Electromagnetism.!

!!

!

!!

!!

!!

!!

!!

!!

!

!

!

!!

!

!!

Quantum.Mechanics..

.

..

..

.

..

..

..

..

..

..

..

..

..

.

..

.

.

Statistical.Physics.and.Thermodynamics..

..

..

..

..

..

.

..

..

.

..

..

.

.

..

.. .

Theoretical.Mechanics..

..

..

..

..

..

..

..

..

..

.

Useful formulas

Moments of inertia:

• homogeneous ring of radius R and mass M : I = MR

2.

• homogeneous disk of radius R and mass M : I = 12MR

2

• homogeneous ball of radius R and mass M : I = 25MR

2.

• homogeneous rectangle with lengths of sides a and b: I = 112M(a2 + b

2).

• The “parallel axis theorem”:

{I} = {ICM

} + {IR

}where ~

R ⌘ (X, Y, Z) is the radius vector of the new origin drawn from the center ofmass (CM), and {I

CM

} is the tensor of inertia computed in that frame. Here, theelements of {I

R

} are: {IR

}XX

= M(Y 2 + Z

2), {IR

}XY

= �MXY , etc, doing allpermutations of the coordinates X ! Y ! Z.

• If a vector ~

A is constant in a frame rotating with angular velocity ~⌦, then it’s rate ofchange with respect to the lab frame is

d

~

A

dt

= ~⌦⇥ ~

A

• Rdx

x

= ln x + C

• Rx

n

dx = x

n+1

n+1 + C for n 6= �1

• Rsin xdx = � cos x + C and

Rcos xdx = sin x + C

• sin(↵ + �) = sin ↵ cos � + cos ↵ sin �

• cos(↵ + �) = cos ↵ cos � � sin ↵ sin �

• sin2↵ = 1

2(1� cos 2↵)

• cos2↵ = 1

2(1 + cos 2↵)

• Solutions of the quadratic equation ax

2 + bx + c = 0 are

x1,2 =�b ±pb

2 � 4ac

2a

• Scalar product: ~a ·~b = a

x

b

x

+ a

y

b

y

+ a

z

b

z

.

• A second order linear di↵erential equation d

2x

dt

2 + !

2x has the solution in the form of

x(t) = A sin(!t) + B cos(!t).

4