2D Kinematics Relative Motion Circular Motion

Transcription

1 2D Kinematics Relative Motion Circular Motion Lana heridan De Anza College Oct 5, 2017

2 Last Time range of a projectile trajectory equation projectile example began relative motion

3 Overview relative motion uniform circular motion

4 on P of the particle relative ative Relative to observer Motion B with the we see One that the veryvectors useful rtechnique P A for physical reasoning is considering n other frames of reference. (4.22) A reference frame is a coordinate system that an observer adopts. e, noting that v BA is con- Different observers may have different perspectives: different frames of reference. Consider a pair of observers, one stationary (A), one moving with constant velocity v BA. Both observe a (4.23) W W Galilean velocity particle P. by observer A and transformation up B is its icle velocity rather than v, o reference frames.) Equation equations. They relate A B P bservers in relative motion. When relative velocities are rpa rpb uter ones (P, A) match the t velocities for the particle, We can verify that by taking A vbat B vba Figure 4.20 A particle located x

5 Frames of Reference to the other. Q uick Quiz 39.1 How do we relate coordinates in different frames of reference? observer in the Two frames and y O vt x y x O x v P (event) Figure 39.2 An event occurs at Galilean transformations: a point P. The event is seen by two observers in inertial frames and x = x 9, + where vt, 9 y moves = y with, z a = velocity z, t = t v relative to. x uppose some observed by an o frame means tha The event s locati nates (x, y, z, t). W of an observer in with uniform rela Consider two in stant velocity v al We assume the or space at some inst and the observer frames in Figure fixed location in

6 Relative Motion (4.23) by observer A and up B is its icle velocity rather than v, o reference frames.) Equation equations. They relate bservers in relative motion. When relative velocities are uter ones (P, A) match the t velocities for the particle, We can verify that by taking A W W Galilean velocity transformation A vbat rpa B B rpb vba P x Figure 4.20 A particle located at P ris PA described = r PB by + two vobservers, BA t one in the fixed frame of refer- that A and v BA the isother a constant: in the Differentiating, notingence conclude that a P A 5 frame a B, which moves to the right P B with a constant velocity v BA. The he acceleration of the partis the same as that measured BA dr PA vector r PA is = dr PB the particle s +v position vector dt relative to A dt, and r P B is its tive to the first frame. position vector relative to B. u PA = u PB + v BA Differentiating again shows that observers in two frames related by a constant relative velocity see the same accelerations.

7 Relative Motion What if v BA is (4.23) not constant? W by observer A and u P B is its icle velocity rather than v, o reference frames.) Equation equations. They relate bservers in relative motion. When relative velocities are uter ones (P, A) match the t velocities for the particle, We can verify that by taking A W Galilean velocity transformation A vbat rpa B B rpb vba P x Figure 4.20 A particle located uppose B acceleratesat relative P is described to by Atwo with observers, a constant acceleration. one in the fixed frame of reference A and the other in the PA = r PB + v BA,i t + a BA t 2 r conclude that a P A 5 frame a B, which moves to the right P B with a constant velocity v BA. The he acceleration Differentiating of the partis the same as that measured vector relative to A, and r P B is its twice: vector r PA is the particle s position tive to the first frame. position vector relative to B. a PA = a PB + a BA

8 Relative Motion Example v BA dt Figure 4.20 A particle located at P is described by two observers, one in the fixed frame of refer- A boat crossing a wide river ence moves A and the with other ain speed the of 10.0 km/h frame relative to the water. The water B, which moves to the right with a constant in the velocity river v BA. has The a uniform speed of 5.00 km/h due east relative vector rthe PA is the Earth. particle s If position the boat heads due vector relative to A, and r P B is its north, determine the velocity position of the vector boat relative relative to B. to an observer standing on either bank. 1 ore, we conclude that a P A 5 a P B hat is, the acceleration of the partirence is the same as that measured city relative to the first frame. r peed of ver has a e Earth. locity of bank. s a river will not end up W a N vbr E you relau vre vbe W b N E vbr v re 2 Page Figure 97, 4.21 erway (Example & Jewett 4.8) (a) A boat aims directly across a u vbe

