Download Physics Lecture Note I and more Study notes Physics in PDF only on Docsity!
General Physics Mechanics
Mechanics æææ...
Newtonian Mechanics
~F = m~a
Lagrangian Mechanics d dt[^
∂L ∂ q˙μ ] =^
∂L ∂qμ
Hamiltonian Mechanics
p ˙μ = −∂H ∂qμ , q˙μ = ∂H ∂pμ , H = H(qμ, pμ)
Electrodynamics
∇ ·^ ~ E~ = 4πρ, c∇ ×~ E~ + ∂ B~
∂t = 0,
∇ ·^ ~ ~B = 0, c∇ ×~ B~ − ∂ E~
∂t = 4π
~j
Schr¨odinger’s Quantum Mechanics −¯h^2 2 m ∇
2 Ψ + U (~r)Ψ = i¯h∂Ψ
∂t
Relativistic Quantum Mechanics (¯h = c = 1) Klein-Gordon Eq. (⊓⊔+m^2 )φ = 0 Dirac Eq. (iγμ∂μ − m)ψ = 0 { c~α ·
( ~p − e c A~
)
} ψ = i¯h∂ψ ∂t
THUgeneralPHYSICS-1.tex -1- gp110707b.tex
General Physics Units
Units SI (Syst`eme International d’Unit´es) International System of Units
7 independent base units length: meter (m) time: second (s) mass: kilogram (kg)
electric current: ampere (A) temperature: kelvin (K) amount of substance: mole (mol) luminosity: candela (cd)
2 supplementary units ordinary angle: radian (rad) solid angle: steradian (sr)
∗ Unit Ðrι¶ Fó(plural)s. ¶¶μêY
THUgeneralPHYSICS-1.tex -2- gp110707a.tex
General Physics Vectors
Vectors
Scalar (A): ö a number and its unit magnitude without direction
Vector (A~): '
magnitude and direction vector algebra +, −, ·, ×
Vector addition tail-to-tip method:
- tail is placed at tip
- resultant: tail(of first) −→ tip(of last)
A^ ~ + B~ = C~
Vector subtraction
- tail is placed at tail
- resultant: tip(of last) −→ tip(of first) ¼' 3
C^ ~ − A~ = B~
THUgeneralPHYSICS-1.tex -3- gp110707c.tex
General Physics Vectors
THUgeneralPHYSICS-1.tex -4- gp110707c.tex
General Physics Vectors in Two Dimentions
Vectors in two dimentions
rectangular components of the vector A~
Ax = A cos θ Ay = A sin θ
magnitude: A =
√ A^2 x + A^2 y
direction: tan θ = A Ayx ,
θ = arctan A Ayx
θ: measured from the +x axis
The components of a vector may be positive or neg- ative numbers.
THUgeneralPHYSICS-1.tex -5- gp110707d.tex
General Physics Vectors in Two Dimentions
THUgeneralPHYSICS-1.tex -6- gp110707d.tex
General Physics Vectors in 3 Dimentions
Vectors in 3 dimentions unit vectors
ˆi, ˆj, ˆk
A^ ~ = Axˆi + Ayˆj + Azˆk
A =
√ A^2 x + A^2 y + A^2 z
THUgeneralPHYSICS-1.tex -7- gp110707e.tex
General Physics Vectors in 3 Dimentions
vector sum(resultant) of several vectors
R^ ~ = A~ + ~B
Rx = Ax + Bx Ry = Ay + By Rz = Az + Bz
R^ ~ = A~ + B~ + ~C + D~ + ~E +...
THUgeneralPHYSICS-1.tex -8- gp110707e.tex
General Physics 1D Kinematics
THUgeneralPHYSICS-1.tex -13- gp110707h.tex
General Physics 1D Kinematics
One-dimensional Kinematics
position x displacement ∆x = xf − xi time interval ∆t = tf − ti
average velocity vav = ∆ ∆xt (= ¯v) instantaneous velocity v = lim∆t→ 0 ∆ ∆xt = dx dt ∗ The slope of the tangent to the x versus t graph.
average acceleration aav = ∆ ∆vt (= ¯a) instantaneous acceleration a = lim∆t→ 0 ∆ ∆vt = dv dt ∗ The slope of the tangent to the v versus t graph.
