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PHYS 1441 – Section 501 Lecture #3

PHYS 1441 – Section 501 Lecture #3. Wednesday, June 9, 2004 Dr. Jae hoon Yu. Chapter two: Motion in one dimension Free Fall Chapter three: Motion in two dimension Coordinate systems Vector and Scalar Two dimensional equation of motion Projectile motion

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PHYS 1441 – Section 501 Lecture #3

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  1. PHYS 1441 – Section 501Lecture #3 Wednesday, June 9, 2004 Dr. Jaehoon Yu • Chapter two: Motion in one dimension • Free Fall • Chapter three: Motion in two dimension • Coordinate systems • Vector and Scalar • Two dimensional equation of motion • Projectile motion • Chapter four: Newton’s Laws of Motion Today’s homework is HW #2, due 6pm, next Wednesday, June 15!! PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  2. Announcements • E-mail distribution list: 25 of you have registered • Remember 3 extra credit points if done by midnight tonight • Next Wednesday is the last day of e-mail registration • -5 extra points if you don’t register by next Wednesday • A test message will be sent out this Friday • Homework: You are supposed to download the homework assignment, solve it offline and input the answers back online. • 44 registered • 40 submitted PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  3. Kinematic Equations of Motion on a Straight Line Under Constant Acceleration Velocity as a function of time Displacement as a function of velocities and time Displacement as a function of time, velocity, and acceleration Velocity as a function of Displacement and acceleration You may use different forms of Kinetic equations, depending on the information given to you for specific physical problems!! PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  4. Falling Motion • Falling motion is a motion under the influence of gravitational pull (gravity) only; Which direction is a freely falling object moving? • A motion under constant acceleration • All kinematic formula we learned can be used to solve for falling motions. • Gravitational acceleration is inversely proportional to the distance between the object and the center of the earth • The gravitational acceleration is g=9.80m/s2on the surface of the earth, most of the time. • The direction of gravitational acceleration is ALWAYS toward the center of the earth, which we normally call (-y); where up and down direction are indicated as the variable “y” • Thus the correct denotation of gravitational acceleration on the surface of the earth is g=-9.80m/s2 PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  5. Example for Using 1D Kinematic Equations on a Falling object Stone was thrown straight upward at t=0 with +20.0m/s initial velocity on the roof of a 50.0m high building, • What is the acceleration in this motion? g=-9.80m/s2 • (a) Find the time the stone reaches at the maximum height. • What happens at the maximum height? The stone stops; V=0 Solve for t • (b) Find the maximum height. PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  6. Example of a Falling Object cnt’d • (c) Find the time the stone reaches back to its original height. • (d) Find the velocity of the stone when it reaches its original height. • (e) Find the velocity and position of the stone at t=5.00s. Velocity Position PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  7. +y y1 r q +x x1 Coordinate Systems • Makes it easy to express locations or positions • Two commonly used systems, depending on convenience • Cartesian (Rectangular) Coordinate System • Coordinates are expressed in (x,y) • Polar Coordinate System • Coordinates are expressed in (r,q) • Vectors become a lot easier to express and compute How are Cartesian and Polar coordinates related? (x1,y1)=(r,q) O (0,0) PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  8. y q x qs (-3.50,-2.50)m Example Cartesian Coordinate of a point in the x-y plane are (x,y)= (-3.50,-2.50)m. Find the polar coordinates of this point. r PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  9. Vector and Scalar Vector quantities have both magnitude (size) and direction Force, gravitational acceleration, momentum Normally denoted in BOLD letters, F, or a letter with arrow on top Their sizes or magnitudes are denoted with normal letters, F, or