Quantitative Physics

Question 1

Signal to noise ratio

Answer 1

SNR = measurement / uncertainty

Question 2

What are the 5 basic dimensions

Answer 2

Mass M, length L, time T, temperature θ, and current I.

Question 3

Standard Error on the mean

Answer 3

S(x̄) = s(x) / √N

Where s(x) is the unbiased standard deviation
N is the number of trials

Question 4

Mean

Answer 4

x̄ = Σx / N

Question 5

sample standard deviation

Answer 5

S(x) = Σ(x-x̄)² / (N-1)

Question 6

% of measurements within 1,2,3 std deviations for normal distribution

Answer 6

For normal distribution with large N centered at x̄:

68% of all mesurements fall w/in 1std deviation
98% of all mesurements fall w/in 2std deviation
99.7% of all mesurements fall w/in 3std deviation

Question 7

Difference between measurements and uncertainty to determine experimental validity

Answer 7

Given tow measurements x1 ± u1 and
x2 ± u2.
Δx = |x2 - x1| , u(Δx) = √(u1² + u2²)
the ratio Δx/u(Δx) is the # of standard deviations the difference in the measurements is away from zero.
Δx/u(Δx) = 0 implies the measurements are completely consistent.

Question 8

Independent event condition

Answer 8

A and B are independent if P(A ∩ B) = P(A)P(B)

Question 9

Binomial theorem conditions

Answer 9

You can model X w/ binomial distribution B(n,p) if:
-There are a fixed number of trials n.
-There are two possible outcomes
-There is a fixed probability of one outcome P
-The trials are independent of one another

Question 10

Binomial Theorem equation

Answer 10

If X has binomial distribution P(n, p) then
P(x=r) = ⁿ C ᵣ p ʳ (1 - p) ⁿ ⁻ ʳ

Question 11

Small diff approx (differential)

Answer 11

for some function f(x), for a small change in Δx:

Δf ≈ f ‘(x)Δx
so f(x + Δx) ≈ f(x) + Δf ≈ f(x) + f ‘(x)Δx

Question 12

Small diff approx (expansion)

Answer 12

For behaviour around x=0 —> evaluate limiting form of the macalurin series expansion
For behaviour around x=a –> evaluate limiting form of taylor series expansion

Question 13

Force-potential equation

Answer 13

u(x) = - ∫ F dx

Question 14

Mass integral

Answer 14

M = ∫ dm = ∫ p dV

Question 15

Moment of inertia integral

Answer 15

I = ∫ r² dm

Question 16

Centre of mass integral

Answer 16

x꜀ₘ = 1/M ∫ x dm

Question 17

odd and even function integral properties

Answer 17

Even function [-a,a] ∫ f(x) dx = 2 [0,a] ∫ f(x) dx
Odd function [-a,a] ∫ f(x) dx = 0

Question 18

How to find the limit of a function at a small value

Answer 18

Use a power series and ignore higher order terms

Question 19

Properties of exponentials w/regard to limiting forms

Answer 19

eˣ will dominate over power law as it tends to a limit faster
for a function xⁿe⁻ˣ as x tends to infinity, xⁿ tends to infinity but e⁻ˣ tends to zero, the exponential dominates hence the function tends to zero.

Question 20

Number distribution integrals

Answer 20

N = [-∞,∞] ∫ N(x) dx

<x> = 1/N [-∞,∞] ∫ x N(x) dx
</x>

Question 21

Probability distribution integrals

Answer 21

1 = [-∞,∞] ∫ f(x) dx

<x> = [-∞,∞] ∫ x f(x) dx
</x>

Question 22

Relationship between number distribution N(x) and probability distribution f(x)

Answer 22

f(x) = 1/N N(X)

Question 23

What is the most probable value of a probability distribution

Answer 23

the x values at the peaks of the distribution i.e when f’(x) and f’‘(x)<0