2D Kinematics: Nonuniform Circular Motion Dynamics: Forces

Transcription

1 2D Kinematics: Nonuniform Circular Motion Dynamics: Forces Lana heridan De Anza College Oct 6, 2017

2 Last Time relative motion uniform circular motion

3 Overview nonuniform circular motion Introduce forces

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, 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 Page(a) 93, Can erway a & particle Jewett moving with instantaneous speed

15 Radial and Tangential Accelerations pg 105, #41 Problems cm inate cenbe at e the te of mall dge). arch ntal, Figin a away. ond, 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 a t = past the m/s lowest 2 ; apoint r = 1.29 on its m/s way 2 up, (calling its total outward acceleration positive) is i^ j^2 m/s 2. For that instant, (a) sketch a vector a = diagram 1.48 m/s showing 2 inward the at an components angle 29.9 of its acceleration, (b) determine the magnitude of its radial acceleration, and (c) determine backward the from speed the and direction velocity of travel of the ball Page(a) 93, Can erway a & particle Jewett moving with instantaneous speed

16 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.

17 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

18 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

19 ummary nonuniform circular motion introduced forces 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)