Chapter 6

1. Chapter 6 Circular Motion and Other Applications of Newton’s Laws

2. Circular Motion and Other Applications of Newton’s Laws

3. Circular Motion • Two analysis models using Newton’s Laws of Motion have been developed. • The models have been applied to linear motion. • Newton’s Laws can be applied to other situations: • Objects traveling in circular paths • Motion observed from an accelerating frame of reference • Motion of an object through a viscous medium • Many examples will be used to illustrate the application of Newton’s Laws to a variety of new circumstances. Introduction

4. A particle moves with a constant speed in a circular path of radius r with an acceleration. The magnitude of the acceleration is given by The centripetal acceleration, , is directed toward the center of the circle. The centripetal acceleration is always perpendicular to the velocity. Uniform Circular Motion, Acceleration Section 6.1

5. Uniform Circular Motion, Force • A force, , is associated with the centripetal acceleration. • The force is also directed toward the center of the circle. • Applying Newton’s Second Law along the radial direction gives Section 6.1

6. Uniform Circular Motion, cont. • A force causing a centripetal acceleration acts toward the center of the circle. • It causes a change in the direction of the velocity vector. • If the force vanishes, the object would move in a straight-line path tangent to the circle. • See various release points in the active figure Section 6.1

7. Centripetal Force • The Force causing ac is sometimes called Centripetal Force (Center directed force). • So far we know forces in nature: • Friction, Gravity, Normal, Tension. • Should we add Centripetal Force to this List? NO!!!!! • Force causing ac is NOTa new kind of force! • It is aNew Role for force!!! • It is simply one or more of the forces we know acting in the role of a force that causes a circular motion. • Book will not use the term Centripetal Force!!!

8. Centripetal Force as New Role • Earth-Sun Motion: • Centripetal Force≡ Gravity • Object sitting on a rotating turntable: • Centripetal Force≡ Friction • Rock-String (horizontal plane): • Centripetal Force≡ Tension • Wall-Person (rotating circular room): • Centripetal Force≡ Normal • Ferris Wheel (lowest point): • Centripetal Force≡ Normal –Gravity • Ferris Wheel (highest point): • Centripetal Force≡ Normal –Gravity • Rock-String (vertical plane): • Centripetal Force≡ Tension ± Gravity

9. Centrifugal Force • Centrifugal Force (Outward)is Another Misconception • Force on the ball is • NEVERoutward≡Centrifugal Force • Force is ALWAYSinward • IfCentrifugal Forceexisted, the ball would Fly Offas in(a)when released. • Ball Flies off as in figure (b). • Similar to sparks flying in straight line from the • edge of rotating grinding wheel. (a) (b)

10. Conical Pendulum • The object is in equilibrium in the vertical direction . • It undergoes uniform circular motion in the horizontal direction. • ∑Fy = 0 → T cos θ = mg • ∑Fx = T sin θ = m ac • v is independent of m Section 6.1

11. Motion in a Horizontal Circle • The speed at which the object moves depends on the mass of the object and the tension in the cord. • The centripetal force is supplied by the tension. • The maximum speed corresponds to the maximum tension the string can withstand. Section 6.1

12. Horizontal (Flat) Curve • Model the car as a particle in uniform circular motion in the horizontal direction. • Model the car as a particle in equilibrium in the vertical direction. • The force of static friction supplies the centripetal force. • The maximum speed at which the car can negotiate the curve is: • Note, this does not depend on the mass of the car. Section 6.1

13. Banked Curve • These are designed with friction equaling zero. • Model the car as a particle in equilibrium in the vertical direction. • Model the car as a particle in uniform circular motion in the horizontal direction. • There is a component of the normal force that supplies the centripetal force. • The angle of bank is found from Section 6.1

14. Banked Curve, 2 • The banking angle is independent of the mass of the vehicle. • If the car rounds the curve at less than the design speed, friction is necessary to keep it from sliding down the bank. • If the car rounds the curve at more than the design speed, friction is necessary to keep it from sliding up the bank. Section 6.1

15. Ferris Wheel • The normal and gravitational forces act in opposite direction at the top and bottom of the path. • Categorize the problem as uniform circular motion with the addition of gravity. • The child is the particle. Section 6.1

16. Ferris Wheel, cont. • At the bottom of the loop, the upward force (the normal) experienced by the object is greater than its weight. Section 6.1

17. Ferris Wheel, final • At the top of the circle, the force exerted on the object is less than its weight. Section 6.1

18. Non-Uniform Circular Motion • The acceleration and force have tangential components. • produces the centripetal acceleration • produces the tangential acceleration • The total force is Section 6.2

19. Vertical Circle with Non-Uniform Speed • The gravitational force exerts a tangential force on the object. • Look at the components of Fg • Model the sphere as a particle under a net force and moving in a circular path. • Not uniform circular motion • The tension at any point can be found. Section 6.2

20. Top and Bottom of Circle • The tension at the bottom is a maximum. • The tension at the top is a minimum. • If Ttop = 0, then Section 6.2

21. Motion in Accelerated Frames • A fictitious force results from an accelerated frame of reference. • The fictitious force is due to observations made in an accelerated frame. • A fictitious force appears to act on an object in the same way as a real force, but you cannot identify a second object for the fictitious force. • Remember that real forces are always interactions between two objects. • Simple fictitious forces appear to act in the direction opposite that of the acceleration of the non-inertial frame. Section 6.3

22. “Centrifugal” Force • From the frame of the passenger (b), a force appears to push her toward the door. • From the frame of the Earth, the car applies a leftward force on the passenger. • The outward force is often called a centrifugal force. • It is a fictitious force due to the centripetal acceleration associated with the car’s change in direction. • In actuality, friction supplies the force to allow the passenger to move with the car. • If the frictional force is not large enough, the passenger continues on her initial path according to Newton’s First Law. Section 6.3

23. “Coriolis Force” • This is an apparent force caused by changing the radial position of an object in a rotating coordinate system. • The result of the rotation is the curved path of the thrown ball. • From the catcher’s point of view, a sideways force caused the ball to follow a curved path. Section 6.3

24. Fictitious Forces, examples • Although fictitious forces are not real forces, they can have real effects. • Examples: • Objects in the car do slide • You feel pushed to the outside of a rotating platform • The Coriolis force is responsible for the rotation of weather systems, including hurricanes, and ocean currents. Section 6.3

25. Fictitious Forces in Linear Systems • The inertial observer models the sphere as a particle under a net force in the horizontal direction and a particle in equilibrium in the vertical direction. • The non-inertial observer models the sphere as a particle in equilibrium in both directions. • The inertial observer (a) at rest sees • The non-inertial observer (b) sees • These are equivalent if Ffictiitous = ma Section 6.3