting on a 2000kg spacecraft when it orbits two Earth radii from the Earth’s center (r E =6380km, m E =5.98x10 24 kg). © 2014 Pearson Education, Inc. 7 th ed. 5-10

Practice Problems Estimate the effective value of g on top of Mt Everest at 8848m above the Earth’s surface. That is, what is the acceleration due to gravity on top of Everest. (r E =6380km, m E =5.98x10 24 kg) © 2014 Pearson Education, Inc. 7 th ed. 5-11

Practice Problems 5-12 Modified At what speed must a geosynchronous satellite be moving if it is to stay in a circular orbit 36,000km above Earth? (Earth’s mass = 5.97x10 24 kg, Earth’s radius = 6380km) 7 th ed. 5-12