9 Relative Motion Example v BA dt Figure 4.20 A particle located at P is described by two observers, one in the fixed frame of refer- A boat crossing a wide river ence moves A and the with other ain speed the of 10.0 km/h frame relative to the water. The water B, which moves to the right with a constant in the velocity river v BA. has The a uniform speed of 5.00 km/h due east relative vector rthe PA is the Earth. particle s If position the boat heads due vector relative to A, and r P B is its north, determine the velocity position of the vector boat relative relative to B. to an observer standing on either bank. 1 v br = 10.0 km/h v re = 5.00 km/h ore, we conclude that a P A 5 a P B hat is, the acceleration of the partirence is the same as that measured city relative to the first frame. r peed of ver has a e Earth. locity of bank. s a river will not end up W a N vbr E you relau vre vbe W b N E vbr v re 2 Page Figure 97, 4.21 erway (Example & Jewett 4.8) (a) A boat aims directly across a u vbe

10 Relative Motion Example v BA dt Figure 4.20 A particle located at P is described by two observers, one in the fixed frame of refer- A boat crossing a wide river ence moves A and the with other ain speed the of 10.0 km/h frame relative to the water. The water B, which moves to the right with a constant in the velocity river v BA. has The a uniform speed of 5.00 km/h due east relative vector rthe PA is the Earth. particle s If position the boat heads due vector relative to A, and r P B is its north, determine the velocity position of the vector boat relative relative to B. to an observer standing on either bank. 1 v br = 10.0 km/h v re = 5.00 km/h ore, we conclude that a P A 5 a P B hat is, the acceleration of the partirence is the same as that measured city relative to the first frame. r peed of ver has a e Earth. locity of bank. s a river will not end up W a N vbr E you relau vre vbe W b N E vbr v re 2 Page Figure 97, 4.21 erway (Example & Jewett 4.8) (a) A boat aims directly across a imply use vector addition to find v be. u vbe v be = = 11.2 km/h ( ) 5 θ = tan 1 =

11 Relative Motion and Rip Currents Life and death application: rip currents. In shallow ocean water, a rip current is a strong flow of water away from the shore.

12 Relative Motion and Rip Currents If you are caught in one, which way should you swim? 1 Diagram from Wikipedia.

13 Relative Motion and Rip Currents If you are caught in one, which way should you swim? 1 Diagram from Wikipedia.

14 Relative Motion and Rip Currents If you are caught in one, which way should you swim? 1 Diagram from Wikipedia.

15 Relative Motion Example, #49, pg 206 A bolt drops from the ceiling of a moving train car that is accelerating northward at a rate of 2.50 m/s 2. (a) What is the acceleration of the bolt relative to the train car? (b) What is the acceleration of the bolt relative to the Earth? (c) Describe the trajectory of the bolt as seen by an observer inside the train car. (d) Describe the trajectory of the bolt as seen by an observer fixed on the Earth. Let the x-axis point north, and the y-axis point up.

16 Relative Motion Example, #49, pg 206 A bolt drops from the ceiling of a moving train car that is accelerating northward at a rate of 2.50 m/s 2. (a) What is the acceleration of the bolt relative to the train car? As the bolt falls it is no longer attached to the train car. It does not move with the car. Its acceleration is down, but the car keeps accelerating northward. a rel = g a = ( 9.8j 2.50i) m/s a in car = ( ) m/s 2 9.8, θ = tan a in car = 10.1 m/s 2, at 75.7 below the horizontal, southward.

17 Relative Motion Example, #49, pg 206 A bolt drops from the ceiling of a moving train car that is accelerating northward at a rate of 2.50 m/s 2. (b) What is the acceleration of the bolt relative to the Earth? (c) Describe the trajectory of the bolt as seen by an observer inside the train car. (d) Describe the trajectory of the bolt as seen by an observer fixed on the Earth.

18 Relative Motion Example, #49, pg 206 A bolt drops from the ceiling of a moving train car that is accelerating northward at a rate of 2.50 m/s 2. (b) What is the acceleration of the bolt relative to the Earth? g = 9.8 j m/s 2 (c) Describe the trajectory of the bolt as seen by an observer inside the train car. (d) Describe the trajectory of the bolt as seen by an observer fixed on the Earth.

19 Relative Motion Example, #49, pg 206 A bolt drops from the ceiling of a moving train car that is accelerating northward at a rate of 2.50 m/s 2. (b) What is the acceleration of the bolt relative to the Earth? g = 9.8 j m/s 2 (c) Describe the trajectory of the bolt as seen by an observer inside the train car. It accelerates from rest, relative to the car. It must follow the direction of the acceleration. A straight line, 75.7 below the horizontal, southward. (d) Describe the trajectory of the bolt as seen by an observer fixed on the Earth. It falls as a projectile with an initial horizontal velocity equal to the velocity of the car relative to the ground at the moment the screw drops.