THUgeneralPHYSICS-1.tex -14- gp110711a.tex
General Physics 1D Kinematics
THUgeneralPHYSICS-1.tex -15- gp110711a.tex
General Physics Constant Acceleration
Constant Acceleration The equations of kinematics for constant acceleration:
a = constant
∆v ∆t
vf − vi tf − ti
v − v 0 t − 0
v − v 0 t v = v 0 + at
x = x 0 + ∆x
= x 0 +
(v 0 + v)t ⇐ ∆x =
(vi + vf )∆t
= x 0 +
(v 0 + v 0 + at)t
= x 0 + v 0 t +
at^2
v^2 = (v 0 + at)^2 = v 02 + 2v 0 at + a^2 t^2
= v 02 + 2a(v 0 t +
at^2 )
= v 02 + 2a(x − x 0 )
v = v 0 + at
x = x 0 +
(v 0 + v)t
- x = x 0 + v 0 t +
at^2
4 .v^2 = v^20 + 2a(x − x 0 )
THUgeneralPHYSICS-1.tex -16- gp110711b.tex
General Physics Constant Acceleration
THUgeneralPHYSICS-1.tex -17- gp110711b.tex
General Physics Constant Acceleration g
Constant Acceleration g
~g = −gˆj
v = −gt + v 0
y = −
gt^2 + v 0 t + y 0
v^2 = v^20 − 2 g(y − y 0 )
THUgeneralPHYSICS-1.tex -18- gp110711c.tex
General Physics Constant Acceleration g
THUgeneralPHYSICS-1.tex -19- gp110711c.tex
General Physics 2- and 3-D Motion
Two- and 3-dimensional motion ∗Equations are in form of 3-dimensional vectors. position
~r = xˆi + yˆj + zˆk
displacement
∆~r = ~r 2 − ~r 1
= ∆xˆi + ∆yˆj + ∆zˆk
average velocity
~¯v = ∆~r
∆t instantaneous velocity
~v = lim
∆t→ 0
∆~r
∆t
d~r
dt
dx dt
ˆi + dy
dt
ˆj + dz
dt
ˆk
= vxˆi + vyˆj + vzˆk
THUgeneralPHYSICS-1.tex -20- gp110711d.tex
General Physics Horizontal Range
Horizontal Range
Setting y = 0,
y = 0 = x tan θ 0 −
g 2 v^20 cos^2 θ 0
x^2
x
( tan θ 0 −
gx 2 v^20 cos^2 θ 0
) = 0
x = 0 or
x =
2 v 02 g
cos^2 θ 0 tan θ 0
2 v 02 g
sin θ 0 cos θ 0
v^20 g
sin 2θ 0 ⇐ sin 2θ 0 = 2 sin θ 0 cos θ 0
(horizontal range) x = max. when θ 0 = 45◦
THUgeneralPHYSICS-1.tex -25- gp110712a.tex
General Physics Horizontal Range
THUgeneralPHYSICS-1.tex -26- gp110712a.tex
General Physics Hunter and Monkey
The Hunter and the Monkey: A classic demon- stration (Young, p.68) A dart aimed at the initial position of the monkey al- ways hit him, no matter what v 0 is. monkey:
x = d at all times
dart:
x = (v 0 cos α 0 )t
When these are equal,
d = (v 0 cos α 0 )t
or
t =
d v 0 cos α 0
At this time,
ymonkey = d tan α 0 −
gt^2
ydart = (v 0 sin α 0 )t −
gt^2
ymonkey − ydart = d tan α 0 − v 0 sin α 0 t
= d tan α 0 − v 0 sin α 0
d v 0 cos α 0 = d tan α 0 − (tan α 0 )d = 0
Q.E.D.
