Recall from last time

Transcription

1 Welcome back to Physics 215 Today s agenda: Relative Motion Special relativity Forces Physics 215 Spring 2017 Lecture Recall from last time If we want to use (inertial) moving frames of reference, then velocities are not the same in different frames However constant velocity motions are always seen as constant velocity There is a simple way to relate velocities measured by different frames. Physics 215 Spring 2017 Lecture

2 Relative Motion in 2D Motion may look quite different in different inertial frames, e.g., ejecting ball from moving cart Cart frame = simple! Earth frame = complicated! Motion of cart Physics 215 Spring 2017 Lecture SG : While standing still, Otto shoots a basketball into the air. The initial x- component of the basketball velocity is 2m/s, while the y-component of the velocity is 1.5 m/s. Just as he is shooting, a cameraman rolls by in a car at moving at 2 m/s (along the x-axis). What is the apparent launch angle in the frame of the camera? A. Less than 90 degrees B. More than 90 degrees C. Exactly 90 degrees D. We don t have enough information to answer this question Physics 215 Spring 2017 Lecture

3 Ask a physicist How does time relativity work? Why does time pass more slowly for one who is travelling at the speed of light than one who travels at a significantly smaller velocity? Physics 215 Spring 2017 Lecture Chapter 36 Preview Physics 215 Spring 2017 Lecture

4 Einstein s Theories of Relativity Albert Einstein ( ) was one of the most influential thinkers in history. Einstein s first paper on Special Relativity, in 1905, dealt exclusively with inertial reference frames. Ten years later, Einstein published a more encompassing theory of General Relativity that considered accelerated motion and its connection to gravity. We are covering only the theory of Special Relativity. Physics 215 Spring 2017 Lecture Reference Frames The figure shows two reference frames called S and S. The coordinate axes in S are x, y, z and those in S are x, y, z. Frame S moves with velocity v relative to S. Physics 215 Spring 2017 Lecture

5 The Galilean Transformations Consider two reference frames S and S. S moves at a speed v relative to S in the +x direction. The origins of S and S coincide at t = 0. At a time t, an event happens at coordinates (x, y, z) in S and (x, y, z ) in S. These coordinates are related by the Galilean transformations of position: Physics 215 Spring 2017 Lecture Slide The Galilean Velocity Transformations Consider two reference frames S and S. S moves at a speed v relative to S in the +x direction. An object has velocity components (u x, u y, u z ) in S and (u x, u y, u z ) in S. These velocity components are related by the Galilean transformations of velocity: Physics 215 Spring 2017 Lecture

6 Einstein s Principle of Relativity According to Maxwell s theory of electromagnetism, light waves travel with speed Maxwell s equations are true in all inertial reference frames. Therefore, light travels at speed c in all inertial reference frames. Physics 215 Spring 2017 Lecture Einstein s Principle of Relativity Because of the principle of relativity, light travels at speed c in all inertial reference frames. Amy, Cathy, and Bill all measure both light waves to be moving at speed c in their own reference frame! Physics 215 Spring 2017 Lecture

7 Einstein s Principle of Relativity To date, every experiment designed to compare the speed of light in different reference frames has found that light travels at m/s in every inertial reference frame. Physics 215 Spring 2017 Lecture Simultaneity Two events 1 and 2 that take place at different positions x 1 and x 2 but at the same time t 1 = t 2, as measured in some reference frame, are said to be simultaneous in that reference frame. Simultaneity is determined by when the events actually happen, not when they are seen or observed. In general, simultaneous events are not seen at the same time because of the difference in light travel times from the events to an experimenter. Physics 215 Spring 2017 Lecture

8 QuickCheck 36.6 SG: A firecracker explodes high overhead. You notice a slight delay between seeing the flash and hearing the boom. At what time does the event firecracker explodes occur? A. At the instant you hear the boom. B. At the instant you see the flash. C. Very slightly before you see the flash. D. Very slightly after you see the flash. E. There s no unique answer because it depends on the observer. Physics 215 Spring 2017 Lecture The Relativity of Simultaneity The figure shows a railroad car traveling to the right with a velocity v. A firecracker is tied to each end of the car, just above the ground. The two events are simultaneous in Ryan s reference frame. Physics 215 Spring 2017 Lecture

9 The Relativity of Simultaneity Below is the sequence of events in Ryan s reference frame. Physics 215 Spring 2017 Lecture The Relativity of Simultaneity The figure shows the sequence of events in Peggy s reference frame. The two events are not simultaneous in Peggy s reference frame! This is called the relativity of simultaneity. Physics 215 Spring 2017 Lecture

10 QuickCheck 36.9 SG: Peggy is standing at the center of a railroad car as it passes Ryan. Firecrackers A and B at the ends of the car explode. A short time later, flashes from the two explosions reach Peggy at the same instant. In Ryan s reference firecracker A explodes firecracker B. A. before B. at the same time as C. after Physics 215 Spring 2017 Lecture Time Dilation The figure shows a special clock called a light clock. The light clock is a box with a light source at the bottom and a mirror at the top, separated by distance h. The light source emits a very short pulse of light that travels to the mirror and reflects back to a light detector beside the source. The clock advances one tick each time the detector receives a light pulse. Physics 215 Spring 2017 Lecture