absolute values: Scalar quantities have magnitude only Can be completely specified with a value and its unit Energy, heat, mass, weight Normally denoted in normal letters, E Both have units!!! PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  10. y D F A B x E C Properties of Vectors sizes directions • Two vectors are the same if their and the are the same, no matter where they are on a coordinate system. Which ones are the same vectors? A=B=E=D Why aren’t the others? C: The same magnitude but opposite direction: C=-A:A negative vector F: The same direction but different magnitude PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  11. A+B A+B A B A+B B B A A A-B A -B A B=2A Vector Operations • Addition: • Triangular Method: One can add vectors by connecting the head of one vector to the tail of the other (head-to-tail) • Parallelogram method: Connect the tails of the two vectors and extend • Addition is commutative: Changing order of operation does not affect the results A+B=B+A, A+B+C+D+E=E+C+A+B+D OR = • Subtraction: • The same as adding a negative vector:A - B = A + (-B) Since subtraction is the equivalent to adding a negative vector, subtraction is also commutative!!! • Multiplication by a scalar is increasing the magnitude A, B=2A PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  12. N Bsinq Bcosq E B 60o 20 q1 r A Example of Vector Addition A car travels 20.0km due north followed by 35.0km in a direction 60.0o west of north. Find the magnitude and direction of resultant displacement. Find other ways to solve this problem… PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  13. y Ay A q x Ax • Unit vectors are dimensionless vectors whose magnitudes are exactly 1 • Unit vectors are usually expressed in i, j, k or • Vectors can be expressed using components and unit vectors Components and Unit Vectors }Components (+,+) Coordinate systems are useful in expressing vectors in their components (Ax,Ay) } Magnitude (-,+) (-,-) (+,-) So the above vector Acan be written as PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  14. Examples of Vector Operations Find the resultant vector which is the sum of A=(2.0i+2.0j) and B =(2.0i-4.0j) Find the resultant displacement of three consecutive displacements: d1=(15i+30j +12k)cm, d2=(23i+14j -5.0k)cm, and d3=(-13i+15j)cm Magnitude PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  15. Displacement, Velocity, and Acceleration in 2-dim • Average Velocity: • Displacement: How is each of these quantities defined in 1-D? • Instantaneous Velocity: • Average Acceleration • Instantaneous Acceleration: PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  16. 2-dim Motion Under Constant Acceleration • Velocity vectors in x-y plane: • Position vectors in x-y plane: Velocity vectors in terms of acceleration vector • How are the position vectors written in acceleration vectors? PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  17. Example for 2-D Kinematic Equations A particle starts at origin when t=0 with an initial velocity v=(20i-15j)m/s. The particle moves in the xy plane with ax=4.0m/s2. Determine the components of velocity vector at any time, t. Velocity vector Compute the velocity and speed of the particle at t=5.0 s. PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  18. Example for 2-D Kinematic Eq. Cnt’d Angle of the Velocity vector Determine the x and y components of the particle at t=5.0 s. Can you write down the position vector at t=5.0s? PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  19. The only acceleration in this motion. It is a constant!! Projectile Motion PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  20. Projectile Motion • A 2-dim motion of an object under the gravitational acceleration with the assumptions • Gravitational acceleration, -g, is constant over the range of the motion • Air resistance and other effects are negligible • A motion under constant acceleration!!!!  Superposition of two motions • Horizontal motion with constant velocity and • Vertical motion under constant acceleration In a projectile motion, the only acceleration is gravitational one whose direction is always toward the center of the earth (downward). Show that a projectile motion is a parabola!!! ax=0 Plug in the t above PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu What kind of parabola is this?

