Units Ali ÖVGÜN EMU Physics Department www.aovgun.com
1 mile = 1609 m
January 22-25, 2013
January 22-25, 2013
Vectors Ali ÖVGÜN EMU Physics Department www.aovgun.com
Example 1: Operations with Vectors C Vector A is described algebraically as (-3, 5), while vector B is (4, -2). Find the value of magnitude and direction of the sum (C) of the vectors A and B. A 3iˆ 5 ˆj B 4iˆ 2 ˆj A B ( 3 4)ˆ i (5 2) ˆj 1ˆ i 3ˆj C 1 3 x 2 C y 2 y 1/ 2 C ( C x C ) C 1 y tan ( ) tan C x 1 (1 3 2 3 2 ) 1/ 2 71.56 3.16 January 21, 2015
Example 2 : January 21, 2015
Example 3 : January 21, 2015
Ex. 4: Find: Solution: Examples of Cross Products A B? A B 0 4ˆ i A (2ˆ i 3 ˆ) j ( iˆ 2 ˆ) j ˆj 3 ˆj iˆ 0 2iˆ 3ˆj B iˆ 2 ˆj 2ˆ i ( iˆ) 2ˆ i 2 ˆj 3 ˆj ( iˆ) 3 ˆj 2 ˆj 4kˆ 3kˆ 7kˆ Ex.5: Calculate r F given a force and its location F ( 2ˆ i 3ˆ) j N r (4ˆ i 5 ˆj ) m Solution: Where: r F (4iˆ 5 ˆj ) (2iˆ 3 ˆj ) 4iˆ 2iˆ 4iˆ 3 ˆj 5 ˆj 2iˆ 5 ˆj 3 ˆj 0 4iˆ 3 ˆj 5 ˆj 2iˆ 0 12kˆ 10kˆ 2 kˆ (Nm) j AB March 16, 2016 i k iˆ ˆj kˆ 4 5 0 2 3 0
Example 6: January 21, 2015
Example 7: January 21, 2015
January 21, 2015
January 21, 2015
Problem 5 Problem 6 Problem 7 January 21, 2015
Problem 8 January 21, 2015
Motion in 1D Ali ÖVGÜN EMU Physics Department www.aovgun.com
Jan. 28-Feb. 1, 2013
Jan. 28-Feb. 1, 2013
Jan. 28-Feb. 1, 2013
September 8, 2008
Jan. 28-Feb. 1, 2013
September 8, 2008
Jan. 28-Feb. 1, 2013
Free Fall for Rookie A stone is thrown from the top of a building with an initial velocity of 20.0 m/s straight upward, at an initial height of 50.0 m above the ground. The stone just misses the edge of the roof on the its way down. Determine (a) the time needed for the stone to reach its maximum height. (b) the maximum height. (c) the time needed for the stone to return to the height from which it was thrown and the velocity of the stone at that instant. (d) the time needed for the stone to reach the ground (e) the velocity and position of the stone at t = 5.00s Jan. 28-Feb. 1, 2013
Example Example Jan. 28-Feb. 1, 2013
Problem1 Problem2 Jan. 28-Feb. 1, 2013
Problem3 Problem4 Jan. 28-Feb. 1, 2013
Problem5 Problem6 Jan. 28-Feb. 1, 2013
Problem7 Problem8 Problem9 Jan. 28-Feb. 1, 2013
Problem10 Jan. 28-Feb. 1, 2013
September 8, 2008
September 8, 2008
September 8, 2008
24.5 m/s and 0.1 m/s September 8, 2008
Motion in 2D and 3D Ali ÖVGÜN EMU Physics Department www.aovgun.com
Example: 1 During volcanic eruptions, chunks of solid rock can be blasted out of the volcano; these projectiles are called volcanic bombs. The figure below shows a cross section of Mt. Fuji, in Japan. From the vent A to the foot of the volcano at B, the vertical distance is h = 3.30km and horizontal distance is d = 940m. Neglecting air resistance, (a) calculate the time of flight, and (4 P) (b) calculate the initial speed of the projectile. (2P) Example: 2 A movie stunt driver on a motorcycle speeds horizontally off a 50m high cliff. If the motorcycle will land 90m from the base of the cliff, (ignore any kind of friction or resistance) (a) Find the time of flight, (b) Find its initial speed in x-direction, (c) Find its acceleration vector just before hitting the ground. February 5-8, 2013
Example: 3 A cliff diver is about to jump down a cliff of height 35.0m, at the bottom of the cliff there is a 5m wide rock bank next to the sea. Calculate the minimum horizontal initial velocity the cliff jumper has to push off. (No initial velocity component in y direction) February 5-8, 2013
