General Physics (Lecture Notes)

General Physics (Lecture Notes)

Textbook: Essential University Physics by

Richard Wolfson

2

Chapter 1

Doing Physics

1.1 Realms of Physics

Physics:

• Mechanics ( Chs. 2 12)

• Oscillations, Waves, and Fluids ( Chs. 13 15)

• Thermodynamics ( Chs. 20 29)

• Electromagnetism ( Chs. 20 29)

• Optics ( Chs. 30 32)

• Modern Physics (Chs. 33 39)

1.2 Measurements and Units

SI unit system: (International System of Units)

• Length: meter (m)

• Time: second (s)

• Mass: kilogram (kg)

• Temperate: kelvin (K)

• Electric current: ampere (A)

• Amount: mole (mol)

• Luminosity: candela (cd)

3

4 CHAPTER 1. DOING PHYSICS

1.3 Working with Numbers

Scientific notation:

e.g.,

4.185 =⇒ 4.185 × 10³ (four significant figures)

1.4 Strategies for Learning Physics

An IDEA strategy:

• Intepret: know the problem.

• Develop: draw a diagram and choose eqs.

• Evaluate: in symbolic form, then plug in numbers at the end.

• Assess: make sense? or reasonable?

Chapter 2

Motion in a Straight Line

2.1 Average Motion

∆x = x2− x1 (displacement), d = |x2− x2| (distance), and

¯ v = ∆x

∆t (average velocity), ∆t = t2− t1, S = d

∆t (average speed).

2.2 Instantaneous Velocity

v = lim∆t→0

∆x

∆t = dx

dt (inst. vel.) Math: (taking derivatives)

e.g.,

x = b tⁿ =⇒ dx

dt = b n tⁿ⁻¹.

2.3 Acceleration

¯ a = ∆v

∆t (ave. acc.), and

a = lim∆t→0∆v

∆t = dv

dt (inst. acc.) 5

6 CHAPTER 2. MOTION IN A STRAIGHT LINE

2.4 Constant Acc.

a = ¯a = ∆v

∆t = v2− v1

t2− t1

= v − v0

t

=⇒ v = v0+ at (for const acc. only) · · · (1)







¯

v = ¹₂(v0+ v)

¯

v = ^∆x_∆t = ^x_t2²⁻₋^x_t1¹ = ^x−x_t ⁰

=⇒ x − x0

t = 1

2(v0+ v)

=⇒ x = x0+¹₂(v0+ v)t · · · (2) (1), = x0+¹₂(v0+ v0+ a t) t

= x0+ v0t + ¹₂a t² (a = const) · · · (3)

(1) =⇒ t = (v − v0)/a · · · (4) (4)(2) =⇒ x = x0+1

2(v0+ v)(v − v0)/a

=⇒ v²= v²₀+ 2a(x − x0)

2.5 The Acceleration of Gravity

(a = const) =⇒















v = v0+ at

x = x0+ v0t +¹₂a t² v² = v₀²+ 2a(x − x0) x → y and a → (−g),

(free fall, g = 9.8 m/s²) =⇒















v = v0− gt

y = y0+ v0t −¹₂g t² v² = v₀²− 2g(y − y0)

Chapter 3

Motion in Two and Three Dimensions

3.1 Vectors

Quantities :

 Scalars: no direction

Vectors: magnitude and direction

Vector Components:

A~ = A~x+ ~Ay, A~x = A cos θ ˆi(or ˆx), A~y = A sin θ ˆj(or ˆy), and

A =q

A²_x+ A²_y, tan θ = Ay

Ax

.

Vector Arithmetic with Unit Vectors:

A = ~~ Ax+ ~Ay, B = ~~ Bx+ ~By,

=⇒ ~A + ~B = (Axˆi + Ayˆj) + (Bxˆi + Byˆj) = (Ax+ Bx)ˆi + (Ay+ By)ˆj

3.2 Velocity and Acceleration Vectors (in 2 or 3-dim.)

~vave =∆r

∆t (ave. velocity vector) 7

8 CHAPTER 3. MOTION IN TWO AND THREE DIMENSIONS

~v = lim∆t→0∆~r

∆t = dr

dt (inst. velocity velocity) If ~r = xˆi + yˆj,

=⇒ ~v = dr dt =dx

dtˆi + dy

dtˆj = vxˆi + vyˆj, and

~aave= ∆~v

∆t (ave. acc. vector),

~a = lim∆t→0∆~v

∆t = dv dt = dvx

dtˆi + dvy

dt ˆj = axˆi + ayˆj. (inst. acc. vector)

3.3 Relative Motion

If ~V = V , V = const ( relative vel.), and x = x⁰+ V t,

=⇒ dx dt = dx⁰

dt + V =⇒ vx= v_x⁰ + V · · · (1) Since y = y⁰,

=⇒ dy dt =dy⁰

dt or vy= v⁰_y· · · (2)

(1)(2) =⇒ ~v = vxˆi + vyˆj = (v_x⁰ + V )ˆi + v⁰_yˆj = (v⁰_xˆi + v_y⁰ ˆj)

| {z }

~v⁰

+V ˆi,

=⇒ ~v = ~v⁰+ ~V _#.