11 Time Dilation Consider a light clock at rest in reference frame S. We call this the rest frame of the clock. Reference frame S moves to the right with velocity v relative to reference frame S. Physics 215 Spring 2017 Lecture Time Dilation The figure shows the light clock as seen in frame S. In this frame the clock must tick more slowly, since light must travel along the hypotenuse at speed c. Physics 215 Spring 2017 Lecture

12 Time Dilation In frame S, light takes time D t to travel along the hypotenuse from one mirror to the other. We can use the Pythagorean theorem to write: Solving for D t gives: If we define the dimensionless number b = v/c: Physics 215 Spring 2017 Lecture Time Dilation The time interval between two events that occur at the same position is called the proper time D t. The proper time is measured with a single clock that is present at both events. An inertial reference frame moving with velocity v = b c relative to the proper-time frame must use two clocks to measure the time interval: one at the position of the first event, the other at the position of the second event. The time interval between the two events in this frame is: Physics 215 Spring 2017 Lecture

13 The Twin Paradox George and Helen are twins. On their 25th birthday, Helen departs on a starship voyage to a distant star. Helen s starship travels at speed 0.95c to a star that is 9.5 light years from earth. Upon arriving, she immediately turns around and heads home at 0.95c. According to George, it takes Helen 20 years to travel out and back. George will be = 45 years old when his sister Helen returns. Physics 215 Spring 2017 Lecture The Twin Paradox In George s reference frame, Helen s moving clock will run more slowly than his stationary clock. Helen s clock travels with her, so it measures the proper time D t : Helen will be = just over 31 years old when she returns, about 14 years younger than her twin brother! Physics 215 Spring 2017 Lecture

14 The Twin Paradox In Helen s reference frame, it is George s clock that is moving, so it runs more slowly than hers. According to Helen s calculations, George will be younger than her when she returns! Who is correct? Physics 215 Spring 2017 Lecture The Twin Paradox Note that Helen s reference frame is not inertial. Helen must accelerate at the beginning, middle, and end of her journey. The principle of relativity applies only to inertial reference frames. Our equations for time dilation are only valid for inertial reference frames. Thus George s analysis and calculations are correct. Helen s analysis and calculations are not correct because she was trying to apply an inertial reference frame result while traveling in a noninertial reference frame. Helen is younger than George when she returns. Physics 215 Spring 2017 Lecture

15 What about lengthscales? Consider a rocket which travels from the sun to Saturn at speed v. In the reference frame S of the solar system, the distance traveled is L, and v = L/D t, where D t is the time for the rocket to make the journey in the S frame. Physics 215 Spring 2017 Lecture In the reference frame S, the rocket is at rest and the solar system moves to the left at speed v. The relative speed between S and S is the same for both reference frames: Physics 215 Spring 2017 Lecture

16 Length Contraction The time interval D t measured in frame S is the proper time D t because both events occur at the same position in frame S and can be measured by one clock. Since the speed is the same in both frames: The D t cancels, and the distance L in frame S is: where is the proper length measured in the frame where the objects are at rest. The length of any object is less when it is measured in a reference frame in which the object is moving. Physics 215 Spring 2017 Lecture Another Paradox? Carmen and Dan each carry meter sticks, and run past each other, in opposite directions, at a relative speed v = 0.9c. Dan s meter stick can t be both longer and shorter than Carmen s meter stick. Is this another paradox? Physics 215 Spring 2017 Lecture

17 Another Paradox? Relativity allows us to compare the same events as they re measured in two different reference frames. But the events by which Dan measures the length of Carmen s meter stick are not the same events as those by which Carmen measures the length of Dan s meter stick. In Dan s reference frame, Carmen s meter stick has been length contracted, and is less than 1 m in length. In Carmen s reference frame, Dan s meter stick has been length contracted, and is less than 1 m in length. There s no conflict between their measurements! Physics 215 Spring 2017 Lecture Recall: acceleration is same for all inertial FOR! We have: v PA = v PB + v BA For velocity of P measured in frame A in terms of velocity measured in B à D v PA /D t = D v PB /D t since v BA is constant à Thus acceleration measured in frame A or frame B is same! Physics 215 Spring 2017 Lecture

18 Physical Laws Since all FOR agree on the acceleration of object, they all agree on the forces that act on that object All such FOR are equally good for discovering the laws of mechanics Physics 215 Spring 2017 Lecture Forces are interactions between two objects (i.e., a push or pull of one object on another) can be broadly categorized as contact or non-contact forces have a direction and a magnitude -- vectors can be used to predict and explain the motion of objects described by Newton s Laws of Motion Physics 215 Spring 2017 Lecture

