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Physics 207: Lecture 6, Pg 1 Lecture 6 l Goals:  Discuss circular motion Chapters 5 & 6  Recognize different types of forces and know how they act on.

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Presentation on theme: "Physics 207: Lecture 6, Pg 1 Lecture 6 l Goals:  Discuss circular motion Chapters 5 & 6  Recognize different types of forces and know how they act on."— Presentation transcript:

1 Physics 207: Lecture 6, Pg 1 Lecture 6 l Goals:  Discuss circular motion Chapters 5 & 6  Recognize different types of forces and know how they act on an object in a particle representation  Identify forces and draw a Free Body Diagram  Begin to solve 1D and 2D problems with forces in equilibrium and non-equilibrium (i.e., acceleration) using Newton’s 1 st and 2 nd laws. Assignment: HW3, (Chapters 4 & 5, due 2/10, Wednesday) Finish reading Chapter 6 Exam 1 Wed, Feb. 17 from 7:15-8:45 PM Chapters 1-7

2 Physics 207: Lecture 6, Pg 2 Concept Check Q1. You drop a ball from rest, how much of the acceleration from gravity goes to changing its speed? A. All of it B. Most of it C. Some of it D. None of it Q2. A hockey puck slide off the edge of the table, at the instant it leaves the table, how much of the acceleration from gravity goes to changing its speed? A. All of it B. Most of it C. Some of it D. None of it

3 Physics 207: Lecture 6, Pg 3 Uniform Circular Motion (UCM) Arc traversed s =  r Tangential speed | v t | =  s/  t or (in the limit) ds/dt = r d  /dt Period T = 2  r / | v t | l Frequency f = 1 / T Angular position  Angular velocity  = d  /dt =  v t | / r Period (T): The time required to do one full revolution, 360 ° or 2  radians Frequency (f): 1/T, number of cycles per unit time Angular velocity or speed  = 2  f = 2  /T, number of radians traced out per unit time (in UCM average and instantaneous will be the same) r  vtvt s

4 Physics 207: Lecture 6, Pg 4 Example Question l A horizontally mounted disk 2 meters in diameter spins at constant angular speed such that it first undergoes 10 counter clockwise revolutions in 5 seconds and then, again at constant angular speed, 2 counter clockwise revolutions in 5 seconds. l 1 What is T the period of the initial rotation? 2 What is  the initial angular velocity? l 3 What is the tangential speed of a point on the rim during this initial period? 4 Sketch the  (angular displacement) versus time plot. l 5 What is the average angular velocity over the 1 st 10 seconds? l 6 If now the turntable starts from rest and uniformly accelerates throughout and reaches the same angular displacement in the same time, what must the angular acceleration be? l 7 What is the magnitude and direction of the acceleration after 10 seconds?

5 Physics 207: Lecture 6, Pg 5 Example l A horizontally mounted disk 2 meters in diameter spins at constant angular speed such that it first undergoes (1) 10 counter clockwise revolutions in 5 seconds and then, again at constant angular speed, (2) 2 counter clockwise revolutions in 5 seconds. l 1 What is T the period of the initial rotation? T = time for 1 revolution = 5 sec / 10 rev = 0.5 s also T = 2  r / | v t | ( just like x = x 0 + v  t   t = (x- x 0 ) / v )

6 Physics 207: Lecture 6, Pg 6 Example l A horizontally mounted disk 2 meters in diameter spins at constant angular speed such that it first undergoes 10 counter clockwise revolutions in 5 seconds and then, again at constant angular speed, 2 counter clockwise revolutions in 5 seconds. l 1 What is T the period of the initial rotation? 2 What is  the initial angular velocity?  = d  /dt =  /  t  = 10 2π radians / 5 seconds = 12.6 rad / s ( also 2  f = 2  / T )

7 Physics 207: Lecture 6, Pg 7 Example l A horizontally mounted disk 2 meters in diameter spins at constant angular speed such that it first undergoes 10 counter clockwise revolutions in 5 seconds and then, again at constant angular speed, 2 counter clockwise revolutions in 5 seconds. l 1 What is T the period of the initial rotation? 2 What is  the initial angular velocity? l 3 What is the tangential speed of a point on the rim during this initial period? | v t | = ds/dt = (r d  /dt = r  | v t | = r  = 1 m 12.6 rad/ s = 12.6 m/s

8 Physics 207: Lecture 6, Pg 8 Example l A horizontal turntable 2 meters in diameter spins at constant angular speed such that it first undergoes 10 counter clockwise revolutions in 5 seconds and then, again at constant angular speed, 2 counter clockwise revolutions in 5 seconds. l 1 What is the period of the turntable during the initial rotation T (time for one revolution) =  t /# of revolutions/ time = 5 sec / 10 rev = 0.5 s l 2 What is initial angular velocity?  = angular displacement / time = 2  f = 2  / T = 12.6 rad / s l 3 What is the tangential speed of a point on the rim during this initial period? | v t | = r d  /dt = r  = 1.0 m x 12.6 rad / s = 12.6 m/s

9 Physics 207: Lecture 6, Pg 9 Angular displacement and velocity Notice that if  ≡ d  / dt and, if  is constant, then integrating  = d  / dt, we obtain:  =   +   t ( In one dimensional motion if v = dx/dt = constant then x = x 0 + v  t ) Counter-clockwise is positive, clockwise is negative r  vtvt s  =   +   t

10 Physics 207: Lecture 6, Pg 10 Example l A horizontally mounted disk 2 meters in diameter spins at constant angular speed such that it first undergoes 10 counter clockwise revolutions in 5 seconds and then, again at constant angular speed, 2 counter clockwise revolutions in 5 seconds. l 1 What is T the period of the initial rotation? 2 What is  the initial angular velocity? l 3 What is the tangential speed of a point on the rim during this initial period? 4 Sketch the  (angular displacement) versus time plot.