20 Another Example, #53, pg106 A science student is riding on a flatcar of a train traveling along a straight, horizontal track at a constant speed of 10.0 m/s. The student throws a ball into the air along a path that he judges to make an initial angle of 60.0 with the horizontal and to be in line with the track. The student s professor, who is standing on the ground nearby, observes the ball to rise vertically. How high does she see the ball rise?

21 Another Example, #53, pg106 A science student is riding on a flatcar of a train traveling along a straight, horizontal track at a constant speed of 10.0 m/s. The student throws a ball into the air along a path that he judges to make an initial angle of 60.0 with the horizontal and to be in line with the track. The student s professor, who is standing on the ground nearby, observes the ball to rise vertically. How high does she see the ball rise? v x = 10.0 m/s and v i = v x cos θ h = v 2 i sin 2 θ 2g or v 2 f,y = v 2 i,y 2gh

22 Another Example, #53, pg106 A science student is riding on a flatcar of a train traveling along a straight, horizontal track at a constant speed of 10.0 m/s. The student throws a ball into the air along a path that he judges to make an initial angle of 60.0 with the horizontal and to be in line with the track. The student s professor, who is standing on the ground nearby, observes the ball to rise vertically. How high does she see the ball rise? v x = 10.0 m/s and v i = v x cos θ h = v 2 i sin 2 θ 2g or v 2 f,y = v 2 i,y 2gh h = 15.3 m

23 it. From the right triangle in Figure 3.2b, we find that sin u y/r and that cos x/r. (A review of trigonometric functions is given in Appendix B.4.) Therefore, Circular Motion: Radial Coordinates y (x, y) r sin u = y r cos u = x r r y O u x y tan u = x u x a e 3.2 (a) The plane polar coordinates of a point are represented by the distance r and the u, where u is measured counterclockwise from the positive x axis. (b) The right triangle used to (x, y) to (r, u). b

24 it. From the right triangle in Figure 3.2b, we find that sin u y/r and that cos x/r. (A review of trigonometric functions is given in Appendix B.4.) Therefore, Circular Motion: Radial Coordinates s conceptualize the problem. We wish to find r and u. We expect r to a y lem and re simply r coordisubstitunot have tution of alize step O r (x, y) Figure 3.3 (Example 3.1) sin u = y r cos u = x r Finding u polar coordinates when y u x tan u = ( 3.50, 2.50) x Cartesian coordinates are given. x e 3.2 (a) The plane polar coordinates of a point are represented by the distance r and the u, where u is measured counterclockwise from the positive x axis. (b) The right triangle used to (x, y) to (r, u). b u r r y (m) y x (m)

25 it. From the right triangle in Figure 3.2b, we find that sin u y/r and that cos x/r. (A review of trigonometric functions is given in Appendix B.4.) Therefore, Circular Motion: Radial Coordinates s conceptualize the problem. We wish to find r and u. We expect r to a y lem and re simply r coordisubstitunot have tution of alize step O r (x, y) Figure 3.3 (Example 3.1) sin u = y r cos u = x r Finding u polar coordinates when y u x tan u = ( 3.50, 2.50) x Cartesian coordinates are given. x e 3.2 Give (a) The position plane polar coordinates of ina terms point are ofrepresented r and θ. by Tothe transform distance r and from the r, θ to x, y: (x, y) to (r, u). x = r cos θ y = r sin θ u, where u is measured counterclockwise from the positive x axis. (b) The right triangle used to b u r r y (m) y x (m)

26 Radial Coordinates ˆ θ ˆ r ˆ θ ˆ r We can define perpendicular radial and tangential unit vectors, but their direction changes with the motion of a particle around a circle.

27 the center of the circle. Let us now find the magnitude of the acceleration of the particle. C diagram of the position and velocity vectors in Figure 4.15b. The figur the vector representing the change in position D r for an arbitrary ti As a car moves around a circular path its velocity changes, if not in The particle follows a circular path of radius r, part of which is shown by magnitude, then in direction. Circular Motion r O Top view v a b c r i v i r qu r f v f v i v f u If the radius remains constant and the speed of the car does as well, then v points toward the center of the circle. v Figure stant s particl, its v tion fo Dv, wh That means the acceleration vector does, too.

