Chapter 2 Dynamics( 动力学） ( Newton’s Laws of Motion )

Chapter 2 Dynamics(动力学） (Newton’s Laws of Motion)

§2-1 Newton’s Laws(牛顿定律) §2-2 General Properties of Forces in Mechanics 力的基本性质 The Fundamental Forces of Nature 四种基本力 §2-3 Applying Newton’s Law of Motion 牛顿定律应用

§2-1 Newton’s laws Constant vector (including zero) when First law Second law In inertial reference frame, --momentum

For m =const. Third law Action-reaction law

Notes The principle of superposition of forces or principle of independence of forces.

 The components of second law Cartesian Coor. Sys. Nature Coor. Sys.  Newton’s law are used only for inertial reference frame

§2-2 General Properties of Forces The Fundamental Forces of Nature 1.Weight (Earth gravity) Gravitational Force

F 0 r d 2. Molecular force and elastic force --The force between atoms ,,s,t-- constant depend on experiment

F--- Restoring force x--- deformation -- Hooke’s law

When --coefficient of static friction 3. Frictional force between two solid surfaces Static frictional force Maximum static frictional force

When --Coefficient of kinetic friction. Kinetic frictional force What is the reason of friction ？？

4. Fluid friction (Drag force) (at low speed) (at high speed) Experienced formula

Action at a distance force 5. The Fundamental Forces of Nature  Gravitational Force  Electromagnetic Force  Strong Force (~104N, acting on nuclear particles, hold the atomic nucleus together) Distance of force <10-15m Short distance force

 Weak Force (~10-2N acting on most elementary particle, only manifest itself in certain kinds of radioactive decay reactions ) Distance of force <10-17m Short distance force

§2-3 Applications of Newton’s Law Solving procedure  Select a body to which Newton’s laws will be applied.  Draw a free-body diagram. Draw all the forces acting on the chosen body. Analyze the accelerations of the chosen body.

 Set up a coordinate system.  Write the component equations of Newton’s Second Law for each body. Solve the equations to find unknown quantities.  Analyze the results if necessary.

[Example1] A incline of mass M is placed on a table and a block of mass m is put on M. Suppose all contact surface are frictionless and angle  is known. Find the Acce. of M , and the Acce.of m with respect to M.

--M with respect to the earth --m with respect to M [Solution] y For M: For m: x and

We get

A ball of mass m is sinking in the water. Suppose the drag force exerted on the ball is , the floating force is F. The initial condition is that the ball is at rest on the surface of water at the beginning. [Example 2] Find (1) the speed of the ball, (2) the sinking distance of the ball at any time. Solution (1) Draw all the forces acting on the ball. Analyze its acceleration

o X Apply New.’s Law to the ball Separate the variables Integrate in both side of “=”

From We can get Discuss When t ，v(t) ， Whent， vT-- terminal speed (2) We have

Make a integration, we can get

[Example 3] A small block of mass m slides on a horizontal frictionless surface as it travels around the inside of radius R. Suppose its initial speed is v0 and the between it and the wall is known. Find its speed at any time. R A v0

Solution Draw forces Analyze accelerations Set up nature coordinate system

From We can get the dynamics equations Normal direction Tangential direction

Solve the equations, we have

 [Example 4 ] One end of a weightless rope of lengthlis fixedon a nail. Its another end connects with a ball of mass m. Pull the ball and make the rope horizontally first. Then let the ball fall down. Calculate the speed of the ball and the tension of the rope at any angle.

 mg Solution Draw forces Analyze accelerations Set up nature coordinate system Normal direction ….. Tangential direction …..

Rewrite Eq. Separate the variables After integration, we have Substitute this solution to Eq.  we have

[例5]均质软绳单位长质量为，开始时盘 绕在桌面。若以恒定加速度a竖直向上提绳，求当提起高度为y时，作用在绳上的力？若以恒定速度v竖直向上提绳，情况又如何？（设t=0时，y=0,v=0）  

F F+dF o r dr [例6]直九型直升机的每片旋翼长L，质量 m，若视其为均匀薄片，求旋翼以角速度 旋转时根部所受拉力？ dm对o轴的拉力: 整个旋翼对o的拉力:

§2-4 非惯性系惯性力 对地对车对地对车 ----牛律不成立

一.惯性系和非惯性系 • 牛律适用的参照系--惯性系, 反之--非惯性系说明:  一参照系是否是惯性系，要靠实验判断

船走吗? 在惯性系内进行任何力学实验均不能确定该系作匀速直线运动的速度 ----力学相对性原理 相对于惯性系作匀速直线运动的参照系都是惯性系，作变速运动的参照系为非惯性系

：---K 对K ：---球对K K：惯性系二. 直线加速参照系 K ：非惯性系(车) 则球对K：在K系

有定义： ----惯性力 ----非惯性系中的牛二律说明： 惯性力与真实力有区别 在非惯性系中研究物体的运动时，才考虑惯性力

[例5]用惯性力的方法解[例3] 解：以劈为参照系劈和木块的受力如图对m 水平方向垂直方向对M 即可解得

一人站在一转盘边缘随盘以转动， 三. 转动参照系 1.匀速转动参考系中的惯性离心力 从地面观察，人作匀速圆周运动，向心力由摩擦力提供，

在转盘上观察，人静止，即 引入： ----惯性离心力则

Chapter 2 Dynamics( 动力学） ( Newton’s Laws of Motion )

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