Chapter 1-12 Formulas

Uncertainty Calculation for Z = X + Y

dZ = dX + dY

Uncertainty Calculation for Z = X - Y

dZ = dX + dY

Uncertainty Calculation for Z = kX

dZ = k dX

Uncertainty Calculation for Z = nX +- mY

dZ = n dX + m dY

Uncertainty Calculation for Z = XY

dZ/Z = dX/X + dY/Y

Uncertainty Calculation for Z = X/Y

dZ/Z = dX/X + dY/Y

Uncertainty calculation of Z = Xⁿ

dZ/Z = |n| dX/X

Uncertainty calculation for Z = nX^a mY^b

dZ/Z = |a| dX/X + |b| dY/Y

Uncertainty calculation for Z = nX^a / mY^b

dZ/Z = |a| dX/X + |b| dY/Y

Uncertainty calculation for trigonometric functions Z

dZ = Z - Z min or Z max - Z (whichever one is larger)

Velocity (formula) with SI units

v = ds/dt
m/s

Acceleration (formula) with SI units

a = dv/dt
m/s²

Average acceleration (formula) with units

<a> = (v_f - v_i) / dt
m/s²

Assuming there is constant acceleration, what equation connects final and initial velocity and time?

v = u + at

Assuming there is constant acceleration, what equation connects displacement, initial velocity and time?

s = ut + ½ at²

Assuming there is constant acceleration, what equation connects final velocity, initial velocity and displacement?

v² = u² + 2as

Linear momentum (formula) with SI units

p = mv
kgm/s or Ns

The change in momentum of a body (formula)

p_f - p_i

Newton’s Second Law of Motion (formula)

Resultant Force proportional to dp/dt

Impulse (formula)

F_ave x t

Conservation of momentum (formula)

p_Ai + p_Bi = p_Af + p_Bf

Head-on collision, relative speed of approach and separation (formula)

u_A - u_B = v_B - v_A

Formula related to elastic collision

½ m_Au_A² + ½ m_Bu_B² = ½ m_Av_A² + ½ m_Bv_B²

Hooke’s Law (formula)

F = kx

Moment of a force (formula) with SI units

moment = force x distance
J

Torque of a couple (formula) with SI units

Torque = One force x distance
J

Pressure (formula) with SI units

p = F/A
N/m² or Pa

Upthrust (formula) with SI units

v_{fluid displaced} p_fluid g
N

Archimedes’ Principle (formula)

U = W_{fluid displaced}

Work done by a force (formula) with units

W = Fscos(theta)
J

Work done by a gas (formula) with SI units

p(V_f - V_i)
J

Gravitational Potential Energy in a uniform gravitational field (formula) with SI units

E_p = mgh
J

Elastic Potential Energy (formula) with SI units

½ kx²
J

Kinetic Energy (formula)

½ mv²

Power (formula) with SI units

P = dW/dt
W or J/s

Average Power (formula) with SI units

<P> = (W_f - W_i) / dt
W or J/s

Formula relating power, velocity and force

P = Fv

Efficiency (formula) [all 3 of them]

1. (Useful energy output / energy input) x 100%

2. (Useful work done / energy input) x 100%

3. (Useful power output / power input) x 100%

Angular Velocity (formula) [all of them]

1. omega = d(theta)/dt

2. omega = 2 pi f

3. omega = 2 pi / T

Linear Velocity in circular motion (formula)

v = r omega

Centripetal acceleration (formula) [all of them]

1. a = v² / r

2. a = r omega²

Centripetal Force (formula) [all of them]

1. F = ma

2. F = mv² / r

3. F = mr omega²

What is the value and unit of universal gravitational constant (G)?

6.67 × 10^-11 Nm²kg^-2

Newton’s Law of Gravitation (formula) with SI units

F = GMm / r²
N

Gravitational Field Strength (formula) with SI units [all of them]

1. g = F / m

2. g = GM / r²

N/kg or m/s²

Gravitational Potential Energy in a non-uniform field (formula) with SI units

U = -GMm / r
J

Relationship between Gravitational Force and Gravitational Potential Energy

F = -dU/dr

Escape Velocity (final and initial formula)

Total Energy at Earth’s Surface = Total Energy at Infinity

-GMm/r + ½ mv² = 0

v = √2GM/r , sub g = GM / r²

v = √2gr

Gravitational Potential (formula) with SI units [all of them]

1. phi = U / m

2. phi = -GM / r

J/kg

Relationship between gravitational potential and gravitation field strength

g = -d(phi) / dr

Kepler’s Third Law (formula) [from derivation to final]

F = ma

GMm / r² = m r omega²

T² = (4 pi² r³) / GM

(Gravitational) Potential Energy of a satellite (formula)