Problem:1 A projectile is fired at an initial velocity of 35.0 m/s at an angle of 30.0 degrees above the horizontal from the roof of a building 30.0 m high, as shown. Find a) The maximum height of the projectile b) The time to rise to the top of the trajectory c) The total time of the projectile in the air d)the velocity of the projectile at the ground e)the range of the projectile Problem:2 aa plane drops a package of supplies to a party of explorers. If the plane is traveling horizontally at 40 m/s and is 100 m above the ground. Where does the package strike the ground? February 5-8, 2013
Problem:3 February 5-8, 2013
Problem:4 February 5-8, 2013
Problem:5 February 5-8, 2013
Problem:6 February 5-8, 2013
Problem:7 At what initial speed must the basketball player in Figure throw the ball, at angle u0 = 37 above the horizontal, to make the foul shot? The horizontal distances are = 30 cm and, = 440cm and the heights are = 220 cm and. = 300 cm. February 5-8, 2013
Problem:8 The bobsled track contains turns with radii of 33 m and 24 m. Find the centripetal acceleration at each turn for a speed of 34 m/s. Express answers as multiples of. February 5-8, 2013
Newton`s Laws Ali ÖVGÜN EMU Physics Department www.aovgun.com
Example1: One or two forces act on a puck that moves over frictionless ice along an x axis, in one-dimensional motion. The puck's mass is m = 0.20 kg. Forces F 1 and F 2 and are directed along the x axis and have magnitudes F 1 = 4.0 N and F 2 = 2.0 N. Force F 3 is directed at angle = 30 and has magnitude F 3 = 1.0 N. In each situation, what is the acceleration of the puck? 1 a) F a x b) F F a x 2 1 F1 F2 m ma x F1 4.0 N 2 20 m/s m 0.2kg ma x 4.0 N 2.0 N 0.2kg 10 m/s 2 F net, x ma x c) F a x 3, x F 2 ma F3 cos F2 m x F 3, x F 1.0 Ncos30 2.0 N 0.2kg 3 cos 5.7 m/s 2 www.aovgun.com
Example2: www.aovgun.com
Example3: www.aovgun.com
Equilibrium, Example 1 A lamp is suspended from a chain of negligible mass The forces acting on the lamp are the downward force of gravity the upward tension in the chain Applying equilibrium gives Fy 0T Fg 0T Fg www.aovgun.com
Equilibrium, Example 2 A traffic light weighing 100 N hangs from a vertical cable tied to two other cables that are fastened to a support. The upper cables make angles of 37 and 53 with the horizontal. Find the tension in each of the three cables. Conceptualize the traffic light Assume cables don t break Nothing is moving Categorize as an equilibrium problem No movement, so acceleration is zero Model as an object in equilibrium x y F 0 F 0 www.aovgun.com
Equilibrium, Example 2 Need 2 free-body diagrams Apply equilibrium equation to light Apply equilibrium equations to knot N F T F T F g g y 100 0 0 3 3 N F T F T F g g y 100 0 0 3 3 N T T N T T T T N T T T T T F T T T T F y y y y x x x 80 1.33 60 1.33 cos53 cos37 0 100 sin 53 sin 37 0 cos53 cos37 1 2 1 1 1 2 2 1 3 2 1 2 1 2 1 www.aovgun.com
Accelerating Objects, Example 1 A man weighs himself with a scale in an elevator. While the elevator is at rest, he measures a weight of 800 N. What weight does the scale read if the elevator accelerates upward at 2.0 m/s 2? a = 2.0 m/s 2 What weight does the scale read if the elevator accelerates downward at 2.0 m/s 2? a = - 2.0 m/s 2 Upward: N mg ma m w g 800 N 9.8 m/s Downward: F y m( g a) 2 N mg ma 81.6kg N 81.6( 2.0 9.8) N mg N 81.6(2.0 9.8) 962.9 N N mg 636.5 N www.aovgun.com N mg N mg
Example4: www.aovgun.com
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 www.aovgun.com