THUgeneralPHYSICS-1.tex -27- gp110712b.tex
General Physics Hunter and Monkey
THUgeneralPHYSICS-1.tex -28- gp110712b.tex
General Physics Uniform Circular Motion
Uniform Circular Motion ∗ Centripetal acceleration
∆~r = ~r 2 − ~r 1
∆~v = ~v 2 − ~v 1
r 1 = r 2 = r v 1 = v 2 = v
From similar triangles,
|∆~r|
r
|∆~v|
v That is,
|∆~v| =
v r
|∆~r|
Arc length:
|∆~r| ≈ v∆t
we can write,
|∆~v| =
v r
|∆~r| ≈
v r
v∆t
a = lim ∆t→ 0
|∆~v|
∆t
=
v^2 r radial acceleration:
~ar = −
v^2 r
ˆr
THUgeneralPHYSICS-1.tex -29- gp110712c.tex
General Physics Uniform Circular Motion
THUgeneralPHYSICS-1.tex -30- gp110712c.tex
General Physics Uniform Circular Motion
Uniform Circular Motion
period T
frequency f = (^1) T
angular frequency ω = 2πf = (^2) Tπ
speed v = 2 πr T = ωr
radial acceleration a = v
2 r =^ ω
(^2) r
THUgeneralPHYSICS-1.tex -31- gp110712d.tex
General Physics Uniform Circular Motion
THUgeneralPHYSICS-1.tex -32- gp110712d.tex
General Physics A Boat Crossing River
A Boat Crossing a River
~vbr = vel. of boat relative to river
~vre = vel. of river relative to Earth
~vbe = ~vbr + ~vre
vbe =
√ v^2 br + v^2 re =
√ (10.0)^2 + (5.0)^2 km/h = 11.2 km/h direction of vbe:
θ = tan−^1 (v vrebr ) = tan−^1 ( 105.^0. 0 ) = 26. 6 ◦
The boat will be traveling at a speed of 11.2 km/h in
the direction 63.4◦^ north of east relative to Earth.
THUgeneralPHYSICS-1.tex -37- gp110713c.tex
General Physics A Boat Crossing River
THUgeneralPHYSICS-1.tex -38- gp110713c.tex
General Physics A Boat Crossing River
Which Way Should We Head?
~vbe = ~vbr + ~vre
vbe =
√ v br^2 − v^2 re
=
√ (10.0)^2 − (5.0)^2 km/h = 8 .66km/h
direction of vbr:
θ = tan−^1 (
vre vbe
) = tan−^1 (
The boat must steer a course 30.0◦^ west of north.
THUgeneralPHYSICS-1.tex -39- gp110713c.tex
General Physics A Boat Crossing River
THUgeneralPHYSICS-1.tex -40- gp110713c.tex
General Physics Newton’s Laws
Dynamics
Newton’s Laws of Motion
∗ valid in inertial frames.
1st law: If ~F = 0, ~v = const.
∗ from Galileo
2nd law:
F = m~a
read as “ F~ CW m~a ”
3rd law: ~FAB = −~FBA
AB: on A by B KÎåæ
Comments:
- First law is a special case of 2nd law.
- 2nd law is an equation of motion.
- Mass is the ratio of coefficients,
since ~Fi ∝ ~ai
- “Action-reaction” was Newton’s law, the 1st and 2nd were due to Galileo’s.
THUgeneralPHYSICS-1.tex -41- gp110713d.tex
General Physics Newton’s Laws
Dynamics Newton’s Laws of Motion ∗ valid in inertial frames.
1st law: If ~F = 0, ~v = const.
∗ from Galileo
2nd law:
F = m~a
read as “ F~ CW m~a ”
{ (^) ~F=m~a ~F−m~a=
} are different.
3rd law: ~FAB = −~FBA
AB: on A by B KÎåæ
Comments:
- First law is a special case of 2nd law.
- 2nd law is an equation of motion.
- Mass is the ratio of coefficients,
since ~Fi ∝ ~ai
- “Action-reaction” was Newton’s law, the 1st and 2nd were due to Galileo’s.
THUgeneralPHYSICS-1.tex -42- gp110713d.tex
General Physics Newton’s Laws
Dynamics
Newton’s Laws of Motion
∗ valid in inertial frames.
1st law: If ~F = 0, ~v = const.
∗ from Galileo
2nd law:
F = m~a
read as “ F~ CW m~a ”
3rd law: ~FAB = −~FBA
AB: on A by B KÎåæ
Comments:
- First law is a special case of 2nd law.
- 2nd law is an equation of motion.
- Mass is the ratio of coefficients,
since ~Fi ∝ ~ai
- “Action-reaction” was Newton’s law, the 1st and 2nd were due to Galileo’s.
THUgeneralPHYSICS-1.tex -43- gp110713d.tex
General Physics Newton’s Laws
THUgeneralPHYSICS-1.tex -44- gp110713d.tex
General Physics Weight
THUgeneralPHYSICS-1.tex -49- gp110713g.tex
General Physics Force Diagram
Applications of Newton’s Laws
- iforce diagram, êÌfree-body diagram.