19 Types of forces Contact forces normal frictional tension Non-contact forces gravitational electric magnetic Physics 215 Spring 2017 Lecture SG: A hovercraft puck is a plastic disk with a built-in ventilator that blows air out of the bottom of the puck. The stream of air lifts up the puck and allows it to glide with negligible friction and at (almost) constant speed on any level surface. After the puck has left the instructor s hands the horizontal forces on the puck are: A. the force of the motion. B. the force of inertia. C. the force of the motion and the force of inertia. D. Neglecting friction and air drag, there are no horizontal forces. Physics 215 Spring 2017 Lecture

20 Newton s First law (Law of inertia) In the absence of a net external force, an object at rest remains at rest, and an object in motion continues in motion with constant velocity (i.e., constant speed and direction). Physics 215 Spring 2017 Lecture Remarks Compatible with principle of relativity All FOR moving with constant velocity will agree that no forces act Since forces are vectors, this statement can be applied to any components -- (x, y, z) separately Only net force required to be zero Physics 215 Spring 2017 Lecture

21 SG: A locomotive is pulling a long freight train at constant speed on straight tracks. The horizontal forces on the train cars are as follows: A. No horizontal forces at all. B. Only a pull by the locomotive. C. A pull by the locomotive and a friction force of equal magnitude and opposite direction. D. A pull by the locomotive and a somewhat smaller friction force in the opposite direction. Physics 215 Spring 2017 Lecture Common forces 1. Weight Gravitational force (weight) Universal force of attraction between 2 massive bodies For object near earth s surface directed downward with magnitude mg Notation: W BE Physics 215 Spring 2017 Lecture

22 Common forces 2. Normal forces Two objects A, B touch à exert a force at 90 o to surface of contact Notation: N AB is normal force on A due to B Physics 215 Spring 2017 Lecture SG A book is at rest on a table. Which of the following statements is correct? The vertical forces exerted on the book (and their respective directions) are 1. A weight force (down) only. 2. A weight force (down) and another force (up). 3. A weight force (down) and two other forces (one up and one down). 4. There is no force exerted on the book;; the book just exerts a force on the table (which is downward). Physics 215 Spring 2017 Lecture

23 Free-body diagram for book on table To solve problem introduce idea of free body diagram Show all forces exerted on the book. Do not show forces exerted by the book on anything else. Physics 215 Spring 2017 Lecture Remarks on free-body diagram for book Use point to represent object On earth, there will always be weight force (downwards, magnitude = mg) Since not accelerating, must be upward force also N BT normal force on book due to table No net force à N BT = mg Physics 215 Spring 2017 Lecture

24 Second Law: Newton s Second Law F on object = m a of object where F net is the vector sum of all external forces on the object considered m = (inertial) mass Acceleration measured relative to inertial FOR. Physics 215 Spring 2017 Lecture Illustration of Newton s Second Law Pull cart with constant force as displayed on force-meter How does cart respond? F on object = m a of object Physics 215 Spring 2017 Lecture

25 Newton s Third law Forces always occur in relation to pairs of objects. If A exerts some force on B, then B will exert a force back on A which is equal in magnitude but opposite in direction Physics 215 Spring 2017 Lecture Notation Force on A due to B = F AB Force on B due to A = F BA 3 rd law states: F AB = -F BA F AB and F BA referred to as 3 rd law pair Physics 215 Spring 2017 Lecture

26 Newton s Laws First Law: In the absence of external forces, an object at rest remains at rest and an object in motion continues in motion with constant velocity. Second Law: F net = F on object = m a Third Law: F AB = - F BA ( action = reaction ) [regardless of type of force and of motion of objects in question] Physics 215 Spring 2017 Lecture SG: A book is at rest on a table. Which of the following statements is correct? 1. The book exerts a force on the table, but the table does not exert a force on the book. 2. The table exerts a force on the book, but the book does not exert a force on the table. 3. The book exerts a large force on the table. The table exerts a smaller force on the book. 4. The book exerts a force on the table, and the table exerts a force of the same magnitude on the book. Physics 215 Spring 2017 Lecture

27 SG Consider a person sitting on a chair. We can conclude that the downward weight force on the person (by the Earth) and the upward normal force on the person (by the chair) are equal and in opposite direction, because 1. the net force on the person must be zero 2. the two forces form a Newton s third-law pair 3. neither of the above explanations 4. both of the above explanations Physics 215 Spring 2017 Lecture Free-body diagrams: Person sitting on chair Chair Person Earth (incomplete) Physics 215 Spring 2017 Lecture

28 SG There are two people facing each other, each on a separate cart. If person A pushes on person B, while person B does nothing, what will be the resulting motion of the carts? 1. Cart A doesn t move and Cart B moves backwards 2. Cart B doesn t move and Cart A moves backwards 3. Both carts move in opposite directions 4. Neither cart moves Physics 215 Spring 2017 Lecture