Physics 103: Mechanics Lecture : Physics 1 03 : Mechanics Lecture
OBJECTIVES: February 24, 2009 OBJECTIVES To show that acceleration can also be due to change in direction of velocity through Uniform Acceleration To consider fluid resistance and friction in the problem and compute for physical quantities To determine the concepts in Uniform Circular Motion
Circular Motion & Centripetal Force: February 24, 2009 Circular Motion & Centripetal Force Newton’s 1st, 2nd, and 3rd laws Forces: gravity, normal, tension Applications of Newton’s laws Frictional Force (Section 4.6) Circular Motion (Section 7.4) Centripetal Force (Section 7.4) Problem-Solving Tactics
Forces of Friction: f: February 24, 2009 When an object is in motion on a surface or through a viscous medium, there will be a resistance to the motion . This resistance is called the force of friction This is due to the interactions between the object and its environment Force of static friction: f s Force of kinetic friction: f k Direction: along the surface, opposite the direction of the intended motion in direction opposite velocity if moving in direction vector sum of other forces if stationary Forces of Friction: f
Forces of Friction: Magnitude: February 24, 2009 Magnitude: Friction is proportional to the normal force Static friction: F f = F μ s N Kinetic friction: F f = μ k N μ is the coefficient of friction The coefficients of friction are nearly i ndependent of the area of contact Forces of Friction: Magnitude
Static Friction: February 24, 2009 Static Friction Static friction acts to keep the object from moving If increases, so does If decreases, so does ƒ s µ s N Remember, the equality holds when the surfaces are on the verge of slipping
Kinetic Friction: February 24, 2009 Kinetic Friction The force of kinetic friction acts when the object is in motion Although µ k can vary with speed, we shall neglect any such variations ƒ k = µ k N
Explore Forces of Friction: February 24, 2009 Explore Forces of Friction Vary the applied force Note the value of the frictional force Compare the values Note what happens when the can starts to move
Inclined Plane: February 24, 2009 Inclined Plane Suppose a block with a mass of 2.50 kg is resting on a ramp. If the coefficient of static friction between the block and ramp is 0.350, what maximum angle can the ramp make with the horizontal before the block starts to slip down?
Inclined Plane: February 24, 2009 Newton 2nd law: Then So Inclined Plane
Slide 11: February 24, 2009 Uniform circular motion Constant speed, or, constant magnitude of velocity Motion along a circle: Changing direction of velocity Uniform Circular Motion: Definition
Slide 12: February 24, 2009 Uniform Circular Motion: Observations Object moving along a curved path with constant speed Magnitude of velocity: same Direction of velocity: changing Velocity : changing Acceleration is NOT zero! Net force acting on an object is NOT zero “ Centripetal force ”
Slide 13: February 24, 2009 Magnitude: Direction: Centripetal Uniform Circular Motion O x y r i R A B v i r f v f Δ r v i v f Δ v = v f - v i
Uniform Circular Motion: February 24, 2009 Uniform Circular Motion Velocity: Magnitude: constant v The direction of the velocity is tangent to the circle Acceleration: Magnitude: directed toward the center of the circle of motion Period: time interval required for one complete revolution of the particle
Centripetal Force: February 24, 2009 Centripetal Force Acceleration: Magnitude: Direction: toward the center of the circle of motion Force: Start from Newton’s 2 nd Law Magnitude: Direction: toward the center of the circle of motion
What provide Centripetal Force ?: February 24, 2009 What provide Centripetal Force ? Centripetal force is not a new kind of force Centripetal force stands for any force that keeps an object following a circular path Centripetal force is a combination of Gravitational force mg : downward to the ground Normal force N : perpendicular to the surface Tension force T : along the cord and away from object Static friction force: f s max = µ s N
What provide Centripetal Force ?: February 24, 2009 a What provide Centripetal Force ? mg N
Problem Solving Strategy: February 24, 2009 Problem Solving Strategy Draw a free body diagram , showing and labeling all the forces acting on the object(s) Choose a coordinate system that has one axis perpendicular to the circular path and the other axis tangent to the circular path Find the net force toward the center of the circular path (this is the force that causes the centripetal acceleration, F C ) Use Newton’s second law The directions will be radial, normal, and tangential The acceleration in the radial direction will be the centripetal acceleration Solve for the unknown(s)
Level Curves: February 24, 2009 Level Curves A 1500 kg car moving on a flat, horizontal road negotiates a curve as shown. If the radius of the curve is 35.0 m and the coefficient of static friction between the tires and dry pavement is 0.523, find the maximum speed the car can have and still make the turn successfully.
Level Curves: February 24, 2009 Level Curves The force of static friction directed toward the center of the curve keeps the car moving in a circular path.
Problem-Solving Tactics: February 24, 2009 Problem-Solving Tactics A roller-coaster car has a mass of 500 kg when fully loaded with passengers. (a) If the vehicle has a speed of 20.0 m/s at point A, what is the force exerted by the track on the car at this point? (b) What is the maximum speed the vehicle can have at point B and still remain on the track?
Slide 22: February 24, 2009 step 1: draw free-body diagram step 2: choose a coordinate system: centripetal acceleration? step 3: Newton’s 2nd law mg N
Slide 23: February 24, 2009 What is the maximum speed at point B? mg N
The Conical Pendulum: February 24, 2009 The Conical Pendulum A small ball of mass m is suspended from a string of length L . The ball revolves with constant speed v in a horizontal circle of radius r . Find an expression for v and a mg T θ
The Conical Pendulum: February 24, 2009 The Conical Pendulum Find v and a
Banked Curves: February 24, 2009 Banked Curves A car moving at the designated speed can negotiate the curve. Such a ramp is usually banked, which means that the roadway is tilted toward the inside of the curve. Suppose the designated speed for the ramp is to be 13.4 m/s and the radius of the curve is 35.0 m. At what angle should the curve be banked?
Banked Curves: February 24, 2009 Banked Curves