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Level 197

Special Relativity & Rotational Mechanics

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distance between two events in space
is the same in any Euclidean frame, Δs² = Δs'²
Galilean Transformation, x'
x - vt
Einstein's 1st Postulate
All of the laws of physics are the same in every inertial frame
Einstein's 2nd Postulate
The speed of light in a vacuum is the same for all observers
Time Dilation, Δt
1 / √(1 - v²/c²)
Shortest time interval between two events
measured by a clock present at both events, known as proper time interval
Length Contraction, L
Lorentz Transformation, t'
γ(t - vx/c²)
Lorentz Transformation, t
γ(t' + vx'/c²)
Lorentz Transformation, x'
γ(x - vt)
Lorentz Transformation, x
γ(x' + vt')
Speed, Ux
(U'x + v)/(1 + (vU'x / c²))
Motion at an angle, Uy
U'y / γ(1 + (vU'x / c²))
v₀ √((c-v)/(c+v))
Frequency of pulses, v₁
Rest Energy, E₀
Mass-energy, E
K + E₀ = γmc²
Kinetic Energy, K
Relativistic Momentum, p
E² - m²c⁴
Centre of Mass, R
(1/M) ∫[body] r dm
In 1 dimension
dm = ρ dl
In 2 dimensions
dm = ρ da
In 3 dimensions
dm = ρ dV
Conditions for Static Equilibrium
Vector sum of all external forces acting is zero and the sum of all turning moments about an axis through any point is zero
Law of the Lever
Gext = ∑Gi = ∑fi li = 0
Moment Vector, G
Iω' = F x r, where F is the Force acting perpendicular to the axis and r is the distance from the point
Centripetal Force, F
-mω²r = -mv²/r (towards centre of circle)
Angular Moment, Moment of Momentum, L
Iω = r x P = r x mv (perpendicular to r and v (or P))
conservation of angular momentum
just as linear momentum is conserved, so is angular momentum
Angular Impulse, J
r x I = ∫ G dt = ΔL
Moment of Inertia I,
∑mi ri² = ∫[body] r² dm (write dm in terms of dr using density)
Parallel Axis Theorem
The moment of inertia of an axis, AB, parallel to n axis, OP through the centre of mass about which the moment of inertia I₀, is given by:
Perpendicular Axis Theorem
Iz = Ix + Iy, where the lamina lies in the xy plane
½ Iω²
Total Rotational Kinetic Energy