3.4 Constant Acceleration (~a = const.)

1-dim. (Ch. 1):

v = v0+ a t

x = x0+ v0t + ¹₂a t²

~a = const, 2 or 3-dim.:

~v = ~v0+ ~a t

~r = ~r0+ ~v0t +¹₂~a t²

or 













x = x0+ v0xt +¹₂axt² y = y0+ v0yt +¹₂ayt² z = z0+ v0zt + ¹₂azt²

3.5. PROJECTION MOTION 9

3.5 Projection Motion

~v = ~v0+ ~a t,

~a = ax

|{z}

0

ˆi + ay

|{z}

−g

ˆj = −g y.

=⇒







vx = v0x+ axt = v0x, vy = v0y+ ayt = v0y− g t.

Since ~r = ~r0+ ~v0t + ¹₂~a t²,

=⇒







x = x0+ v0xt +¹₂axt²= x0+ v0xt_#· · · (1) y = y0+ v0yt +¹₂ayt²= y0+ v0yt −¹₂g t²

#· · · (2)

(1) =⇒ x − x0= v0x

|{z}

v0cos θ0

t =⇒ t = x − x0

v0cos θ0· · · (3)

(2)(3), v0y = v0sinθ0=⇒ y − y0= v0sinθ0

 x − x0

v0cos θ0



−1

2g  x − x0

v0cos θ0

2

= (x − x0) tan θ0− g

2v₀²cos θ²₀(x − x0)²

#

· · · (∗)

The Range of a Projectile:

y = y0in (*),

=⇒ 0 = (x − x0) tan θ0− g

2v²₀ cos θ₀²(x − x0)²

=⇒ x − x0= tan θ0

| {z }

sin θ0 cos θ0

×2v²₀ cos θ₀² g = 2v²₀

g sin θ0cos θ0, sin 2θ0= 2 sin θ0cos θ0

=⇒ x − x0=2v₀² g sin 2θ0

#

(horizontal range)

Since sin 2θ0|max= 1,

=⇒ (x − x0)max=2v₀²

g (at 2θ0= 90⁰or θ0= 45⁰)

#

10 CHAPTER 3. MOTION IN TWO AND THREE DIMENSIONS

3.6 Uniform Circular Motion

∆v v =∆r

r · · · (1) If θ ↓ =⇒ ∆r = v∆t,

(1) =⇒ ∆v

v =v ∆t

r =⇒ ∆v

|{z}∆t

a

= v²

r =⇒ a =v² r _#

Chapter 4

Force and Motion

4.1 The Wrong Question ( =⇒ What keeps things moving?)

The Right Question =⇒ What causes changes in moving?

4.2 Newton’s First and Second Laws

Ans.: =⇒ Force causes change in moving.

Newton’s 1^st Law: A body in uniform motion remains in uniform motion, and a body at rest remains at rest, unless acted on by a nonzero net force.

Newton’s 2^ndLaw: force ↔ changing in motion.

=⇒ ~p^(def.)= m ~v (momentum)

=⇒ ~Fnet= d~p

dt_# (2^nd law) If m = const,

=⇒ ~Fnet= d~p

dt = d (mv)

dt = m d~v

|{z}dt

~ a

=⇒ ~Fnet = m ~a_# (m = const, 2^ndlaw)

If ~Fnet = 0, =⇒ ~a = 0 =⇒ ~v = const# (=⇒ 1^st law) Mass, Inertia, and Force:

1^st law = law of inertia (=⇒ resists changes in motion) 11

12 CHAPTER 4. FORCE AND MOTION If F = m amand F = M aM,

=⇒ m am= M aM

=⇒ m

M =aM

a_{m #} (definition of mass) Inertial Reference Frames: (where Newton’s laws are valid)

=⇒ Test: check whether Newton’s 1^st law is obeyed.

Noninertial Reference Frames: e.g., Earth, an accelerating car, · · ·

4.3 Forces

The Fundamental Forces:

Theory of Everything:

• Gravity

• Grand Unified Force:

1. Strong 2. Electroweak:

(a) Weak

(b) Electromagnetism:

i. Electricity ii. Magnetism

4.4 The Force of Gravity

Mass: a measure of a body’s resistance to change in motion.

Weight: the force that gravity exerts on a body.

