1443 501 Spring 2002 Lecture 24 Dr Jaehoon Yu Review of Chap 1 15 Final Exam at 5 30pm Monday May 6 covers Ch 1 16 2 dim Motion Under Constant Acceleration Position vectors in xy plane ri xi i yi j rf x f i y f j Velocity vectors in xy plane vi v xi i v yi j v f v xf i v yf j v xf v xi a x t v yf v yi a y t v f v xi a x t i v yi a y t j vi at How are the position vectors written in acceleration vectors 1 1 a x t 2 y f yi v yi t a y t 2 2 2 1 1 r f xi v xi t a x t 2 i yi v yi t a y t 2 j 2 2 1 ri vt at 2 2 x f xi v xi t May 1 2002 1443 501 Spring 2002 Dr J Yu Lecture 24 2 Example 2 12 A stone was thrown straight upward at t 0 with 20 0m s initial velocity on the roof of a 50 0m high building g 9 80m s2 1 Find the time the stone reaches at maximum height v 0 2 3 4 5 Find the maximum height Find the time the stone reaches its original height Find the velocity of the stone when it reaches its original height Find the velocity and position of the stone at t 5 00s 1 v f vyi ayt 20 0 9 80t 0 00 20 0 t 2 04 s 9 80 3 t 2 04 2 4 08s 2 y yi vyit 1 ayt 2 2 f 1 50 0 20 2 04 9 80 2 04 2 2 50 0 20 4 70 4 m 4 vyf vyi ayt 20 0 9 80 4 08 20 0 m s 5Velocity vyf vyi ayt 20 0 9 80 5 00 29 0 m s May 1 2002 yf yi vyit 1 ayt 2 2 50 0 20 0 5 00 1443 501 Spring 2002 Dr J Yu Lecture 24 5 Position 1 9 80 5 00 2 27 5 m 2 3 Uniform Circular Motion A motion with a constant speed on a circular path The velocity of the object changes because the direction changes Therefore there is an acceleration r vj r1 r2 r vi vj v vi Angle is The acceleration pulls the object inward Centripetal Acceleration Average Acceleration Instantaneous Acceleration May 1 2002 a vf vi v t f ti t v v r v r v v a r r t r r v v v2 ar lim a lim v t 0 t 0 t r r r r Is this correct in dimension What story is this expression telling you 1443 501 Spring 2002 Dr J Yu Lecture 24 4 Relative Velocity and Acceleration The velocity and acceleration in two different frames of references can be denoted Frame S v0 Frame S r r O v0t O r r v0 t d v d v d v0 dt dt dt Galilean transformation equation r r v0 t d r d r v0 dt dt v v v0 What does this tell you The accelerations measured in two frames are the same when the frames move at a constant velocity with respect to each other The earth s gravitational a a when v0 is constant acceleration is the same in a frame moving at a constant velocity wrt the earth May 1 2002 1443 501 Spring 2002 5 Dr J Yu Lecture 24 Example 4 9 A boat heading due north with a speed 10 0km h is crossing the river whose stream has a uniform speed of 5 00km h due east Determine the velocity of the boat seen by the observer on the bank vBB vBR vR N vBB vR vBR 2 vR vBR 10 0 2 5 00 2 11 2km h vBR 10 0 j and vR 5 00 i vBB 5 00 i 10 0 j vBB E vBBy vBBx tan 1 How long would it take for the boat to cross the river if the width is 3 0km May 1 2002 2 5 00 tan 1 26 6 10 0 vBB cos t 3 0km t 3 0 3 0 0 30hrs 18 min vBB cos 11 2 cos 26 6 1443 501 Spring 2002 Dr J Yu Lecture 24 6 Newton s Laws 1st Law Law of Inertia 2nd Law Law of Forces F ma i i 3rd Law Law of Action and Reaction F21 F12 May 1 2002 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 The acceleration of an object is directly proportional to the net force exerted on it and inversely proportional to the object s mass If two objects interact the force F12 exerted on object 1 by object 2 is equal magnitude to and opposite direction to the force F21 exerted on object 1 by 1443 501 Spring 2002 7 objectDr 2 J Yu Lecture 24 Applications of Newton s Laws Suppose you are pulling a box on frictionless ice using a rope What are the forces being exerted on the M T box Gravitational force Fg n F Normal g Freeforce n body T Tension diagra force T F T Ma a T Total force M m Fg Mg F Fg n T F F n 0 a 0 T T v v a t v t If T is a M constant force 1 T x x x v t t 2 M ax is constant x y xf xi x x x g y xi 2 f May 1 2002 i 1443 501 Spring 2002 Dr J Yu Lecture 24 xi 8 Example 5 4 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 T2 37o Free body Diagram 53o T3 i 3 Fx Tix 0 F T1 T2 T3 x i 1 i 3 Fy Tiy 0 T sin 53 mg 0 T cos 37 T cos 53 0 cos 53 T T 0 754T cos 37 T sin 53 0 754 sin 37 1 25T T1 sin 37 i 1 2 1 2 1 2 2 2 2 125N T2 100 N T1 0 754T2 75 4 N May 1 2002 1443 501 Spring 2002 Dr J Yu Lecture 24 9 Example 5 12 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 c one can determine coefficient of static friction s M n a n Free body Diagram fs kn Fg y F Ma x F Mg F Fg n f s Fy Ma …
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