28 be modeled as a particle. If it moves Uniform Circular Motion onstant speed v, the magnitude of its The velocity vector points along a tangent to the circle (4.14) otion is given by is (4.15) (4.16) For uniform circular motion: the radius is constant the speed is constant the magnitude of the acceleration is constant ipetal Acceleration of the Earth AM r a c v Examples: of constan fectly circu form magn nucleus in hydrogen

29 Uniform Circular Motion The magnitude of the acceleration is given by a c = v 2 r If the constant speed is v, then the time period for one complete orbit is T = 2πr v

30 Uniform Circular Motion Pitfall Prevention 10.1 Remember the Radian In rotational equations, you must use angles expressed in radians. angle u from the reference line, ev the same angle u. Therefore, we object as well as with an individua We can also consider the Don t rate fall into atthe which trap of using the angular coordinate is angles measured in degrees in changing: rotational equations. O y r,t f ui u f,t i Figure 10.2 A particle on a rotating rigid object moves from to along the arc of a circle. In the time interval Dt 5 t 2 t i, the radial line of θ length = r moves θ f through θ i an angular displacement Du 5 u f 2 u i. Then we can define the angular speed, ω, as Average angular speed ω = dθ dt x position of a rigid object in its rot the object, such as a line connect angular position of the rigid obje the object and the fixed reference uch identification is similar to the lational motion as the distance x which is the origin, x 5 0. Therefo motion that the position x does in As the particle in question on o tion in a time interval Dt as in F sweeps out an angle Du 5 u f 2 u i. placement of the rigid object: The rate at which this angular di spins rapidly, this displacement c slowly, this displacement occurs in rates can be quantified by definin omega) as the ratio of the angular val Dt during which the displaceme v avg

31 Uniform Circular Motion ω gives the amount by which the angle θ advances in radians, per unit time. Therefore, ω = 2π T where T is the period (time for one revolution).

32 Uniform Circular Motion ω gives the amount by which the angle θ advances in radians, per unit time. Therefore, ω = 2π T where T is the period (time for one revolution). Putting in the expression for T (T = 2πr v ): ω = 2π v 2πr ω = v r

33 Uniform Circular Motion ω gives the amount by which the angle θ advances in radians, per unit time. Therefore, ω = 2π T where T is the period (time for one revolution). Putting in the expression for T (T = 2πr v ): ω = 2π v 2πr ω = v r This gives us another expression for the centripetal acceleration: a c = ω 2 r

34 Radial and Tangential Accelerations Quick Quiz A particle moves in a circular path of radius r with speed v. It then increases its speed to 2v while traveling along the same circular path. (i) The centripetal acceleration of the particle has changed by what factor? Choose one: A 0.25 B 0.5 C 2 D 4 1 Page 93, erway & Jewett

35 Radial and Tangential Accelerations Quick Quiz A particle moves in a circular path of radius r with speed v. It then increases its speed to 2v while traveling along the same circular path. (i) The centripetal acceleration of the particle has changed by what factor? Choose one: A 0.25 B 0.5 C 2 D 4 1 Page 93, erway & Jewett

36 Radial and Tangential Accelerations Quick Quiz A particle moves in a circular path of radius r with speed v. It then increases its speed to 2v while traveling along the same circular path. (ii) From the same choices, by what factor has the period of the particle changed? A 0.25 B 0.5 C 2 D 4 1 Page 93, erway & Jewett

37 Radial and Tangential Accelerations Quick Quiz A particle moves in a circular path of radius r with speed v. It then increases its speed to 2v while traveling along the same circular path. (ii) From the same choices, by what factor has the period of the particle changed? A 0.25 B 0.5 C 2 D 4 1 Page 93, erway & Jewett

38 ummary relative motion circular motion Quiz start of class Monday, Oct 9. Collected Homework! due tomorrow, Oct 6. (Uncollected) Homework erway & Jewett, Ch 4, onward from page 104. Problems: 45, 51 (relative motion probs, set last time) Ch 4, Problems: 37 (uniform circular motion, new) Ch 4, onward from page 104. Problems: 83 (rel. motion, new) Ch 4, onward from page 104. Problems: 40, 43 (nonuniform circular motion, can wait to do)