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Transcript
PHYS 1443 – Section 001 Lecture #6 Thursday June 9, 2005 Dr. Nurcan Ozturk for Dr. Brandt • • • Newton’s Laws/Force Friction Circular Motion Thursday June 9, 2005 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt 1 Announcements • Homework: – HW3 on ch4 due Monday 6/13 at 2pm – HW4 on ch5 due Tuesday 6/14 at 8pm Thursday June 9, 2005 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt 2 Newton’s Laws of Motion Newton’s 1st law of motion (Law of Inertia): In the absence of external forces, an object at rest remains at rest and an object in motion continues in motion with a constant velocity. Newton’s 2nd law of motion: The acceleration of an object is directly proportional to the net force exerted on it and is inversely proportional to the object’s mass. Newton’s 3rd law of motion: Whenever one object exerts a force on a second object, the second exerts an equal and opposite force on the first. Fi ma i since Force is a vector… For simplicity, we define a new derived unit called, a Newton (N) Thursday June 9, 2005 F ix max i F iy may F iz i maz i 1 1N 1kg m / s lb 4 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt 2 3 Some Basic Information When Newton’s laws are applied, external forces only are of interest!! Gravitational Force (Weight) r r W FG M g Mg Normal Force, n: Reaction force that reacts to gravitational force due to the surface structure of an object. Its direction is perpendicular to the surface. Tension, T: The reactionary force by a stringy object against an external force exerted on it. Free-body diagram Thursday June 9, 2005 A graphical tool which is a diagram of external forces on an object and is extremely useful in analyzing forces and motion!! Drawn only on an object. PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt 4 Applications of Newton’s Laws Suppose you are pulling a box on frictionless ice, using a rope. M What are the forces being exerted on the box? T Gravitational force: Fg n= -Fg Free-body diagram n= -Fg Normal force: n T Fg=Mg T Fg=Mg Thursday June 9, 2005 Tension force: T Total force: F=Fg+n+T=T If T is a constant force, ax, is constant Fx T Ma x F y ax Fg n Ma y 0 T M ay 0 v xf vxi a xt v xi T t M 1 T 2 x x v t t x f i xi PHYS 1443-001, I 2005 in the Is Summer there motion Dr. Andrew Brandt 2M y-direction? 5 Example Using Newton’s Laws A traffic light weighing 125 N hangs from a cable tied to two other cables fastened to a support. The upper cables make angles of 37.0o and 53.0o with the horizontal. Find the tension in the three cables. 37o y 53o T1 Free-body Diagram T2 37o 53o T3 r ur ur ur ur F T 1 T 2 T 3 ma 0 i 3 x-comp. of Tix 0 net force Fx i 1 y-comp. of net force Fy i 3 T i 1 iy Thursday June 9, 2005 0 x Newton’s 2nd law T1 cos 37 T2 cos 53 0 T1 T sin 53 0.754 sin 37 1.25T T cos 37 cos 53o o 2 0.754T2 T1 sin 37o T2 sin 53o mg 0 2 2 125N T1 I0.754 T2 75.4 N T2 PHYS 1001443-001, N ; Summer 2005 Dr. Andrew Brandt T3 125N 6 Example without Friction A crate of mass M is placed on a frictionless inclined plane of angle q. a) Determine the acceleration of the crate after it is released. y n x q F Fg n ma n Fx Ma x Fgx Mg sin q Free-body Diagram q Fg y b) Supposed the crate was released at the top of the incline, and the length of the incline is d. How long does it take for the crate to reach the bottom and what is its speed at the bottom? ax g sin q x F= -Mg F Ma n F n mg cosq 0 y y gy 1 2 1 v t a x t g sin q t 2 t d ix 2 2 v xf vix axt g sin q 2d g sin q 2d 2dg sin q g sin q vxf 2dg sin q Thursday June 9, 2005 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt 7 Forces of Friction Resistive force exerted on a moving object due to viscosity or other types of frictional properties of the medium in, or surface on, which the object moves. These forces are either proportional to the velocity or the normal force. Force of static friction, fs: The resistive force exerted on the object until just before the beginning of its movement Empirical Formula f s s n What does this formula tell you? Frictional force is variable and will increase until it reaches the limit Beyond the limit, the object moves, and there is NO MORE static friction but kinetic friction takes over. Force of kinetic friction, fk fk k n The resistive force exerted on the object during its movement Which direction does kinetic friction apply? Thursday June 9, 2005 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt Opposite to the motion! 