PHYS1121 and MECHANICS

Transcription

1 PHYS1121 and MECHANICS Lecturer weeks 1-6: John Webb, Dept of Astrophysics, School of Physics Multimedia tutorials Where can I find the lecture slides? There will be a link from MOODLE but the files can be accessed directly here: There are pptx, ppt, and pdf directories The notes may be modified as we go through.

2 Joe Wolfe s lecture notes on Mechanics: MOODLE

3 Assumed mathematical knowledge: Plotting and reading graphs Appropriate use of significant figures Solving quadratic equations Trigonometric functions and some identities Exponential and log Solving simultaneous equations Differentiation and integration Solving simple differential equations The following sections will be rapid: Introduction to vector addition and subtraction Vector components and resolving vectors

4 Assumed physics knowledge: Officially none. However, we'll go more quickly through some parts of mechanics (eg. projectiles) compared to others. Homework Homework books are part of the course pack. Buy the lab book at the bookshop, then show it to staff at the FY lab to get the rest of the course pack. Labs Run independently from lecture syllabus.

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7 Chapter 2 Mo,on in One Dimension

8 Kinema,cs Describes mo,on while ignoring the agents that caused the mo,on For now, will consider mo,on in one dimension Along a straight line Will use the par,cle model A par,cle is a point-like object, has mass but infinitesimal size

9 The object s posi,on is its loca,on with respect to a chosen reference point Consider the point to be the origin of a coordinate system In the diagram, allow the road sign to be the reference point Posi,on

10 Posi,on-Time Graph The posi,on-,me graph shows the mo,on of the par,cle (car) The smooth curve is a guess as to what happened between the data points

11 Mo,on of Car Note the rela,onship between the posi,on of the car and the points on the graph Compare the different representa,ons of the mo,on

12 Data Table The table gives the actual data collected during the mo,on of the object (car) Posi,ve is defined as being to the right

13 Alterna,ve Representa,ons Using alterna,ve representa,ons is oren an excellent strategy for understanding a problem For example, the car problem used mul,ple representa,ons Pictorial representa,on Graphical representa,on Tabular representa,on Goal is oren a mathema,cal representa,on

14 Displacement Defined as the change in posi,on during some,me interval Represented as Δx Δx x f - x i SI units are meters (m) Δx can be posi,ve or nega,ve Different than distance the length of a path followed by a par,cle

15 Distance vs. Displacement An Example Assume a player moves from one end of a football field to the other and back Distance is twice the length of the field Distance is always posi,ve Displacement in this case is zero Δx = x f x i = 0 because x f = x i

16 Vectors and Scalars Vector quan,,es need both magnitude (size or numerical value) and direc,on to completely describe them Will use + and signs to indicate vector direc,ons Scalar quan,,es are completely described by magnitude only

17 Average Velocity The average velocity is rate at which the displacement occurs Δx xf xi v x, avg = Δt Δt The x indicates mo,on along the x-axis The dimensions are length /,me [L/T] The SI units are m/s Is also the slope of the line in the posi,on,me graph

18 Average Speed Speed is a scalar quan,ty same units as velocity d total distance / total,me: vavg t The speed has no direc,on and is always expressed as a posi,ve number Neither average velocity nor average speed gives details about the trip described

19 Instantaneous Velocity The limit of the average velocity as the,me interval becomes infinitesimally short, or as the,me interval approaches zero The instantaneous velocity indicates what is happening at every point of,me

20 Instantaneous Velocity, graph The instantaneous velocity is the slope of the line tangent to the x vs. t curve At point A this is the green line The light blue lines show that as Δt gets smaller, they approach the green line

21 Instantaneous velocity The general equa,on for instantaneous velocity is Δx dx vx = = Δt dt lim0 Δ t The instantaneous velocity can be posi,ve, nega,ve, or zero

22 Instantaneous Speed The instantaneous speed is the magnitude of the instantaneous velocity The instantaneous speed has no direc,on associated with it

23 Vocabulary Note Velocity and speed will indicate instantaneous values Average will be used when the average velocity or average speed is indicated

24 Par,cle Under Constant Velocity Constant velocity indicates the instantaneous velocity at any instant during a,me interval is the same as the average velocity during that,me interval v x = v x, avg The mathema,cal representa,on of this situa,on is the equa,on Δx xf xi vx = = or xf = xi + vxδt Δt Δt Common prac,ce is to let t i = 0 and the equa,on becomes: x f = x i + v x t (for constant v x )

