physics 218: mechanics instructor: dr. tatiana erukhimova lecture 4
TRANSCRIPT
Physics 218: Mechanics Instructor: Dr. Tatiana Erukhimova
Lecture 4
Mechanics
• Various forms of motion:
- mechanical
- electromagnetic
- thermal, etc.
Mechanical form of motion is connected with displacements of various bodies relative to each other and with changes of the shapes of the bodies
Historical Notes
• History of mechanics linked with history of human culture
• Aristotle (384-322 B.C.); Physics
• Archimedes (3rd century B.C.), the law of lever, the law of equilibrium for floating bodies
• Galileo Galilei (1564-1624), the basic law of motion
Archimedes (3rd century B.C.), the law of lever, the law of equilibrium for floating bodies
GIVE ME A PLACE TO STAND AND I WILL MOVE THE EARTH
Buoyancy
VgF objectfluid )(
VgF objectgasgsurroundin )(
When a body is completely or partially immersed in a fluid, the fluid exerts an upward force on the body equal to the weight of the fluid displaced by the body.
“Father of modern science”
Was the first to apply a scientific method: Put forward a hypothesis, verify it by experiment, describe it with a mathematical model
Insisted that language of mathematics should describe the laws of nature and experiments should prove it.
No place for arguments based on beauty, religion etc.
Stephen Hawking: Galileo probably bears more of the responsibility for the birth of modern science than anybody else.
Albert Einstein
Achievements in physicsVerified that free-fall acceleration is independent on masses of bodies. This fact inspired Einstein’s General Relativity.
Formulated the Principle of Relativity, which laid the framework for Newton’s laws and inspired Einstein’s Special Relativity.
Proposed the Principle of Inertia, which was used (borrowed?) by Newton as his First Law.
Found that the period of a pendulum is independent on its amplitude.
He discovered it by observing swings of a bronze chandelier in the Cathedral of Pisa and using his pulse to measure the time!
Imagine you drop a light feather and a heavy coin from Albritton Bell Tower (138 ft.) Will they reach the ground at the same time?
384-322 B.C. Aristotle says : “No! The coin will land first because heavier objects fall faster than the lighter ones, in direct proportion to weight”.
1800 years later Galileo says : “Yes! A coin and a feather will land together if there is no air resistance!”
1564-1642
Free fall
g-positive!
On planet Earth, if you neglect
air resistance, any body which is dropped will experience a constant acceleration, called g, independent of its size or weight.
g=9.8 m/s2=32 ft/s2
a
v
a = g = const for all bodies independently on their masses
Galileo Galilei (1564-1624), the basic law of motion
22 /32/8.9 sftsmg
Galileo's “Law of Falling Bodies” distance (S) is proportional to time (T) squared
Galileo’s notes
Free fall
Falling with air resistance
A New Era of Science
Newton’s law of gravitation
Clockwork universe
1905 Albert Einstein
"Gravitation cannot be held responsible for people falling in love.“ Albert Einstein
Motion in One Dimension (Chapter 2)
We consider a particle
)(tx - as time goes, the position of the particle changes
Velocity is the rate at which the position changes with time
Average velocity:
t
x
tt
txtxv
initialfinal
initialfinalave
)()(
dt
tdxv
)(
You travel from CS to Houston. First 20 miles to Navasota you cover in 20 min. You make a 10 min stop in Navasota and continue for another 20 min until you reach Hempstead which is 20 miles from Navasota. There you make a 15 min stop for lunch. Then you continue the remain 50 miles to Houston and reach it in 35 min. Find your average velocity.
Acceleration is the rate at which the velocity changes with time
Average acceleration
t
v
tt
tvtva
initialfinal
initialfinalave
)()(
dt
tdva
)(
)0()0(2
1)( 2 xtvtatx c
)0()( vtatv c
))()((2)()( 1212
22 txtxatvtv c
€
v(t) =dx(t)
dt
a(t) =dv(t)
dt
€
x(t) = v(t)dt∫v(t) = a(t)dt∫
or
x(t) = x0 + v(t)dt0
t
∫
v(t) = v0 + a(t)dt0
t
∫
If a=ac=Const:
)0()0(2
1)( 2 xtvtatx c
)0()( vtatv c
€
v 2(t2) − v 2(t1) = 2ac (x(t2) − x(t1))
A “police car” problem
x=0 x1 x2
V3=20m/sa=0
ap=kt
x2 – x1 = 3.5 km
V1=30m/s V2=40m/s
You start moving from rest with constant acceleration. There is a police car hiding behind the tree. The policeman has a metric radar. He measures your velocity to be 30 m/s. While the policeman is converting m/s to mph, you continue accelerating. You meet another police car. This policeman measures your velocity to be 40 m/s. You also notice the police, drop your velocity to 20 m/s and start moving with a constant velocity. However, it is too late. This police car starts chasing you with acceleration kt (k is a constant). After some distance he catches you.
a=const
V(t=0)=0
A “police car” problem
x=0 x1 x2
V3=20m/sa=0
ap=kt
x2 – x1 = 3.5 km
V1=30m/s V2=40m/s
1. What was your acceleration before you met the second police car?2. How long did you travel from x1 to x2? 3. Find x1
4. At which distance does the police car catch you?5.Convert the velocity from m/s to mph
a=const
V(t=0)=0
Have a great day!
Reading: Chapter 2