8 Example w/ Friction Suppose a block is placed on a rough surface inclined relative to the horizontal. The inclination angle is increased till the block starts to move. Show that by measuring this critical angle, qc, one can determine coefficient of static friction, s. y n n Free-body Diagram Fg fs=sn q q Net force x F= -Mg F M a F g n f s On the verge of sliding, block is stationary, acceleration is zero, friction is static friction. x comp. Fx Fgx f s Mg sin q f s 0 f s s n Mg sin q c y comp. Fy Ma y n Fgy n Mg cosqc 0 n Fgy Mg cos q c If block moves at constant speed, acceleration Mg sin q c Mg sin q c s Mg cos q tan q c is zero, friction is kinetic friction. Coefficient c n of kinetic friction can be calculated as k tan q Thursday June 9, 2005 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt 9 Newton’s Second Law & Uniform Circular Motion The centripetal acceleration is always perpendicular to the velocity vector, v, for uniform circular motion. v2 ar r Are there forces in this motion? If so, what do they do? The force that causes the centripetal acceleration acts toward the center of the circular path and causes a change in the direction of the velocity vector. This force is called centripetal force. v2 Fr mar m r What do you think will happen to the ball if the string that holds the ball breaks? Why? Based on Newton’s 1st law, since the external force no longer exist, the ball will continue its motion without change and will fly away following the tangential direction to the circle. Thursday June 9, 2005 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt 10 Example of Uniform Circular Motion A ball of mass 0.500kg is attached to the end of a 1.50m long cord. The ball is moving in a horizontal circle. If the string can withstand a maximum tension of 50.0 N, what is the maximum speed the ball can attain before the cord breaks? Fr m Centripetal acceleration: When does the string break? v2 ar r v2 Fr mar m r T when the centripetal force is greater than the sustainable tension. v2 m T r v Tr 50.0 1.5 12.2 m / s m 0.500 Calculate the tension of the cord when speed of the ball is 5.00m/s. Thursday June 9, 2005 v2 5.00 8.33 N 0.500 T m r 1.5 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt 2 11 Example of Banked Highway (a) For a car traveling with speed v around a curve of radius r, determine a formula for the angle at which a road should be banked so that no friction is required to keep the car from skidding. y x x comp. F FN sin q mar y comp. F FN cos q mg 0 x y v2 FN sin q m r FN cos q mg 2 mg mg sin q mv FN sin q FN mg tan q sin q cos q r v2 tan q gr (b) What is this angle for an expressway off-ramp curve of radius 50m at a design speed of 50km/h? v 50km/ hr 14m / s Thursday June 9, 2005 tan q 142 50 9.8 0.4 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt q tan 1 0.4 22o 12 Forces in Non-uniform Circular Motion The object has both tangential and radial accelerations. What does this statement mean? Fr F The object is moving under both tangential and radial forces. Ft F Fr Ft These forces cause not only the velocity but also the speed of the ball to change. The object undergoes a curved motion under the absence of constraints, such as a string. How does the acceleration look? Thursday June 9, 2005 a ar2 at2 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt 13 Example for Non-Uniform Circular Motion A ball of mass m is attached to the end of a cord of length R. The ball is moving in a vertical circle. Determine the tension of the cord at any instant when the speed of the ball is v and the cord makes an angle q with vertical. q T R m What are the forces involved in this motion? •The gravitational force Fg •The radial force, T, providing tension. Fg=mg tangential comp. Radial comp. F t mg sin q mat 2 v Fr T mg cosq mar m R at g sin q v2 T m g cos q R At what angles does the tension become maximum and minimum. What are the max and min tensions? Thursday June 9, 2005 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt 14