2. ×ÍÎ×Í%im~a.
- &ÎXå²æÄ 6 b¼Ù
- ¯ 2 ýâ´ 94 ݲæ 6 ÄÝ 5
- ¶ ∑ %¶ 2 P ∑ Fx^ =^ max, Fy = may. ×ÍP©b×Ím.
- O¥Ñr
THUgeneralPHYSICS-1.tex -50- gp110713h.tex
General Physics Force Diagram
THUgeneralPHYSICS-1.tex -51- gp110713h.tex
General Physics Dynamics
Dynamics æææ...
∑ (^) ~
F = m~a
F:
W^ ~ ¥æ Gravitational force
(gravity, weight)
N^ ~ °'æ Normal force
Ñ'æ
T^ ~ ùæ Tension
F^ ~s ¬Ææ Restoring force
(Ä) (spring)
~f `æ Frictional force
THUgeneralPHYSICS-1.tex -52- gp110713e-1.tex
General Physics Putt-Putt Physics
Putt-Putt Physics (Crummett, p.134) Q1: The normal force between the track and the ball at the top and bottom of the 1st loop.
Q2. Speed at top of the 2nd loop so it is “just barely in contact” with the surface.
THUgeneralPHYSICS-1.tex -53- gp110714a.tex
General Physics Putt-Putt Physics
Solution:
(1a)
F = m~a
−N − W = ma = m(−v
2 r ) N = mv
2 r −^ W = m(v
2 r −^ g)
(1b)
F = m~a
+N − W = ma = m(+v
2 r ) N = mv
2 r +^ W = m(v
2 r +^ g)
(2) N = 0 ⇐ “just barely in contact” −N − W = m(−v
2 r ) −W = −mv
2 r mg = mv
2 r v = √rg
THUgeneralPHYSICS-1.tex -54- gp110714a.tex
General Physics Dynamics
Dynamics æææ...
∑ (^) ~
F = m~a
F:
W^ ~ ¥æ Gravitational force
(gravity, weight)
N^ ~ °'æ Normal force
Ñ'æ
T^ ~ ùæ Tension
~Fs ¬Ææ Restoring force
(Ä) (spring)
~f `æ Frictional force
THUgeneralPHYSICS-1.tex -55- gp110713e-2.tex
General Physics Dynamics
THUgeneralPHYSICS-1.tex -56- gp110713e-2.tex
General Physics Dynamics
THUgeneralPHYSICS-1.tex -61- gp110713e-3.tex
General Physics Dynamics
Dynamics æææ...
∑ (^) ~
F = m~a
F:
W^ ~ ¥æ Gravitational force
(gravity, weight)
N^ ~ °'æ Normal force
Ñ'æ
T^ ~ ùæ Tension
F^ ~s ¬Ææ Restoring force
(Ä) (spring)
~f `æ Frictional force
THUgeneralPHYSICS-1.tex -62- gp110713e-3.tex
General Physics Dynamics
THUgeneralPHYSICS-1.tex -63- gp110713e-3.tex
General Physics Pulling A Crate
THUgeneralPHYSICS-1.tex -64- gp110714d.tex
General Physics Pulling A Crate
Pulling a Crate(Young, p.124)
Keep the crate moving with constant velocity.
W = 500 N, μk = 0.40, find T =?
Solution: ∑ Fx = T cos 30◦^ − Fk = T cos 30◦^ − 0. 40 N = 0 (1) ∑ Fy = T sin 30◦^ + N − W = 0 N = W − T sin 30◦
Substitute this into eq. (1),
T cos 30◦^ − 0 .40(W − T sin 30◦) = 0 T cos 30◦^ + 0. 40 T sin 30◦^ − 0. 40 × 500 = 0 T = 188 N N = W − T sin 30◦^ = 406 N
For least T , θ = arctan μ = 21. 8 ◦.
THUgeneralPHYSICS-1.tex -65- gp110714d.tex
General Physics Pulling A Crate
THUgeneralPHYSICS-1.tex -66- gp110714d.tex
General Physics Friction
Friction
Find the minimun value of the coefficient of friction
such that m 1 does not slides on m 2.
