example problems for jhu-pha diagnostic...
TRANSCRIPT
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