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Physics 115 2013

Physics 115 2013. Final Review. Calculus is about “rates of change”. A TIME RATE is anything divided by time. CHANGE is expressed by using the Greek letter, Delta, D . For example: Average SPEED is simply the “RATE at which DISTANCE changes”. The MEANING?.

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Physics 115 2013

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  1. Physics 115 2013 Final Review

  2. Calculus is about “rates of change”. ATIME RATEis anything divided by time. CHANGE is expressed by using the Greek letter, Delta, D. For example: Average SPEED is simply the “RATE at which DISTANCE changes”.

  3. The MEANING? For example, if t = 2 seconds, using x(t) = kt3=(1)(2)3= 8 meters. The derivative, however, tell us how our DISPLACEMENT (x) changes as a function of TIME (t). The rate at which Displacement changes is also called VELOCITY. Thus if we use our derivative we can find out how fast the object is traveling at t = 2 second. Since dx/dt = 3kt2=3(1)(2)2= 12 m/s

  4. Derivative of a power function

  5. Unit Vector Notation The proper terminology is to use the “hat” instead of the arrow. So we have i-hat, j-hat, and k-hat which are used to describe any type of motion in 3D space. How would you write vectors J and K in unit vector notation?

  6. Example A boat moves with a velocity of 15 m/s, N in a river which flows with a velocity of 8.0 m/s, west. Calculate the boat's resultant velocity with respect to due north. 8.0 m/s, W 15 m/s, N Rv q The Final Answer :

  7. Dot Products in Physics Consider this situation: A force F is applied to a moving object as it transverses over a frictionless surface for a displacement, d. As F is applied to the object it will increase the object's speed! But which part of F really causes the object to increase in speed? It is |F|Cos θ ! Because it is parallel to the displacement d In fact if you apply the dot product, you get (|F|Cos θ)d, which happens to be defined as "WORK" (check your equation sheet!) Work is a type of energy and energy DOES NOT have a direction, that is why WORK is a scalar or in this case a SCALAR PRODUCT (AKA DOT PRODUCT).

  8. Example Suppose a person moves in a straight line from the lockers( at a position  x = 1.0 m) toward the physics lab(at a position x = 9.0 m). “To the right” is taken as positive, as shown below The answer is positive so the person must have been traveling horizontallyin the positive direction.

  9. Example Suppose the person turns around! The answer is negative so the person must have been traveling horizontallyin the negative direction What is the DISPLACEMENT for the entire trip? What is the total DISTANCE for the entire trip?

  10. Instantaneous Velocity Instantaneous velocity is a measure of an object’s displacement per unit time at a particular point in time. Example: A body’s position is defined as:

  11. Instantaneous Acceleration Instantaneous velocity is a measure of an object’s velocity per unit time at a particular point in time. If the velocity of an object is defined as:

  12. What do the “signs”( + or -) mean?

  13. The 3 Kinematic equations There are 3 major kinematic equations than can be used to describe the motion in DETAIL. All are used when the acceleration is CONSTANT.

  14. Kinematics for the VERTICAL Direction All 3 kinematics can be used to analyze one dimensional motion in either the X direction OR the y direction.

  15. Examples A stone is dropped at rest from the top of a cliff. It is observed to hit the ground 5.78 s later. How high is the cliff? Which variable is NOT given and NOT asked for? Final Velocity! -163.7 m H =163.7m

  16. Examples A pitcher throws a fastball with a velocity of 43.5 m/s. It is determined that during the windup and delivery the ball covers a displacement of 2.5 meters. This is from the point behind the body when the ball is at rest to the point of release. Calculate the acceleration during his throwing motion. Which variable is NOT given and NOT asked for? TIME 378.5 m/s/s

  17. Examples How long does it take a car at rest to cross a 35.0 m intersection after the light turns green, if the acceleration of the car is a constant 2.00 m/s/s? Which variable is NOT given and NOT asked for? Final Velocity 5.92 s

  18. Examples A car accelerates from 12.5 m/s to 25 m/s in 6.0 seconds. What was the acceleration? Which variable is NOT given and NOT asked for? DISPLACEMENT 2.08 m/s/s

  19. Summary There are 3 types of MOTION graphs • Displacement(position) vs. Time • Velocity vs. Time • Acceleration vs. Time There are 2 basic graph models • Slope • Area

  20. Summary v (m/s) a (m/s/s) slope = v x (m) slope = a area = v area = x t (s) t (s) t (s)

  21. Comparing and Sketching graphs One of the more difficult applications of graphs in physics is when given a certain type of graph and asked to draw a different type of graph List 2 adjectives to describe the SLOPE or VELOCITY The slope is CONSTANT slope = v x (m) The slope is POSITIVE t (s) v (m/s) How could you translate what the SLOPE is doing on the graph ABOVE to the Y axis on the graph to the right? t (s)

  22. Example v (m/s) x (m) t (s) t (s) • 1st line • 2nd line The slope is constant The slope is “0” The slope is “-” • 3rd line The slope is “+” The slope is constant

  23. Example – Graph Matching What is the SLOPE(a) doing? a (m/s/s) The slope is increasing v (m/s) t (s) a (m/s/s) t (s) a (m/s/s) t (s) t (s)

  24. Force Diagrams FN • Weight(mg)– Always drawn from the center, straight down • Force Normal(FN)– A surface force always drawn perpendicular to a surface. • Tension(T or FT)– force in ropes and always drawn AWAY from object. • Friction(Ff)-Always drawn opposing the motion. T Ff T W1,Fg1 or m1g m2g A pictorial representation of forces complete with labels.

