2D Kinematics: Nonuniform Circular Motion Dynamics: Laws of Motion Newton s 1st & 2nd Laws

Transcription

1 2D Kinematics: Nonuniform Circular Motion Dynamics: Laws of Motion Newton s 1st & 2nd Laws Lana heridan De Anza College Oct 6, 2017

2 Last Time relative motion uniform circular motion

3 Overview nonuniform circular motion Introduce forces Newton s Laws! (1st & 2nd)

4 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 ipetal Acceleration of the Earth AM r a c v Examples: of constan fectly circu form magn nucleus in hydrogen the magnitude of the acceleration is constant, a c = v 2 = ω 2 r r

5 ngential acceleration component causes a change in the speed v of the part omponent is parallel to the instantaneous velocity, and its magnitude is give Non-Uniform Circular Motion a t 5 ` dv dt ` (4 Path of particle a t a r a a a r a r a t a t a

6 Radial and Tangential Accelerations a t 5 ` dt ` Path of particle a t ved e nt on ts - ial a a r a a t a r a = a t + a r a = a t ˆθ acˆr Let ˆθ(t) be a unit vector in the direction of the velocity. Note that its direction changes with time! v(t) = v(t) ˆθ(t)

7 Radial and Tangential Accelerations a = dv dt ; v(t) = v(t) ˆθ(t) Find the acceleration using the product rule: a = dv dt ˆθ + v dˆθ dt

8 Radial and Tangential Accelerations a = dv dt ; v(t) = v(t) ˆθ(t) Find the acceleration using the product rule: a = dv dt ˆθ + v dˆθ dt ( The term dv ˆθ ) dt is all in the tangential component of the acceleration. ( ) But how to find what is? We need to find how ˆθ changes v dˆθ dt with time. (It rotates, but at what rate?)

9 Radial and Tangential Accelerations: How do the perpendicular axes change? Let s find out! ˆθ is changing, so let us say that ˆθ i is the initial tangential unit vector and ˆθ f is the final tangential unit vector.

10 Radial and Tangential Accelerations: How do the perpendicular axes change? ˆθ is changing, so let us say that ˆθ i is the initial tangential unit vector and ˆθ f is the final tangential unit vector. Both vectors are 1 unit long (the same length). Both remain perpendicular to the radial direction. Therefore we have two similar triangles! r i vθ ˆ ii r qu r f vθ ˆ f f vθ ˆ i u ˆ vθ ff v Δθˆ

11 Radial and Tangential Accelerations: How do the perpendicular axes change? ˆθ is changing, so let us say that ˆθ i is the initial tangential unit vector and ˆθ f is the final tangential unit vector. Both vectors are 1 unit long (the same length). Both remain perpendicular to the radial direction. Therefore we have two similar triangles! r i vθ ˆ ii r qu r f vθ ˆ f f vθ ˆ i u ˆ vθ ff v Δθˆ (ˆθ) = r ˆθ i r d dt ˆθ = 1 r dr dt = v r This tells us how fast the tangential unit vector changes in direction.

12 Radial and Tangential Accelerations: How do the perpendicular axes change? d dt ˆθ = v r This tells us how fast the tangential unit vector changes in direction. Now consider that the direction of change must be radial! d dt ˆθ = v r ˆr

13 Radial and Tangential Accelerations a = dv dt = d (v ˆθ) dt Find the acceleration using the product rule: We said a = a t ˆθ acˆr so, a = dv dt ˆθ + v dˆθ dt = dv dt ˆθ + v ( v ) r ˆr = dv dt ˆθ v 2 r ˆr tangen. radial a c = v 2 r

14 Radial and Tangential Accelerations pg 105, #41 Problems cm inate cenbe at e the te of mall dge). arch ntal, Figin a away. ond, lera- 41. A train slows down as it rounds a sharp horizontal M turn, going from 90.0 km/h to 50.0 km/h in the 15.0 s it takes to round the bend. The radius of the curve is 150 m. Compute the acceleration at the moment the train speed reaches 50.0 km/h. Assume the train continues to slow down at this time at the same rate. 42. A ball swings counterclockwise in a vertical circle at the end of a rope 1.50 m long. When the ball is 36.9 past the lowest point on its way up, its total acceleration is i^ j^2 m/s 2. For that instant, (a) sketch a vector diagram showing the components of its acceleration, (b) determine the magnitude of its radial acceleration, and (c) determine the speed and velocity of the ball. 43. (a) Can a particle moving with instantaneous speed 1 Page , erway m/s on & Jewett a path with radius of curvature 2.00 m

15 Forces! A force is a push or a pull that an object experiences. Forces are connected to acceleration of an object that has mass. Unbalanced forces cause an acceleration. Forces are vectors.

16 Forces Two types of forces contact forces another object came into contact with the object field forces a kind of interaction between objects without them touching each other

17 Forces Force type examples: er 5 The Laws of Motion es of e, a force hin the the e boxed bject. Contact forces a b c Field forces m M q Q Iron N d e f orbit around the Earth. This change in velocity is caused by the gravitational force exerted by the Earth on the Moon. When a coiled spring is pulled, as in Figure 5.1a, the spring stretches. When a stationary cart is pulled, as in Figure 5.1b, the cart moves. When a football is kicked, as 1 erway in Figure & 5.1c, Jewett it is both deformed and set in motion. These situations are all

18 Diagrams of Forces This is a physical picture. (a) ketch the forces Physical picture We need to identify the system we want to study. Here: the chair. 1 (b) Isolate the object of interest (c) Choose a convenient coordinate sy Diagrams from Walker, Physics.

