Part One Mechanics 力学 Part One Mechanics 力学

Chapter 1 Kinematics ( 质点 ) 运动学

第一章 质点运动学 (Kinematics) 第一章 质点运动学 (Kinematics) §1-1 参考系 质点 Frame of reference particle §1-2 位置矢量 位移 Position vector and displacement §1-3 速度 加速度 Velocity and acceleration §1-4 两类运动学问题 Two types of Problems §1-6 运动描述的相对性 Relative motion §1-5 圆周运动及其描述 Circular motion

§1-1 Frame of reference Particle 1. Frame of Reference !!Choose different objects as the reference frames to describe the motion of a given body, the indications will be different.

Coordinate system: Cartesian coordinate system( 直角坐标系） Mathematical reference frame For describing the motion of a given body quantitatively Nature coordinate system （自然坐标系）

2. Particle Ignore the size and shape of a body, only think of its mass Ideal model  Translational motion( 平动） An object can be simplified a particle when…  Its size << moving size

§1-2 Position Vector and Displacement P(x,y,z)P(x,y,z) z Y X 1. Position Vector Express the position of particle Magnitude: Direction:

In the two dimension: Its two components Path equation eliminating t (Position function) (Moving equation)

2. Displacement( 位移 ): Describe the change of position during a given time interval  t: In Cartesian coordinate system:

Its magnitude are different. Note: On the condition of limitation:

§1-3 Velocity and Acceleration Average velocity ： 1.Velocity Its direction is same with that of Average speed( 速率） :

（ Instantaneous 瞬时） velocity at time t ： In the tangent( 切线） of the path, to point at the advance direction. Direction:

Magnitude ： 速率 In Cartesian coordinate system:

2.Acceleration Average acceleration Instantaneous acceleration In the coordinate system:

Example 1.1: The of a particle is where  and  are constants. Find the velocity and acceleration. Solution:

Example 1. 2 The position function of a particle in SI unit is x=2t, y=19  2t 2. Calculate （ 1 ） Path function. （ 2 ） Velocity and acceleration at t=1s. (2) Take time derivation of position function Solution （1）（1） (1) Eliminate t from position function

(2) Take time derivation of velocity, we have Substitute t=1s into Eq.(1) and (2) The magnitude and direction of velocity: (Formed by and +x direction)

x h Example 1-3 Someone stands on a dam, pulls a boat with constant speed Find: Speed and acceleration of boat at any position x

Take time derivation of it Solution  Set up a coordinate system shown in Fig. Then the position of boat depend by r =- v 0 =v=v

§ 1-4 Two Types Problems in Kinematics (2) Given acceleration(or velocity) and initial condition, find the velocity and position by means of integration method 积分法. There are two kinds of problems to be solved: (1) Given position function, find the velocity and acceleration by using derivation method 微分法. See the examples above.

Solution Separate variables Integrate in both side of “=” Example1.4 (k is constant) and its speed is at t=0. Find An object moves along a straight line. Its Acce. is

Example1.5 A particle moves in a plane with an Acce., When t=0, its Vel. is at a initial point (0,0). Find its velocity at time t and path equation. Solution, we can obtain: Using, we have From Integrate

Using and the initial condition(0,0), we have

§1-5 Circular Motion Many of circular motions in our world

 Take any point on path as origin point of coordinate system  The position of a particle depend on the path length from O to P  Two coordinate axes are on the moving particle. 1. The nature coordinate system （自然坐标系） S= S(t) S :tangential direction :normal direction Both are unit vectors.

 Position ：  Velocity ： 2. Description of circular motion with nature coordinate system  Acce. =?

---- Normal acceleration Changes the direction of the velocity tangential acceleration Changes the magnitude of the velocity.

3. General curvilinear motion on a plane  -- Radius of curvature at any point on the curve  

4. Angular variables in circular motion Angular position Position function Angular displacement Angular velocity Angular acceleration

 － rad  － rad.s -1  － rad.s -2 Units: Counterclockwise( 反时针） : positive direction Clockwise （顺时针） : negative direction Two directions:

Special Example  Uniform circular motion  Circular motion with constant angular Acce.

5. Relation between linear & angular variables r 请自己推导！

§1-6 Relative motion 1. Relative motion

2.Relativity of the description about a motion :K 相对 K 的牵连坐标 K K’ A particle is moving in the space. K’ moves with respect to K Measure particle in K: Measure particle in K’: P

： K 测得质点速度 --- 绝对速度 ： K 测得质点速度 --- 相对速度 ： K 相对 K 速度 --- 牵连速度

Assuming that O and O’ coincide at t=0 and moves along the x-axis at speed of u, we have Galilean transformations ( 伽利略变换）