Newton’s Laws

Agenda today

1. Newton’s Laws

2. Force on a body

3. Centripetal force

4. Non-inertial reference system

Newton’s First Law of motion:

Any object will keep its state of

movement when it is left along.

）（ 0 Fconstv

Inertia(惯性）：the property to keep one’s state

of movement

Mass（质量）: the measure of the inertia of a

body

Newton’s second Law of Motion

The acceleration of an object is directly

proportional to the net force and is inversely

proportional to its mass,its direction is the

direction of the net force.

amF

或dt

vdmF

Force（力）: The first derivative of linear

momentum

Linear momentum（动量）: the product of object’s

mass and velocity

Newton’s third law of motion

Whenever one object exert a force on another

one ,the another exert a force on the one same

in magnitude and opposite in direction.

FF

These forces( action and reaction) appears in pairs.

This law implies the conservation law of momentum .

the action force and reaction force are in the same

nature

According to Newton’s third law, the force I exert

on the cart will cancel the force that the cart exert

on me,so there is no need to pull the carriage

a wise horse’s reason

Forces in NatureBasic interactions

The gravitation force（引力）

The electromagnetic force(电磁力）

The strong nuclear force（强相互作用力）

The weak nuclear force（弱相互作用力）

Contact Forces

Elastic force（弹力）

xkF

0 x

F

Hook’s Law

K : force constant

Friction

Kinetic Friction（滑动摩擦力）

Nf

: the coefficient of kinetic Friction

v

fN

gm

A wonder caused by friction Leonid meteor shower

Static friction（静摩擦力）

The Maximum static friction

Nfs

max

F

f

N

gm

Ff

Air drag force

2

2

1AvCD

Where C is drag coefficient

Terminal speed

AC

Fv

g

t

2

The way to solve problem

amF

steps：

（1）draw a sketch of the situation and free-body diagram.

（3）Resolve the force into component。

（2）choose the coordinate system

（4）apply the Newton’s law to x axis.y axis.

（5）solve the equation, and find the unknown

As the figure shows，two blocks have masses:

mA=1.0kg and mB=2.0kg。The coefficient of

friction between A and B:1=0.20,The coefficient of

B and ground is 0.30. The magnitude of each block

is 0.15m·s-2。What is the Force exert on the objects？

mAg

Tf 1

N1

mBg N1f 1

f 2

BF

T

N2

A

Ay

xF

B

amTf A1

01 gmN A

111 Nf

amTffF B 21

012 gmNN B

222 Nf

A： B：

By equation a, we have amTgm AA 1By equation b, we have：

amTgmmgmF BBAA )(21 By solving them ： NF 2.13

y

x

mAg

Tf 1

N1

mBgN1f 1

f 2

BF

T

N2

A

Centripetal force in circular movement

The force gives rise to centripetal acceleration in

circular movement

r

vmF

2

F

x

y

A ball falls from point A along the half sphere with radius r, find the ball’s velocity and

force exert on the sphere at any position given.

A

mg

N

Solution：

dt

dvmmg cos

R

vmmgN

2

sin

Rd

dvv

dsdt

dvds

dt

dv

dRgvdv cos

x

yA

mg

N

00

cos dRgvdvv

sin2

1 2 Rgv

sin2Rgv

R

RgmmgN

sin2sin

R

vmmgN

2

sin

sin3mg

Force in a non-inertial reference system

mg

F

T

Inertial reference

system

A system that Newton’s Law holds true

Non-inertial reference system

A system that Newton’s law does not hold

ay´

x´

mg

T

Q=ma

A force invented to make

Newton’s law true in a non-inertial

system

pseudoforce

amQ

a

Newton’s second law in a non-inertial system

amQF

Example:

Suppose a disk rotating with a constant angular velocity ,

a man is walking along the radical way with constant speed

v from the center to the edge, what is friction force exerted

on the man? If we choose the disk as the reference system,

what is the pseudoforce?

Solution:Choose a rectangular coordinate system,we can

describe the man ‘s position function as:

By derivative action, we have

Thus, the friction force exert on the man is ma

The pseudoforce is

The first term is called centrifugal force（离心力）, the

second one Coriolis force（科里奥利力）.

Why the shape of roller roaster not a circle?

Using computer to solve physical problem

Modeling progress

Algorithm （算法）choosing

Use physical formula to write

differential equations

Write flowing chart or pseudocode

programming

The way to make your calculation faster

or more accurate

An example revisit：

Example A ball bump into some kind of liquid with

initial velocity: its acceleration

is given by: . Find how far can

it go before it is at rest？

)(100 1

0

smiv

)(10 1 smiva

Euler’s method:

A Solution not very good:

#include <stdlib.h>

#include <stdio.h>

#include <string.h>

void main()

{

double v0=100,a,r=0;

double rlast, err;

do{

a=-10.0*v0;

v0+=a*1E-6;

rlast=r;

r+=v0*1E-6;

err=r-rlast;

}while(err>=1E-7);

printf( "The displacement is %3.4f m before stopped\n",

r);

}

Solve problem with matlab

Example A ball bump into some kind of liquid with

initial velocity: its acceleration

is given by: . Find how far can

it go before it is at rest？

Solution:By the definition of velocity and

acceleration, we have:

Build m-file test1.m as following:

function dydt = test1(t,y)

dydt=[y(2)

-10*y(2)];

Save it in the working directory, then type these sentences

in the working space

tspan=[0,2];

y0=[0;10];

[t,y]=ode45(@test1,tspan,y0);

axis([0 2 0 2]);

plot(t,y(:,1))

Or even simpler:

y2=dsolve('D2y=-10*Dy','y(0)=0','Dy(0)=10');

ezplot(y2,[0,2])

Another Example by Matlab:

Project one : Draw the trajectory curve of the ball in

projection experiment at different height; with different

angle; considering the air resistance. (the ball’s initial speed

exceeds 200m/s)