  25. Force Diagrams mg Ff FN

  26. New’s 1st Law and Equilibrium Since the Fnet = 0, a system moving at a constant speed or at rest MUST be at EQUILIBRIUM. TIPS for solving problems • Draw a FBD • Resolve anything into COMPONENTS • Write equations of equilibrium • Solve for unknowns

  27. Example FN Fa Fay Ff 30 Fax mg 10-kg box is being pulled across the table to the right at a constant speed with a force of 50N at an angle of 30 degrees above the horizontal. Calculate the Force of Friction Calculate the Normal Force

  28. Springs – Hooke’s Law One of the simplest type of simple harmonic motion is called Hooke's Law. This is primarily in reference to SPRINGS. The negative sign only tells us that “F” is what is called a RESTORING FORCE, in that it works in the OPPOSITE direction of the displacement.

  29. Hooke’s Law from a Graphical Point of View Suppose we had the following data: k =120 N/m

  30. Example A load of 50 N attached to a spring hanging vertically stretches the spring 5.0 cm. The spring is now placed horizontally on a table and stretched 11.0 cm. What force is required to stretch the spring this amount? 110 N 1000 N/m

  31. Newton’s Second Law • Tips: • Draw an FBD • Resolve vectors into components • Write equations of motion by adding and subtracting vectors to find the NET FORCE. Always write larger force – smaller force. • Solve for any unknowns The acceleration of an object is directly proportional to the NET FORCE and inversely proportional to the mass.

  32. Newton’s 2nd Law In which direction, is this object accelerating? The X direction! So N.S.L. is worked out using the forces in the “x” direction only FN Fa Ff mg A 10-kg box is being pulled across the table to the right by a rope with an applied force of 50N. Calculate the acceleration of the box if a 12 N frictional force acts upon it.

  33. Example A mass, m1 = 3.00kg, is resting on a frictionless horizontal table is connected to a cable that passes over a pulley and then is fastened to a hanging mass, m2 = 11.0 kg as shown below. Find the acceleration of each mass and the tension in the cable. FN T T m1g m2g

  34. Example (cont.)

  35. Where does the calculus fit in? There could be situations where you are given a displacement function or velocity function. The derivative will need to be taken once or twice in order to get the acceleration. Here is an example. You are standing on a bathroom scale in an elevator in a tall building. Your mass is 72-kg. The elevator starts from rest and travels upward with a speed that varies with time according to: When t = 4.0s , what is the reading on the bathroom scale (a.k.a. Force Normal)? 4.6 m/s/s 1036.8 N

  36. TWO types of Friction • Static – Friction that keeps an object at rest and prevents it from moving (no sliding) • Kinetic – Friction that acts during motion (two surfaces sliding)

  37. Force of Friction • The Force of Friction is directly related to the Normal Force. The coefficient of friction is a unitless constant that is specific to the material type and usually less than one.

  38. Example a) What is the coefficient of kinetic friction between the crate and the floor? A 1500 N crate is being pushed across a level floor at a constant speed by a force F of 600 N at an angle of 20° below the horizontal as shown in the figure. Fa FN Fay 20 Fax Ff mg

  39. Example FN Fa Fay 20 If the 600 N force is instead pulling the block at an angle of 20° above the horizontal as shown in the figure, what will be the acceleration of the crate. Assume that the coefficient of friction is the same as found in (a) Fax Ff mg

  40. Inclines q q q Ff FN q q • Tips • Rotate Axis • Break weight into components • Write equations of motion or equilibrium • Solve mg q

  41. Example Masses m1 = 4.00 kg and m2 = 9.00 kg are connected by a light string that passes over a frictionless pulley. As shown in the diagram, m1 is held at rest on the floor and m2 rests on a fixed incline of angle 40 degrees. The masses are released from rest, and m2 slides 1.00 m down the incline in 4 seconds. Determine (a) The acceleration of each mass (b) The coefficient of kinetic friction and (c) the tension in the string. T FN m2 Ff m2gcos40 40 T m2g m1 40 m2gsin40 m1g

  42. Example

  43. Horizontally Launched Projectiles To analyze a projectile in 2 dimensions we need 2 equations. One for the “x” direction and one for the “y” direction. And for this we use kinematic #2. Remember, the velocity is CONSTANT horizontally, so that means the acceleration is ZERO! Remember that since the projectile is launched horizontally, the INITIAL VERTICAL VELOCITY is equal to ZERO.

  44. Horizontally Launched Projectiles Example: A plane traveling with a horizontal velocity of 100 m/s is 500 m above the ground. At some point the pilot decides to drop some supplies to designated target below. (a) How long is the drop in the air? (b) How far away from point where it was launched will it land? 1010 m 10.1 seconds

  45. Vertically Launched Projectiles There are several things you must consider when doing these types of projectiles besides using components. If it begins and ends at ground level, the “y” displacement is ZERO: y = 0

  46. Vertically Launched Projectiles You will still use kinematic #2, but YOU MUST use COMPONENTS in the equation. voy vo q vox

  47. Example A place kicker kicks a football with a velocity of 20.0 m/s and at an angle of 53 degrees. (a) How long is the ball in the air? (b) How far away does it land? (c) How high does it travel? vo=20.0 m/s q = 53

  48. Example A place kicker kicks a football with a velocity of 20.0 m/s and at an angle of 53 degrees. (a) How long is the ball in the air? 3.26 s

  49. Example A place kicker kicks a football with a velocity of 20.0 m/s and at an angle of 53 degrees. (b) How far away does it land? 39.24 m

  50. Example A place kicker kicks a football with a velocity of 20.0 m/s and at an angle of 53 degrees. (c) How high does it travel? CUT YOUR TIME IN HALF! 13.01 m

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