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Understanding Particle Motion in Mechanics: Kinematics and Dynamics

Explore the fundamental concepts of particle motion in mechanics through kinematics and dynamics. Dive into topics like position vectors, displacement, velocity, and acceleration to understand how and why objects move in a structured manner.

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Understanding Particle Motion in Mechanics: Kinematics and Dynamics

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  1. Part 1 Mechanics Space shuttle

  2. History

  3. Kinematics Dynamics Particle Motion Rotation Oscillation Trains of Thingking MECHANICS How does the matter move? Why does the matter move? 1. Kinematics: description of motion 1.1 Frame of reference and coordinate system 1.2 Physical quantities 1.3 Ideal model and motion

  4. Trains of Thingking MECHANICS Kinematics Dynamics How does the matter move? Why does the matter move? 2. Dynamics: relation of motion to its causes 2.1 Newton’s laws of motion 2.2 Work and energy 2.3 Momentum and impulse

  5. Structure Particle motion Reference of frame quantities to describe motion method to describe motion calculate method linear quantities Angular quantities Project motion circular motion curve motion

  6. scalar vector unit vector magnitude direction length coordinate axis displacement distance vector addition component vectors components Chapter1-3 Particle Motion Key word: positive negative scalar product vector product time interval instant curved line line-segment arrow origin point parallel perpendicular

  7. Key word: particle frame of reference position displacement average (/instantaneous) velocity average (/instantaneous ) acceleration speed free fall acceleration due to gravity projectile trajectory derivative normal component tangential component

  8. a pingpong the earth Which one is a particle? 1. Basic Concepts 1.1 Ideal Model • Particle: It is the body that has only the mass, but not its shape and size. Ideal Models: Simple pendulum, rigid body, point charge, harmonical oscillator…

  9. o x 1.2 Frame of Reference and Coordinate Axis • Frame of Reference: relative, usually refer to earth

  10. Cartesian natural • The Coordinate System: math conception • attached to the real-word bodies Other coordinates: polar, spherical, cylindrical, elliptical…

  11. Scalar: described by a single number with a unit, such as 1kg(mass), 103kg/m3(density), 1A(electrical current). Vector: has both magnitude and a direction, such as Represent by: 1.3 Scalars and Vectors

  12. :represent unit vectors in direction of +x-axis or +y-axis 1) Components of a vector:

  13. 2) Vector Addition (1) adding with components; (2) adding by geometrical way.

  14. Suppose: Then: Suppose: Then: Example: 3) Scalar Product (Dot Product)

  15. Suppose: Then: c Example: 4) Vector Product (Cross Product) Direction: determined by right-hand rule

  16. z Magnitudeis determined by: P(x,y,z)  r  z Direction is determined by:  C  y o x y A B x 2. Physics quantities to describe the particle motion 2.1 Position Vector , Displacement and Motional Equation 1) position vectors

  17. Displacement Vector: Caution! 2) displacement vectors Displacement is different from distance.

  18. Let: Caution! Discussion: A very small displacement during a small time interval A very small displacement: A very small distance: When time interval approaches to 0:

  19. Example: a radar station detects an airplane approaching directly from the east. At first observation, the range to plane is 360m at 400 above the horizon. The plane is tracked for another 1230 in the vertical east-west plane, the range at final contact being 790m. Find displacement of the airplanes during the period of observation.

  20. Solution:

  21. Example: Path equation Path graph y x2+y2=62 x 3) Motional equation Motional equation

  22. 1) Average Velocity z C A D S B o y x 2.2 Velocity and Speed 2) Average Speed

  23. Caution! z C A D S B o y x 3) Instantaneous Velocity 4) Instantaneous Speed

  24. Example: Chose the correct equation

  25. Example: How to determine the direction of V in the curved-line motion? y x

  26. vQ=tg2=0 Q VQ=? B VA=? VB=? tangent t2 vp= tg1 A x Example: How to find Velocity on an x-t graph? Vp=? P O t1 t Slope of tangent = instantaneous speed

  27. Average acceleration • Instantaneous acceleration 2.3 Acceleration 1) acceleration in Cartesian coordinates

  28. Components of velocity and acceleration Principle of superposition

  29. x D E F G O t Inflection point A B C Example :direction of acceleration Concave side of the path

  30. Example : Chose the correct equation:

  31. Example:The position of a particle is given by (1) calculate: when t=2s. (2)when is the velocity perpendicular to acceleration. Solution: (1) (2)

  32. Example: The motion of a particle is described by the function What kind of motion does it undergo? Self-test

  33. derivative integral Tow kinds of problems in kinematics Calculus-based-physics!

  34. Example: deduce the following equation if particle move in straight line with a=c, and t=0, v=v0, x=x0 . Self-test

  35. Example: Suppose the position of an object is given by x = t3-9t2+15t+1(SI). • Find the initial velocity. When does the object turn around? • Find the displacement and the distance traveled for the time interval t=0 to t=2s. Solution: Because condition for turning around is: v=0, the object turns around at t=1,t=5

  36. Example: The position of a particle is given by . • What kind of motion does it undergo? • Find the displacement and the distance traveled for the time interval t=/ to t= 2/. Solution: The particle moves along a circle with constant speed

  37. Example: A radio-controlled model car is moving on a plane (xy-plane). The car has x- and y-coordinates that vary with time according to x=2t, y=19-2t2(SI). Find the car’s coordinates at time t=1s and t=2s, thenfind the displacement and average velocity during the time interval. Find the instantaneous velocity and acceleration at t=1s. Find the path equation of the car. When the car is nearest to the origin point of xy-plane? What is the distance for t=0s to t=1s. Solution:

  38. Solution: This is a parabola

  39. Example:The motion of an object falling from rest in a resisting medium is described by the equation dv/dt=A-Bv, Where A and B are constants. In terms of A and B, find The initial acceleration. The velocity at which the acceleration becomes zero (the terminal velocity). Show that the velocity at any given t is given by Solution:

  40. h y x o x Example: The man on the bank drag the boat with constant velocity. Try to find the velocity and acceleration of the boat, When the distance between the boat and the bank is x. Set up coordinate axis in the picture, then draw the position vector of the boat. Solution:

  41. R tangential direction normal direction P O 2) Tangential and Normal Acceleration Can we represent the acceleration of a particle moving in a curved path in terms of components parallel and perpendicular to the velocity at each point?

  42. R O Q P for a very small time interval

  43. Magnitude: Magnitude: Direction: normal direction Direction: tangential direction Describe the change rate of direction of velocity with time Describe the change rate of the magnitude of velocity with time

  44. Example：correct the following formula Self-test

  45. Example：what is the character of 1) In straight line motion 2) In free fall motion 3) In projectile motion 4) In uniform circular motion 5)In nonuniform circular motion

  46. Example: A particle moves in a circle of radius R. The distance is described by the equation (b,c are constants, b2>Rc) When an= at? When a= c? Solution: For an = | at |

  47. y x Example8: Find an , at and  of projectile motion at any time. Suppose t=0, v=v0 , and makes an angle  with +x. Set up x,y coordinate axis Solution: Projectile motion can be considered as a combination of horizontal motion and vertical motion Self-test

  48. y x Another solution:

  49. y A R  o x 3. Angular quantities to describe the particle motion 1) Angular Displacement, Velocity and Acceleration Suppose a particle moves in a circle of radius R. We can use the single quantity  as a coordinate, Suppose a particle moves in a circle of radius R. We can use the single quantity  as a coordinate,  is called angular coordinate, and usually measured in radians. (1) Angular Displacement (2) Angular velocity: Average angular velocity Instantaneous angular velocity

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