Newton 2nd law: Then So F F F y x y mgsin N N mgcos N mgcos s mgsin mg cos 0 s 0 tan 0.350 tan 1 (0.350) Inclined Plane s 0 19.3 www.aovgun.com
Multiple Objects A block of mass m1 on a rough, horizontal surface is connected to a ball of mass m2 by a lightweight cord over a lightweight, frictionless pulley as shown in figure. A force of magnitude F at an angle θ with the horizontal is applied to the block as shown and the block slides to the right. The coefficient of kinetic friction between the block and surface is μ k. Find the magnitude of acceleration of the two objects. www.aovgun.com
Multiple Objects m 1 : m 2 : T F F F x y F cos T T m2g m2a y m a y 2 f 1 m a N F sin m g 0 N m1 g F sin f N ( m 1 g F sin ) k m2( a g) k k k 1 x m a 1 F cos k ( m1 g F sin ) m2 ( a g) m1a F(cos k sin ) ( m a m m 1 2 2 km1 ) www.aovgun.com g
www.aovgun.com Example1: Example2:
Problem1: www.aovgun.com
Problem2: www.aovgun.com
Problem3: www.aovgun.com
Problem4: www.aovgun.com
Problem5: www.aovgun.com
Problem6: www.aovgun.com
Problem7: www.aovgun.com
Example3: Example4: www.aovgun.com
Circular Motion Ali ÖVGÜN EMU Physics Department www.aovgun.com
Equilibrium, Example 1 What is the smallest value of the force F such that the 2.0-kg block will not slide down the wall? The coefficient of static friction between the block and the wall is 0.2.? N f F F mg March 16, 2016
Ex2: The Conical Pendulum A small ball of mass m = 5 kg is suspended from a string of length L = 5 m. The ball revolves with constant speed v in a horizontal circle of radius r = 2 m. Find an expression for v and a. T θ mg March 16, 2016
March 16, 2016 The Conical Pendulum 0.44 tan 0.4 sin sin cos 0 cos 2 5 5 2 2 2 r L r L r r mv T F mg T mg T F m r m L kg m x y 2 2 2 2 4.3 m/s tan 2.9 m/s tan sin tan tan cos sin g r v a Lg v rg v gr v mg T r mv T Find v and a
Ex:3 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. v rg March 16, 2016
Level Curves The force of static friction directed toward the center of the curve keeps the car moving in a circular path. f N v s,max max F y 2 v sn m r N mg 0 mg snr m (0.523)(9.8m / s v rg max smgr m 2 )(35.0m) gr s 13.4m / s March 16, 2016
Ex:4 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? March 16, 2016
Banked Curves v 13.4 m/s tan F F r y n cos mg tan mv nsin mac r n cos mg 0 2 v rg 1 ( r 35.0 m 13.4 m/s (35.0 m)(9.8 m/s 2 2 ) ) 27.6 March 16, 2016
Problem 1: March 16, 2016
Problem 2: Problem 3:
Problem 4: Problem 5: March 16, 2016
Problem 6: A string under a tension of 50.0 N is used to whirl a rock in a horizontal circle of radius 2.50 m at a speed of 20.4 m/s on a frictionless surface as shown in Figure. As the string is pulled in, the speed of the rock increases. When the string on the table is 1.00 m long and the speed of the rock is 51.0 m/s, the string breaks. What is the breaking strength, in newtons, of the string? Problem 7: A puck of mass m1 is tied to a string and allowed to revolve in a circle of radius R on a frictionless, horizontal table. The other end of the string passes through a small hole in the center of the table, and an object of mass m2 is tied to it Fig. The suspended object remains in equilibrium while the puck on the tabletop March 16, 2016 revolves. Find symbolic expressions for (a) the tension in the string, (b) the radial force acting on the puck, and (c) the speed of the puck.
Problem 8: Problem 9: March 16, 2016
Problem 10: Problem 11: March 16, 2016
Problem 12: March 16, 2016