11 Physics 207: Lecture 6, Pg 11 Sketch of  vs. time time (seconds)        (radians)  =   +   t  =  +  5  rad  =   +   t  =   rad + (  5) 5 rad  = 24 rad

12 Physics 207: Lecture 6, Pg 12 Example l A horizontally mounted disk 2 meters in diameter spins at constant angular speed such that it first undergoes 10 counter clockwise revolutions in 5 seconds and then, again at constant angular speed, 2 counter clockwise revolutions in 5 seconds. l 1 What is T the period of the initial rotation? 2 What is  the initial angular velocity? l 3 What is the tangential speed of a point on the rim during this initial period? 4 Sketch the  (angular displacement) versus time plot. l 5 What is the average angular velocity over the 1 st 10 seconds?

13 Physics 207: Lecture 6, Pg 13 Sketch of  vs. time time (seconds)        (radians)  =   +   t  =  +  5  rad  =   +   t  =   rad + (  5) 5 rad  = 24 rad 5 Avg. angular velocity =  /  t = 24  /10 rad/s

14 Physics 207: Lecture 6, Pg 14 Example l A horizontally mounted disk 2 meters in diameter spins at constant angular speed such that it first undergoes 10 counter clockwise revolutions in 5 seconds and then, again at constant angular speed, 2 counter clockwise revolutions in 5 seconds. l 6 If now the turntable starts from rest and uniformly accelerates throughout and reaches the same angular displacement in the same time, what must be the angular acceleration ?

15 Physics 207: Lecture 6, Pg 15 Key point ….. Angular acceleration is associated with tangential acceleration.

16 Physics 207: Lecture 6, Pg 16 Then angular velocity is no longer constant so d  /dt ≠ 0 Define tangential acceleration as a t = dv t /dt = r d  /dt So s = s 0 + (ds/dt) 0  t + ½ a t  t 2 and s =  r We can relate a t to d  /dt  =  o +  o  t +  t 2  =  o +  t l Many analogies to linear motion but it isn’t one-to-one Remember: Even if  is constant, there is always a radial acceleration. What if  is linearly increasing … atat r 1 2 atat r

17 Physics 207: Lecture 6, Pg 17 Circular motion also has a radial (perpendicular) component Centripetal Acceleration a r = v t 2 /r Circular motion involves continuous radial acceleration vtvt r arar Uniform circular motion involves only changes in the direction of the velocity vector, thus acceleration is perpendicular to the trajectory at any point, acceleration is only in the radial direction. Quantitatively (see text)

18 Physics 207: Lecture 6, Pg 18 Tangential acceleration?  =  o +  o  t +  t 2 (from plot, after 10 seconds) 24  rad = 0 rad + 0 rad/s  t + ½ (a t /r)  t 2 48  rad 1m / 100 s 2 = a t r  vtvt s 1 2 atat r l 6 If now the turntable starts from rest and uniformly accelerates throughout and reaches the same angular displacement in the same time, what must the “tangential acceleration” be? l 7 What is the magnitude and direction of the acceleration after 10 seconds?

19 Physics 207: Lecture 6, Pg 19 Non-uniform Circular Motion For an object moving along a curved trajectory, with non-uniform speed a = a r + a t (radial and tangential) arar atat a r = v2v2 r dt d| | v a t =

20 Physics 207: Lecture 6, Pg 20 Tangential acceleration? a t = 0.48  m / s 2 and  r =  o r + r  t = 4.8  m/s = v t a r = v t 2 / r= 23  2 m/s 2 r  vtvt s atat r l 7 What is the magnitude and direction of the acceleration after 10 seconds? Tangential acceleration is too small to plot!

21 Physics 207: Lecture 6, Pg 21 Angular motion, sign convention l If angular displacement velocity accelerations are counter clockwise then sign is positive. l If clockwise then negative

22 Physics 207: Lecture 6, Pg 22 What is the path? l Circular motion demo.

23 Physics 207: Lecture 6, Pg 23 What causes motion? (Actually changes in motion) What are forces ? What kinds of forces are there ? How are forces and changes in motion related ?

24 Physics 207: Lecture 6, Pg 24 Newton’s First Law and IRFs An object subject to no external forces moves with constant velocity if viewed from an inertial reference frame (IRF). If no net force acting on an object, there is no acceleration. l The above statement can be used to define inertial reference frames.

25 Physics 207: Lecture 6, Pg 25 IRFs  An IRF is a reference frame that is not accelerating (or rotating) with respect to the “fixed stars”.  If one IRF exists, infinitely many exist since they are related by any arbitrary constant velocity vector!  In many cases (i.e., Chapters 5, 6 & 7) the surface of the Earth may be viewed as an IRF

26 Physics 207: Lecture 6, Pg 26 Newton’s Second Law The acceleration of an object is directly proportional to the net force acting upon it. The constant of proportionality is the mass. l This expression is vector expression: F x, F y, F z l Units The metric unit of force is kg m/s 2 = Newtons (N) The English unit of force is Pounds (lb)

27 Physics 207: Lecture 6, Pg 27 Lecture 6 Assignment: HW3, (Chapters 4 & 5, due 2/10, Wednesday) Read rest of chapter 6


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