19 Diagrams of Forces: Free-Body Diagram N Physical picture This is a free-body diagram. We represent the chair as a point-particle with force vectors pointing outward. interest (c) Choose a convenient coordinate system (d) Resolve fo y N N N x = 0 N y = N W F W W x = 0 W y = O x Free-body diagram We also picked a coordinate system (x, y axes).

20 Forces are Vectors +y N + W = (N W ) j = 0 or sometimes written F n + F g = (F n F g ) j = 0 1 Figure from

21 Forces are Vectors A downward force F 1 elongates the spring 1.00 cm. A downward force F 2 elongates the spring 2.00 cm. When F 1 and F 2 are applied together in the same direction, the spring elongates by 3.00 cm. When F 1 is downward and F 2 is horizontal, the combination of the two forces elongates the spring by 2.24 cm F 1 u F 2 F a F 1 b F 2 c F 1 F 2 d Figure 5.2 of a force is t scale. and its direction is u 5 tan 21 (20.500) Because forces have been experimentally verified to behave as vectors, you must use the rules of vector addition to obtain the net force on an object.

22 examples of a class of between two objects. molecules on the wall The simple rules that govern the way in which forces act Another and effect class of f motion. Newton s Laws Isaac Newton English physicist and mathematician Bridgeman-Giraudon/Art Resource, NY between two objects. of attraction between of this class of force. the planets in orbit ar that one electric char force between an elec of a field force is the f The distinction be have been led to beli level, all the forces w (field) forces of the ty els for macroscopic ph The only known fund forces between object forces between subato tive decay processes. and electromagnetic

23 Newton s First Law Newton I (as commonly stated) In an inertia reference frame, an object in motion tends to stay in motion (with constant velocity) and an object at rest tends to stay at rest, unless acted upon by a net force. Newton I (textbook version) If an object does not interact with other objects, it is possible to identify a reference frame in which the object has zero acceleration. A zero-acceration reference frame is called an inertial reference frame. An object for these purposes is something with mass.

24 Newton s First Law This does not really correspond with our expectation from daily life. In our everyday environment, everything seems to naturally slow to a stop. 0 Figure from JPL.

25 Newton s First Law This does not really correspond with our expectation from daily life. In our everyday environment, everything seems to naturally slow to a stop. But we now know of other environments where we see this behavior. 0 Figure from JPL.

26 ng that v BA is con- Different Observers Observer A is at rest and observer B is moving with velocity v BA. uppose observer A sees the particle P at rest. Observer B sees it (4.23) W W Galilean velocity moving, with velocity v BA. ver A and transformation up B is its city rather than v, nce frames.) Equaations. They relate A B P in relative motion. lative velocities are rpa rpb es (P, A) match the ies for the particle, erify that by taking A vbat B vba Figure 4.20 A particle located Both agree that Newton s first law holds for P! at P is described by two observers, one in the fixed frame of reference A and the other in the x

27 Newton s First Law Implications Quick Quiz Which of the following statements is correct? I. It is possible for an object to have motion in the absence of forces on the object. II. It is possible to have forces on an object in the absence of motion of the object. A I. only B II. only C Neither I. or II. D Both I. and II. 2 &J page 114

28 Newton s econd Law The really important one. Newton II In an inertial reference frame, the sum of the forces (net force) on an object is equal to the mass of the object times its acceleration: F net = m a F net = i F i where F i are individual separate forces that we sum to get the net force.

29 Newton s econd Law F net = m a Acceleration is directly proportional to the net force and in the same direction. The constant of proportionality is the mass, m. F net a Alternatively, given a net force, the magnitude of the acceleration is inversely proportional to the mass of the object. a m This expression assumes the mass of the object is constant!

30 Units of Force Newton s second law gives us units for force. F = ma Newtons, N = (kg) (ms 2 ) 1N = 1 kg m s 2 : on Earth s surface there are roughly 10 N per kg. Why?

31 Newton s econd Law Implications Question. If an object with mass 16 kg is acted upon by two forces, F 1 = (10N)i and F 2 = (2N)i, what is the object s acceleration? A 1 2 ms 2 i. B ms 2 i. C 3 4 ms 2 i. D 2 ms 4 i.

32 Newton s econd Law Implications Question. If an object is not accelerating, can there be forces acting on it? A Yes. B No. C I don t pay attention in class. D I choose randomly because I ve no idea what s going on.

33 Newton s econd Law Implications Quick Quiz You push an object, initially at rest, across a frictionless floor with a constant force for a time interval t, resulting in a final speed of v for the object. You then repeat the experiment, but with a force that is twice as large. What time interval is now required to reach the same final speed v? A 4 t B 2 t C t 2 D t 4 4 &J page 116.

34 ummary nonuniform circular motion Newton s laws! First Test Friday, Oct 13. Quiz start of class, Monday. (Focus on Ch 4.) (Uncollected) Homework erway & Jewett, Ch 4, onward from page 104. Problems: 40, 43 (nonuniform circular motion, set last time) Ch 5, onward from page 139. Problems: 3, 5, 9, 19