2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation1 Chapter 3. Theorem of momentum and the law of momentum conservation 3.3 Some Particular Forces 几种特殊的力 Mechanics

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation2  Newton’s Laws tell us how an object moves given the forces applied  What are the causes of these forces? Need to have a detailed microscopic understanding of the interactions of the objects with their environments 3.3 Some Particular Forces

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation3  Four fundamental forces in nature: 1.Gravitational Force( 引力 )  Attractive force between all objects with mass;  Originates with the presence of matter. 3.3 Some Particular Forces 2.Electromagnetic Force( 电磁力 )  Includes basic electric and magnetic interactions;  acts on all objects with electrical charge;  responsible for binding of atoms and structure of material

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation4 3.Weak force( 弱力 )  responsible for radioactivity, fusion 4.Strong Force( 强力 )  Holds protons/neutrons together in the atomic nucleus Unified theory of electromagnetic and weak forces developed ~ Some Particular Forces  Four fundamental forces in nature:

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation5 In classical mechanics, only two forces are involved:  Gravitational force: Apparent in the Earth’s attraction for objects, which gives them their weight.  Electromagnetic force: All the other forces we normally consider are ultimately electromagnetic in origin Tension in string( 绳的张力 ) ； Contact force: normal force, friction,.. Elastic force: in spring 3.3 Some Particular Forces

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation6 Chapter 3. Theorem of momentum and the law of momentum conservation 3.3 Some Particular Forces Universal gravitational force and weight Spring force Tension in a rope Normal or Reaction Force Friction Mechanics

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation7  万有引力定律 (Law of universal gravitation) Universal gravitational force and weight

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation8  万有引力定律 (Law of universal gravitation) G : 万有引力常数 = 6.67 x Nm 2 kg -2, m 、 M : 质点的引力质量 (gravitational masses) r : m 相对于 M 的位置矢量 Newton published the law of gravitation in 1687 r M m F 两个有一定质量的质点沿它们的连线相互吸引，吸 引力的大小与两质点的质量的乘积成正比，与它们 之间的距离的平方成反比： - indicates attractive Universal gravitational force and weight

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation Universal gravitational force and weight 半径为 R ，质量为 M 且均匀分布的球体对球外一质量为 m 的质 点的引力为： r M m

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation Universal gravitational force and weight 引力质量和惯性质量 The force lawThe law of motion 不同性质的物 理定律中的量 考虑地球附近的自由落体运动：物体只受地球万有引力的作用  实验结果：在真空中，所有物体的自由落体加速度都为重力加 速度 g, 与物体的质量无关 最新实验结果：

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation Universal gravitational force and weight How did Newton deduce the law of universal gravitation? Both a falling apple and the distant orbiting Moon accelerate toward the Earth; The force causing these accelerations is gravitational force a moon g RERE R  Dependence on Distance: R = 3.84 x 10 8 m R E = 6.37 x 10 6 m

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation Universal gravitational force and weight How did Newton deduce the law of universal gravitation?  Dependence on Mass: The magnitude of the gravitational force should also depend on the common physical property of the Earth and the Moon  their masses Newton’s third law: M: mass of the Earth, m: mass of the Moon  If the magnitude of gravitational force depends on mass, it must involve the masses of both the Earth and Moon in a symmetric way.

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation Universal gravitational force and weight How did Newton deduce the law of universal gravitation?  Dependence on Mass: If n =1, a grav  M Experiment: a grav does not depend on m

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation14  重力 (Weight): 当物体位于地球表面附近时，可只考虑地球对该物体的万 有引力而忽略所有其它的引力。 地球模型：质量为 m E 、半径为 R E 的均质球； 地球表面附近质量为 m 的物体的重力的大小 ( 物体到地球中心的距离为 R E ) : Universal gravitational force and weight 重力：地球对地球表面附近的物体的万有引力。 RERE mEmE m RERE

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation15 重力加速度 : g 依赖于地球的质量而与所研究的物体质量无关。 Mass of the Earth, M E = 5.97 x kg Radius of the Earth = 6.37 x 10 6 m 在重力的作用下，所 有物体的加速度的大 小都为 g! Universal gravitational force and weight

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation16 在地球表面的不同地方 g 的数值稍有不同 ( ms -2 ) Note: 质量和重力的区别： 质量 : 标量，是物体的固有属性，与相互作用无关 重力 : 矢量，是地球对物体的引力的量度 ( 或物体对地球的引 力的量度 ).  地球不是一个严格的球体；  地球的自转和轨道运动的影响 其大小 ( 重量 ) 依赖于重力加速度的大小 g. 当物体远离地球时，重力为零。 Universal gravitational force and weight

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation17 Chapter 3. Theorem of momentum and the law of momentum conservation 3.3 Some Particular Forces Universal gravitational force and weight Spring force Tension in a rope Normal or Reaction Force Friction Mechanics

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation18 Magnitude: proportional to distance that spring is compressed or stretched Direction: opposite to direction that spring is compressed or stretched Force exerted by a spring The direction of the force is toward the equilibrium position.  “restoring force”. x = 0 x F L0L0 L0L0 L 0 : the free(undeformed) length of the spring Spring force x = 0 L0L0 F