=⇒ ~W = m ~g (weight)

Weightlessness: (a) a falling elevator (b) an orbiting spacecraft.

4.5 Using Newton’s 2^nd Law

Draw a Free-Body Diagram:

1. Identify the object of interest and all the forces acting on it.

2. Reprent the object as a dot.

3. Draw the vectors for only those forces acting on the object, with their tails all starting on the dot.

4.6. NEWTON’S3^RD LAW 13

4.6 Newton’s 3^rd Law

3^rdlaw: If object A exerts a force on object B, then object B exerts an oppositely directed force of equal magnitude on A.

Measuring Force:

=⇒ ~Fsp= −k x (Hooke’s law), k : spring const

14 CHAPTER 4. FORCE AND MOTION

Chapter 5

Using Newton’s Laws

5.1 Using Newton’s 2^nd Law 5.2 Multiple Objects

Understanding tension forces:

5.3 Circular Motion 5.4 Friction

Applications of Friction: e.g., walking, stopping a car, · · ·

5.5 Drag Forces

=⇒ terminal speed (e.g., rain drops of a falling cat)

Objects moving through fluids like water or air experience drag forces that oppose the relative motion of object and fluid.

15

16 CHAPTER 5. USING NEWTON’S LAWS

Chapter 6

Work, Energy, and Power

6.1 Work

1-dim. displacement:

=⇒ W = Fx∆x

Work and the Scalar Product Scalar Product: (or dot product)

A · ~~ B^(def.)= A B cos θ, A = | ~A|, B = | ~B|.

If ~A = Axˆi + Ayˆj + Azˆk and ~B = Bxˆi + Byˆj + Bzˆk, A · ~~ B = AxBx+ AyBy+ AzBz.

For an arbitryry displacement ∆~r,

F · ∆~r = F ∆r cos θ~ _# (const force)

6.2 Forces That Vary

If F 6= const, =⇒ W 6= F ∆x.

=⇒ W ' F (x1)∆x + F (x2)∆x + · · · + F (xN)∆x = XN i=1

F (xi)∆x

=⇒ W = lim

∆x→0

X∞ i=1

F (xi)∆x = Z x2

x1

F (x) dx

formula:

17

18 CHAPTER 6. WORK, ENERGY, AND POWER

Z x2

x1

xⁿdx = xⁿ⁺¹ n + 1

x2

x1

= xⁿ⁺¹₂

n + 1− xⁿ⁺¹₁ n + 1 Stretching a Spring

F = − ~~ Fsp, Fsp= −kx (Eq. 4.9)

=⇒ F = kx

=⇒ W = Z x

0

F (x) dx = Z x

0

k x dx = k Z x

0

x dx

| {z }

x2 2

x

0=^x2₂

=1 2k x²

#

Force and Work in Two and Three Dimensions

∆W = ~F · ∆~r =⇒ dw = ~F · d~r

=⇒ W = Z

dW = Z ~r2

~r1

F · d~r~

#

Work Done Against Gravity

∆W = Fy∆y, Fy= mg,

=⇒ W = Z h

0

Fydy = mg int^h₀dy = mgh_#

6.3 Kinetics Energy

K.E. =⇒ K = 1

2m v², m : mass, v = speed.

Work-energy thm: (1-dim.)

=⇒ Wnet= Z

Fnetdx, Fnet = m a ( = mdv dt)

= Z

mdv

dt dx, v = dx dt

= Z v2

v1

m v dv = m Z v2

v1

v dv = m

2 v₂²−m

2 v₂²= ∆K =⇒ Wnet = ∆K_#

6.4. POWER 19

Energy Units

1joule = 1 newton-meter 1 erg = 10⁻⁷J

electron-volt (eV), calorie, foot-pound, British thermal unit (Btu), kilowatt-hours (kw · h).

See App. C.

6.4 Power

Ave. power: ¯P = ∆W

∆t Inst. power: P = lim

∆t→0

∆W

∆t = dW

d t (J/s = Watt) Since dW = P dt,

=⇒ W = Z t2

t1

P dt

#

If P = const,

=⇒ W = P Z t2

t1

dt = P ∆t, ∆t = t2− t1.

Power and Velocity Since dW = ~F · d~r,

=⇒ P = dW

d t = ~F · d~r

d t = ~F · ~v_#

20 CHAPTER 6. WORK, ENERGY, AND POWER

Chapter 7

Conservation of Energy

7.1 Conservative and Nonconservative Forces

Conservative force: When the total work done by a force ~F acting as an object moves over any closed path is zero, the force is conservative, i.e.,

H ~F · d~r = 0

=⇒ The work done by a conservative force is path-independent.