E_p = -GMm / r

Kinetic energy of a satellite (formula)

E_k = GMm / 2r

Total Energy of a satellite

E_T = - GMm / 2r

Temperature determination using thermometric properties (formula)

T = (X_? - X₀) / (X₁₀₀ - X₀) x 100 degrees

Conversion between Celsius scale and Kelvin scale

T (K) = T (C) + 273.15

Avogadro’s constant with symbol and units

N_A = 6.02 × 10²³ mol^-1

Formula connecting number of moles (n) and number of particles (N)

n = N / N_A

Formula connecting number of moles (n) and mass (m)

n = m / M_r

Formula connecting pressure (p) and temperature (T) when volume is constant

p = kT

Formula connecting volume (V) and temperature (T) when pressure is constant

T = kV

Formula connecting pressure (p) and volume (V) when temperature is constant

p = k/V

Molar Gas Constant with symbol and units

R = 8.31 J / (K mol)

Ideal Gas Equation with R

pV = nRT

Boltzmann constant with symbol and units

k = 1.38 × 10^-23 J / K

Ideal Gas Equation with k

pV = NkT

Kinetic Theory Equation

pV = 1/3 Nm<c²>

Mean Kinetic Energy of one particle

3/2 kT

Kinetic energy of a gas [all formulas]

1. 3/2 NkT

2. 3/2 nRT

3. 3/2 pV

Root mean square speed (formula) in terms of k [just knowledge]

√(3kT/m)

(Ideal gas) Change in Internal Energy (formula)

dU = 3/2 Nk dT
dU = 3/2 nR dT

Heat supplied (formula) [sec school]

Q = mc d(theta)

Heat supplied during a change of state (formula)

L_f = ml_f
L_v = ml_v

First Law of Thermodynamics (formula)

dU = Q + W

Simple Harmonic Motion (formula)

a = - omega² x

(S.H.M.) Formula for magnitude of maximum acceleration

a_o = omega² x_o

(S.H.M.) Formula connecting velocity (v) and displacement (x)

v = +- omega √x_o² - x²

(S.H.M.) Formula for magnitude of maximum speed

v_o = omega x_o

(S.H.M.) Equation for variation of displacement (x) with time (t) when particle starts from equilibrium position

x = +- x_o sin(omega t)

(S.H.M.) Equation for variation of displacement (x) with time (t) when particle starts from amplitude position

x = +- x_o cos(omega t)

(S.H.M.) Equations for derivation of v t equation from x t graph

v = dx / dt

v_o = omega x_o

(S.H.M.) Equation for variation of velocity (v) with time (t) when particle starts from equilibrium position

v = -+ v_o cos(omega t)

(S.H.M.) Equation for variation of velocity (v) with time (t) when particle starts from amplitude position

v = -+ v_o sin(omega t)

(S.H.M.) What does the sign of vt graph depend on?

The sign for x t graph

(S.H.M.) Equations for derivation of a t graph from v t graph

a = dv/dt
a_o = omega² x_o

(S.H.M.) Equation for variation of acceleration (a) with time (t) when particle starts from equilibrium position

a = -+ a_o sin(omega t)

(S.H.M.) Equation for variation of acceleration (a) with time (t) when particle starts from amplitude position

a = -+ a_ocos(omega t)

(S.H.M.) Kinetic energy (formulae)

E_k = ½ m omega² (x_o² - x²)

E_k = ½ m v_o² cos²(omega t)

E_k = ½ m v_o² sin²(omega t)

( ½ m v_o² can be replaced by max KE)

(S.H.M.) Maximum Kinetic energy (formulae)

E_ko = ½ m v_o²

E_ko = ½ m omega² x_o²

Generally, how are U and E_k related?

U = -E_k

Relationship between period (T) and frequency (f)

T = 1/f

Wave speed (formula) with SI units

v = f lambda
m/s

Phase difference (formula) with SI units for displacement distance graph

between

1. Two particles on the same wave

2. Two waves of the same frequency

d(phi) = (dx / lambda) x 2 pi
rad

Phase difference (formula) with SI units for displacement time graph between two waves of the same frequency

d(phi) = (dt / T) x 2 pi

Wave Intensity (formula) with SI units

I = P_source / A_spread
W / m²

Relationship between Intensity (I) and Amplitude (A)

I proportional to A²

Inverse Square Law (formula)

I proportional to 1/r²

Intensity of waves with spherical wavefronts (formula)

I = P / (4 pi r²)

(Wave) Relationship between amplitude (A) and distance (r)

A proportional to 1/r

If a wave is originally unpolarised, what is the Intensity after it passes through the polariser?

I = I_o / 2