  21. Example for Projectile Motion A ball is thrown with an initial velocity v=(20i+40j)m/s. Estimate the time of flight and the distance the ball is from the original position when landed. Which component determines the flight time and the distance? Flight time is determined by y component, because the ball stops moving when it is on the ground after the flight. Distance is determined by x component in 2-dim, because the ball is at y=0 position when it completed it’s flight. PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  22. vi h q Horizontal Range and Max Height What happens at the maximum height? • Based on what we have learned in the previous pages, one can analyze a projectile motion in more detail • Maximum height an object can reach • Maximum range At the maximum height the object’s vertical motion stops to turn around!! Since no acceleration in x, it still flies even if vy=0 PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  23. Maximum Range and Height This formula tells us that the maximum height can be achieved when qi=90o!!! • What are the conditions that give maximum height and range of a projectile motion? This formula tells us that the maximum range can be achieved when 2qi=90o, i.e., qi=45o!!! PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  24. Only physical answer Example for Projectile Motion • A stone was thrown upward from the top of a building at an angle of 30o to horizontal with initial speed of 20.0m/s. If the height of the building is 45.0m, how long is it before the stone hits the ground? • What is the speed of the stone just before it hits the ground? PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  25. Maximum Range and Height This formula tells us that the maximum height can be achieved when qi=90o!!! • What are the conditions that give maximum height and range of a projectile motion? This formula tells us that the maximum range can be achieved when 2qi=90o, i.e., qi=45o!!! PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  26. F1 F2 Force We’ve been learning kinematics; describing motion without understanding what the cause of the motion was. Now we are going to learn dynamics!! FORCEs are what cause an object to move Can someone tell me what FORCE is? The above statement is not entirely correct. Why? Because when an object is moving with a constant velocity no force is exerted on the object!!! FORCEs are what cause any changes in the velocity of an object!! When there is force, there is change of velocity. Forces cause acceleration. What does this statement mean? What happens there are several forces being exerted on an object? Forces are vector quantities, so vector sum of all forces, the NET FORCE, determines the motion of the object. When net force on an objectis 0, it has constant velocity and is at its equilibrium!! NET FORCE, F= F1+F2 PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  27. More Force There are various classes of forces Contact Forces: Forces exerted by physical contact of objects Examples of Contact Forces: Baseball hit by a bat, Car collisions Field Forces: Forces exerted without physical contact of objects Examples of Field Forces: Gravitational Force, Electro-magnetic force What are possible ways to measure strength of Force? A calibrated spring whose length changes linearly with the exertedforce. Forces are vector quantities, so addition of multiple forces must be done following the rules of vector additions. PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  28. Newton’s First Law and Inertial Frames Aristotle (384-322BC): A natural state of a body is rest. Thus force is required to move an object. To move faster, ones needs higher force. Galileo’s statement on natural states of matter: Any velocity once imparted to a moving body will be rigidly maintained as long as the external causes of retardation are removed!! What does this statement tell us? • When no force is exerted on an object, the acceleration of the object is 0. • Any isolated object, the object that do not interact with its surroundings, is either at rest or moving at a constant velocity. • Objects would like to keep its current state of motion, as long as there is no force that interferes with the motion. This tendency is called the Inertia. Galileo’s statement is formulated by Newton into the 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. A frame of reference that is moving at constant velocity is called anInertial Frame PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  29. Mass Mass: A measure of the inertia of a body or quantity of matter • Independent of the object’s surroundings: The same no matter where you go. • Independent of method of measurement: The same no matter how you measure it. The heavier an object gets the bigger the inertia!! It is harder to make changes of motion of a heavier object than the lighter ones. The same forces applied to two different masses result in different acceleration depending on the mass. Note that mass and weight of an object are two different quantities!! Weight of an object is the magnitude of gravitational force exerted on the object. Not an inherent property of an object!!! Weight will change if you measure on the Earth or on the moon. PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  30. Newton’s Second 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. How do we write the above statement in a mathematical expression? Since it’s a vector expression, each component should also satisfy: From the above vector expression, what do you conclude the dimension and unit of force are? The unit of force in SI is For ease of use, we define a new derived unit called, a Newton (N) PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  31. Example 4.2 What constant net force is required to bring a 1500kg car to rest from a speed of 100km/h within a distance of 55m? Acceleration!! What do we need to know to figure out the force? What are given? Initial speed: Final speed: Displacement: This is a one dimensional motion. Which kinetic formula do we use to find acceleration? Acceleration Thus, the force needed to stop the car is • Linearly proportional to the mass of the car • Squarely proportional to the speed of the car • Inversely proportional to the force by the brake Given the force how far does the car move till it stops? PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  32. q2=-20o F2 F1 q1=60o Example for Newton’s 2nd Law of Motion Determine the magnitude and direction of acceleration of the puck whose mass is 0.30kg and is being pulled by two forces, F1 and F2, as shown in the picture, whose magnitudes of the forces are 8.0 N and 5.0 N, respectively. Components ofF1 Components ofF2 Components of total force F Magnitude and direction of acceleration a Acceleration Vector a PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  33. Gravitational Force and Weight The attractive force exerted on an object by the Earth Gravitational Force, Fg Weight of an object with mass M is Since weight depends on the magnitude of gravitational acceleration, g, it varies depending on geographical location. By measuring the forces one can determine masses. This is why you can measure mass using spring scale. PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  34. Newton’s Third Law (Law of Action and Reaction) If two objects interact, the force, F21, exerted on object 1 by object 2 is equal in magnitude and opposite in direction to the force, F12, exerted on object 1 by object 2. F21 F12 The action force is equal in magnitude to the reaction force but in opposite direction. These two forces always act on different objects. What is the reaction force to the force of a free fall object? The force exerted by the ground when it completed the motion. Stationary objects on top of a table has a reaction force (normal force) from table to balance the action force, the gravitational force. PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

  35. F12 F21=-F12 Example of Newton’s 3rd Law A large man and a small boy stand facing each other on frictionless ice. They put their hands together and push against each other so that they move apart. a) Who moves away with the higher speed and by how much? m M b) Who moves farther while their hands are in contact? Given in the same time interval, since the boy has higher acceleration and thereby higher speed, he moves farther than the man. PHYS 1441-501, Summer 2004 Dr. Jaehoon Yu

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