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation19 F = – k  x = – k (x – x 0 ) F = – k x if choose x 0 = 0 k is spring constant or stiffness ( 劲度系数 )stiffer spring  larger k x 0 is equilibrium position of spring The magnitude of the force is given by Hooke’s Law: x = 0 x F Spring force

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation20 Chapter 3. Theorem of momentum and the law of momentum conservation 3.3 Some Particular Forces Universal gravitational force and weight Spring force Tension in a rope Normal or Reaction Force Friction Mechanics

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation Tension in a rope ←The force you would feel if you cut the rope and grabbed the ends. ←An action-reaction pair. Rope can be used to pull from a distance. TensionTension (T) at a certain position in a rope is the magnitude of the force acting across a cross-section of the rope at that position. cut T T T The rope is in “tension” as the two people pull on it. This stretching puts the rope in tension

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation22 T T´T´ mgmg T Consider a short segment of the rope. T, T´ : the forces exerted by the adjoining portion of the rope at each end of the segment mg : the weight of the segment T + T´ + mg = ma If the rope is massless. T = - T´ or T = T´ The tension T has the same magnitude at both ends of the small segment Tension in a rope  An ideal (massless) rope has constant tension along the rope.

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation Tension in a rope mg T m Since a y = 0 (box not moving), T = mg The direction of the force provided by a rope is along the direction of the rope:

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation Tension in a rope FF1FF1 ideal peg or pulley FF2FF2 FF | F 1 | = | F 2 | Pegs & Pulleys are used to change the direction of forces An ideal massless pulley or ideal smooth peg will change the direction of an applied force without altering the magnitude:

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation Tension in a rope Pegs & Pulleys are used to change the direction of forces An ideal massless pulley or ideal smooth peg will change the direction of an applied force without altering the magnitude: mg T m T = mg F W,S = mg

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation26 Chapter 3. Theorem of momentum and the law of momentum conservation 3.3 Some Particular Forces Universal gravitational force and weight Spring force Tension in a rope Normal or Reaction Force Friction Mechanics

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation27 If a block of mass m 1 is at rest and in contact with a horizontal surface, a force provided by the surface supports the block, holding it at rest. l Since the block’s acceleration is zero in the vertical direction, the net force on the block is zero. The contact force is the normal force, N, because it is directed perpendicular, or normal, to the surface. N = m 1 g m1m1 N m1gm1g Normal or Reaction Force

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation28 Chapter 3. Theorem of momentum and the law of momentum conservation 3.3 Some Particular Forces Universal gravitational force and weight Spring force Tension in a rope Normal or Reaction Force Friction Mechanics

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation Friction What does it do?  It opposes relative motion! How do we characterize this in terms we have learned? amaama F F APPLIED f f FRICTION gmggmg N i j  Friction results in a force in the direction opposite to the direction of relative motion!

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation Friction Friction is caused by the “microscopic” interactions between the two surfaces: gmggmg amaama f f FRICTION N  Force of friction acts to oppose relative motion:  Parallel to surface  Perpendicular to Normal force

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation Friction Two distinct cases:  Kinetic (sliding) friction( 滑动摩擦力 )  Static Friction( 静摩擦力 ) ： Surfaces move relative to one another. Surfaces do not actually move.

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation Friction Model for Sliding Friction N The direction of the frictional force vector is perpendicular to the normal force vector N. f N The magnitude of the frictional force vector | f F | is proportional to the magnitude of the normal force | N |. f =  K N  The “heavier” something is, the greater the friction will be...makes sense  K : coefficient of kinetic friction. amaama F gmggmg N f

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation33 object in motion: F friction will oppose F push with a magnitude of  k F N =  k mg Since F push >=  k F N object can move. F g = mg F N = mg F push/pull F friction F push =  k F N F push >  k F N Object moves with constant v. Object accelerates with a = (F push -  k F N ) /m Friction

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation34 Static Friction A friction force also acts between two objects when there is no relative motion. force of friction opposes applied force magnitude equals applied force maximum magnitude proportional to the normal force f max =  s F N f FaFa  s is the static coefficient of friction we can’t calculate it but it is easy to measure as long as F a  f max the object doesn’t move Friction

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation35 object at rest: F g = mg F N = mg F push/pull F friction F friction will oppose F push up to a maximum magnitude of  s F N =  s mg. Since F push =<  s F N object can remain at rest Friction

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation36 Measuring  s with an inclined plane.  FNFN FgFg  F gy F gx f f max =  s F N block doesn’t slide if F gx < f max m g sin  <  s m g cos  Just slides when  s = tan  Friction

2006 年 11 月 9 日 8:00-9:50 Ch.3 Theorem of momentum and the law of moentum conservation37 Summary: Kinetic versus Static friction Graph of Frictional force vs Applied force: f Applied force F f = F f F =  K N f =  S N f =  K N at rest accelerating Friction