Solution: ∑ (^) ~
F = m~a
Block 1 : f = m 1 a 1 (1) Block 2 : F 0 − f = m 2 a 2 (2)
“m 1 does not slide on m 2 ” ⇐ a 1 = a 2 = a
Adding (1) and (2),
F 0 = (m 1 + m 2 )a
Since
fs(max) = μsN 1 = μs(m 1 g)
we have
μs(m 1 g) = m 1 a
= m 1
F 0
m 1 + m 2
μs =
F 0
(m 1 + m 2 )g
[minimum value]
Example : μs =
30N
(2kg + 4kg) × 9 .8m/s^2
THUgeneralPHYSICS-1.tex -67- gp110714e.tex
General Physics Friction
THUgeneralPHYSICS-1.tex -68- gp110714e.tex
General Physics Resistive Media
Motion in Resistive Media
Drag force: FD = γv ∑
F~ = m~a
F^ ~g + ~FD = m~a
mg − γv = m
dv dt
At terminal speed vt, dv dt = 0
mg − γvt = 0 vt =
mg γ
Drag force: FD = kv^2 (turbulent)
mg − kv^2 = m
dv dt At terminal speed vt,
mg − kv^2 t = 0
vt =
√mg
k
THUgeneralPHYSICS-1.tex -73- gp110720a.tex
General Physics Resistive Media
THUgeneralPHYSICS-1.tex -74- gp110720a.tex
General Physics Noninertial Frames
Noninertial Frames
Fictitious force: (no reaction)
(1) Inertial force
T^ ~ + m~g = m~a
T^ ~ + m~g + (−m~a) = 0
(2) Centrifugal force
T^ ~ = m~a = mv^2
r (−ˆr)
T^ ~ + (−mv^2
r )(−ˆr) = 0
THUgeneralPHYSICS-1.tex -75- gp110720b.tex
General Physics Noninertial Frames
THUgeneralPHYSICS-1.tex -76- gp110720b.tex
General Physics Rotational Frames
Noninertial Frames
Position vector of a particle:
~r = rxˆi + ryˆj + rzˆk
Velocity in Moving System
d~r
dt
∣∣ ∣∣ ∣ F
d~r
dt
∣∣ ∣∣ ∣ M
+ ~ω × ~r
Acceleration in Moving System
d^2 ~r
dt^2
∣∣ ∣∣ ∣ F
d^2 ~r
dt^2
∣∣ ∣∣ ∣ M
d~ω dt
∣∣ ∣∣ ∣ M
× ~r + 2~ω ×
d~r
dt
∣∣ ∣∣ ∣ M
+ ~ω × (~ω × ~r)
Motion of a Particle Relative to the Earth
Earth: d~ dtω = 0
Newton’s 2nd Law:
~F = md
2 ~r
dt^2
∣∣ ∣∣ ∣F
= m
d^2 ~r
dt^2
∣∣ ∣∣ ∣ M
d~r
dt
∣∣ ∣∣ ∣ M
+ m~ω × (~ω × ~r)
~F − 2 m~ω × ~v − m~ω × (~ω × ~r) = md
2 ~r
dt^2
∣∣ ∣∣
∣M^ =^ m~a
Coriolis force = − 2 m~ω × ~v
Centrifugal force = −m~ω × (~ω × ~r)[i.e. − mv
2 r ]
THUgeneralPHYSICS-1.tex -77- gp110720c.tex
General Physics Rotational Frames
THUgeneralPHYSICS-1.tex -78- gp110720c.tex
General Physics Work and Kinetic Energy
Work and Kinetic Energy
Work
Work on the object is done by the others.
W = ~F · ~s = F s cos φ
~F ~s !' W ÑÂ
~F ~s D' W Â
~F ~s kà W ëÂ
Work done by a varing force in one dimension
W ≃
∑^ N
i=
Wi =
∑^ N
i=
F (xi)∆x
W = lim ∆x→ 0
∑
i
F (xi)∆x
W =
∫ (^) x 2
x 1
F (x)dx
Force and Work in 3 dimensions
∆W = ~F · ∆~r
W ≃
∑ ∆W =
F · ∆~r
W =
∫ (^) r 2
r 1
~F · d~r
THUgeneralPHYSICS-1.tex -79- gp110720d.tex
General Physics Work and Kinetic Energy
THUgeneralPHYSICS-1.tex -80- gp110720d.tex