Pf:

WAB,1+ WBA,2= 0 =⇒ WAB,1= −WBA,2· · · (1) Since WAB,2= −WBA,2,

(1) =⇒ WAB,1= WAB,2

#

7.2 Potential Energy

Potential energyU :

∆UAB

| {z }

UB−UA

= − Z B

A

F · d~r =⇒ ∆U = −W~

 If W > 0 =⇒ ∆U < 0 If W < 0 =⇒ ∆U > 0





1-dim.: ∆U = − Z x2

x1

F (x) dx



Gravitational P.E.: ∆U = mgh Elastic P.E.:

=⇒ ∆U = − Z x2

x1

F (x) dx = − Z x2

x1

(−k x) dx = 1

2k x²₂−1 2k x²₁

If choose U = 0 as x = 0 (i.e., the spring is neither stretched nor compressed),

=⇒ ∆U = 1 2k x²

#

(elastic P.E.)

21

22 CHAPTER 7. CONSERVATION OF ENERGY

7.3 Conservation of Mechanical Energy

In Ch. 6,

∆K = Wnet

| {z }

W^c+W^nc

· · · Eq. (6.14)

Since Wc = −∆U ,

=⇒ ∆K = −∆U + Wnc =⇒ ∆K + ∆U = Wnc · · · (∗) If Wnc= 0,

(∗) =⇒ ∆K + ∆U = 0

=⇒ K + U

| {z }

E(mech)

= Const

or E(mech)= Const

# (Conservation of mech. energy)

7.4 Potential-Energy Curves

Force and P.E.:

Since ∆U = −W ,

1-dim. : =⇒ ∆U = −Fx∆x (for small displacement)

=⇒ Fx = −^∆U_∆x

∆x → 0 =⇒ Fx = −^dU_dx^(x)

#

3 dim.:

Fx= −∂U (x, y, z)

∂x , Fy= −∂U (x, y, z)

∂y , Fz= −∂U (x, y, z)

∂z .

Chapter 8

Gravity

8.1 Toward a Law of Gravity

Kepler’s laws: (based on observation, no theoretical explanation) (See Fig. 8.1)

1^st law: The orbit is elliptical, with the Sun at one focus.

2^ndlaw: If the shaded areas are equal, so is the time to go from A to B and from C to D.

3^rd law: The square of the orbital

T

z }| {

period is proportional to the cube of the semimajor axis

| {z }

a

. (=⇒ T³ ∝ a³)

8.2 Universal Gravitation

=⇒ F = G m1m2

r² , (attractive)

where G = 6.67 × 10⁻¹¹N · m²/kg² (the const of universal gravitation) The Cavendish Experiment: Weighing the Earth

(See Fig. 8.4)

Since F = ^{G m M}_r2 , measure F and given m, M , and r,

=⇒ calculate G_#

In Example 8.1, use g = G ME/R_E² =⇒ M_{E #} (i.e., weighing the Earth) 23

24 CHAPTER 8. GRAVITY

8.3 Orbital Motion

(See Fig. 8.5)

G M m

r² = mv²

r =⇒ v =

rG M

r _# (circular orbit) · · · (1)

Since T = 2π r

v =⇒ v = 2π r

T · · · (2)

(1)(2) =⇒ 2π r T =

rG M

r =⇒  2π r T

2

=G M

r =⇒ T²=4π²r³

G M # (or T²∝ r³) Elliptical Orbits (see Fig. 8.7)

=⇒ Projectile trajectories are actually elliptical (if no air resistance).

Open orbits (see Fig. 8.8)

8.4 Gravitational Energy

(See Fig. 8.9)

∆U12= − Z r2

r1

F · d~r,~ F · d~r = −~

 GM m

r²

 dr

=⇒ ∆U12= Z r2

r1

GM m

r² dr = GM m 1 r1 − 1

r2



#

· · · (∗)

The Zero of Potential Energy: (Choose r1= ∞, r2= r) (∗) =⇒ U (r) = −GM m

r _# Escape Speed:

E = K + U = 1

2mv²−GM m r If E = 0,

=⇒ 0 = ¹₂mv²−^{GM m}_r ,

=⇒ vesc=q

2GM r #

(escape speed)

At r = RE, =⇒ vesc= 11.2 km/s.

8.5. THE GRAVITATIONAL FIELD 25

Energy in Circular Orbits:

For a circular orbit,

GM m

r² = mv²

r =⇒ v²= GM r

=⇒ K = 1

2mv2= GM m

2r , U = −GM m

r = −2K and

E = U + K = −2K + K = −K_# or

E = U

2 = −GM m 2r _#

8.5 The Gravitational Field

Action at a distance:

F = G~ M m r² (−ˆr) Gravitational field:

~g = −GM r² r,ˆ

=⇒ ~F = m ~g = −GM m r² (ˆr) Near Earth’s surface, ~g ' −g ˆy, where g = 9.8 m/s².