Report

Chapter 6 Circular Motion and Other Applications of Newton’s Laws Circular Motion and Other Applications of Newton’s Laws 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 Uniform Circular Motion, Acceleration 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 ac v 2 r The centripetal acceleration, a c , is directed toward the center of the circle. The centripetal acceleration is always perpendicular to the velocity. Section 6.1 Uniform Circular Motion, Force A force, Fr , 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 F m ac m v 2 r Section 6.1 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 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 NOT a new kind of force! It is a New 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!!! 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 Centrifugal Force Centrifugal Force (Outward) is Another Misconception Force on the ball is NEVER outward ≡ Centrifugal Force Force is ALWAYS inward If Centrifugal Force existed, the ball would Fly Off as 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) 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 v L g s in ta n Section 6.1 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. v Tr m The maximum speed corresponds to the maximum tension the string can withstand. Section 6.1 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: v s gr Note, this does not depend on the mass of the car. Section 6.1 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 ta n v 2 rg Section 6.1 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 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 Ferris Wheel, cont. At the bottom of the loop, the upward force (the normal) experienced by the object is greater than its weight. F n bot n bot m g mv 2 r 2 v mg 1 rg Section 6.1 Ferris Wheel, final At the top of the circle, the force exerted on the object is less than its weight. F n to p m g n to p v2 mg 1 rg mv 2 r Section 6.1 Non-Uniform Circular Motion The acceleration and force have tangential components. F r produces the centripetal acceleration Ft produces the tangential acceleration The total force is F F r F t Section 6.2 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. v2 T mg co s Rg Section 6.2 Top and Bottom of Circle The tension at the bottom is a maximum. v b2o t T mg 1 R g The tension at the top is a minimum. v to2 p T mg 1 Rg If Ttop = 0, then v top gR Section 6.2 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 “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 “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 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 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 F F x T sin m a y T co s m g 0 The non-inertial observer (b) sees F' F' x T sin Ffictitio u s m a y T co s m g 0 These are equivalent if Ffictiitous = ma Section 6.3