25 Par,cle Under Constant Velocity, Graph The graph represents the mo,on of a par,cle under constant velocity The slope of the graph is the value of the constant velocity The y-intercept is x i

26 Average Accelera,on Accelera,on is the rate of change of the velocity Δvx vxf vxi axavg, = Δt t t Dimensions are L/T 2 SI units are m/s² In one dimension, posi,ve and nega,ve can be used to indicate direc,on f i

27 Instantaneous Accelera,on The instantaneous accelera,on is the limit of the average accelera,on as Δt approaches 0 a x Δ = lim x = x = Δ t 0 Δt dt dt 2 v dv dx The term accelera,on will mean instantaneous accelera,on If average accelera,on is wanted, the word average will be included 2

28 Instantaneous Accelera,on -- graph The slope of the velocity-,me graph is the accelera,on The green line represents the instantaneous accelera,on The blue line is the average accelera,on

29 Graphical Comparison Given the displacement-,me graph (a) The velocity-,me graph is found by measuring the slope of the posi,on-,me graph at every instant The accelera,on-,me graph is found by measuring the slope of the velocity-,me graph at every instant

30 Accelera,on and Velocity, 1 When an object s velocity and accelera,on are in the same direc,on, the object is speeding up When an object s velocity and accelera,on are in the opposite direc,on, the object is slowing down

31 Accelera,on and Velocity, 2 Images are equally spaced. The car is moving with constant posi,ve velocity (shown by red arrows maintaining the same size) Accelera,on equals zero

32 Accelera,on and Velocity, 3 Images become farther apart as,me increases Velocity and accelera,on are in the same direc,on Accelera,on is uniform (violet arrows maintain the same length) Velocity is increasing (red arrows are gehng longer) This shows posi,ve accelera,on and posi,ve velocity

33 Accelera,on and Velocity, 4 Images become closer together as,me increases Accelera,on and velocity are in opposite direc,ons Accelera,on is uniform (violet arrows maintain the same length) Velocity is decreasing (red arrows are gehng shorter) Posi,ve velocity and nega,ve accelera,on

34 Accelera,on and Velocity, final In all the previous cases, the accelera,on was constant Shown by the violet arrows all maintaining the same length The diagrams represent mo,on of a par,cle under constant accelera,on A par,cle under constant accelera,on is another useful analysis model

35 Graphical Representa,ons of Mo,on Observe the graphs of the car under various condi,ons Note the rela,onships amongst the graphs Set various ini,al veloci,es, posi,ons and accelera,ons

36 Kinema,c Equa,ons summary

37 Kinema,c Equa,ons The kinema,c equa,ons can be used with any par,cle under uniform accelera,on. The kinema,c equa,ons may be used to solve any problem involving one-dimensional mo,on with a constant accelera,on You may need to use two of the equa,ons to solve one problem Many,mes there is more than one way to solve a problem

38 Kinema,c Equa,ons, specific For constant a, v = v + a t xf xi x Can determine an object s velocity at any,me t when we know its ini,al velocity and its accelera,on Assumes t i = 0 and t f = t Does not give any informa,on about displacement

39 Kinema,c Equa,ons, specific For constant accelera,on, vxi + vxf v x, avg = 2 The average velocity can be expressed as the arithme,c mean of the ini,al and final veloci,es

40 Kinema,c Equa,ons, specific For constant accelera,on, 1 x ( ) f = xi + vx, avg t = xi + vxi + vfx t 2 This gives you the posi,on of the par,cle in terms of,me and veloci,es Doesn t give you the accelera,on

41 Kinema,c Equa,ons, specific For constant accelera,on, 1 xf = xi + vxit + axt 2 Gives final posi,on in terms of velocity and accelera,on Doesn t tell you about final velocity 2

42 Kinema,c Equa,ons, specific For constant a, ( ) v = v + 2a x x 2 2 xf xi x f i Gives final velocity in terms of accelera,on and displacement Does not give any informa,on about the,me

43 When a = 0 When the accelera,on is zero, v xf = v xi = v x x f = x i + v x t The constant accelera,on model reduces to the constant velocity model