Rotation
Linear motion   Newton's Laws   Orbital Mechanics   Relativistic Motion

ω = V/r
   ω is angular velocity in radians/sec
        1 radian/sec = 9.55 rev/min
   r is radius of circle in meters
   V is the tangental velocity in m/s
Angular position θ = ωt


Centripetal force f = mV²/r = mrω²
   ω is angular velocity in radians/sec
        1 radian/sec = 9.55 rev/min
   m is mass in kg
   r is radius of circle in meters
   V is the tangental velocity in m/s = ωr
   f is in Newtons

Centripetal acceleration is
a = Vdθ/dt = ω²r = Vω = V²/r
   V is the tangental velocity in m/s
   r is radius of circle in meters
   ω is angular velocity in radians/sec = v/r
   a is in m/s²

Angular momentum in kg•m²/s
   L = Iω
   I is moment of inertia in kg•m²
   ω is angular velocity in radians/sec

For a small particle moving in a circle:
Angular momentum = ½ωmr² = rmV
   ω is angular velocity in radians/sec
   m is mass in kg
   V is the tangental velocity in m/s
   r is radius of circle in meters

linear momentum converted to angular momentum
   for example a bullet hitting a door
   L = Pr where P = mV is linear momentum

angular acceleration in rad/s²
  α = τ/I for constant α
     I = moment of inertia in kg•m²
     τ = torque in N•m
  τ = αI  torque required for that accel.
  P = τω  power to hold that speed with τ friction
  θ = θ₀ + ω₀t + ½αt²
  θ = θ₀ + ½(ω₀ + ω)t
  ω = ω₀ + αt
  ω² = ω₀² + 2α(θ – θ₀)
  θ = (Δω)(Δt)/2
  ω = dθ/dt
  α = dω/dt
  α = Δω/Δt

linear tangental acceleration, αt
   α = αt/r

Angular kinetic energy E in Joules
E = ½Iω²
   ω is angular velocity in radians/sec
   I = moment of inertia in kg•m²

moment of inertia
A bodies moment of inertia about any given axis is the
moment of inertia about the centroid plus the mass of
the body times the distance between the point and the
centroid squared.
   I₀ = Ic + md²
   I₀ is moment of inertia about point O
   Ic is moment of inertia about the centroid
   d is distance between the point and the centroid

The moment of inertia of a point mass with respect to an
axis is defined as the product of the mass times the
distance from the axis squared. The moment of inertia of any
extended object is built up from that basic definition. More
that one point, they add.

I is moment of inertia in kg•m²
I = cMR²
    M is mass (kg), R is radius (meters)
    c = 1 for a ring or hollow cylinder
    c = 2/5 solid sphere around a diameter
    c = 7/5 solid sphere around a tangent
    c = ⅔ hollow sphere around a diameter
    c = ½ solid cylinder or disk around its center
    c = 1/4 solid cylinder or disk around a diameter
    c = 1/12 rod around its center, R = length
    c = ⅓ for a rod around its end, R = length
    c = 1 for a point mass M at a distance R from
       the axis of rotation
    c = 1/3 for a door, where r is the width
for a wheel with r inner radius and R outer radius
    I = ½M(R²–r²)
for a sphere on the end of a rod:
    Sphere radius R, mass M, rod length = L, rod mass m
    I = (2/5)MR² + M(L+R)² + (1/3)mL²


Theorem of parallel axes: the moment of inertia of a body
about any axis is equal to the sum of the moment of inertia
of the body about a parallel axis passing through its centre
of mass and the product of its mass and the square of the
distance between the two parallel axes.

cylinder or ball rolling at speed V
linear KE = ½mV²
rotational KE = ½Iω² = ½cmR²ω² = ½cmR²(V/R)² = ½cmV²
Total KE = ½mV² + ½cmV²
solid sphere total KE = ½mV² + (1/5)mV² = (7/10)mV²
hollow sphere total KE = ½mV² + (1/3)mV² = (5/6)mV²
solid cylinder total KE = ½mV² + (1/4)mV² = (3/4)mV²
ring or hollow cylinder total KE = ½mV² + ½mV² = mV²

cylinder or ball rolling down a slope

PE at top results in both linear KE and rotational KE
   at the bottom.
mgh = ½mV² + ½Iω²
V = rω   r is radius
V² = (2gh) / (1 + I/mr²)
for a solid cylinder that reduces to V² = (4/3)gh (1.33)
for a solid sphere, that is V² = (10/7)gh (1.43)
for a ring or hollow cylinder, that is V² = 2gh
for a hollow sphere, that is V² = (6/5)gh (1.2)

parallel axis theorem:

It can be used to determine the second moment of area
or the mass moment of inertia of a rigid body about any
axis, given the body's moment of inertia about a
parallel axis through the object's centre of mass and
the perpendicular distance (r) between the axes.

The moment of inertia about the new axis z is given by:
Iz = Icm + mr²
where:
Icm is the moment of inertia of the object about an axis
passing through its centre of mass;
m is the object's mass;
r is the perpendicular distance between the two axes.

consider a body of mass m, moving in a circle of
  radius r, with an angular velocity of ω.
speed is v = r·ω.
centripetal (inward) acceleration is a = rω² = v²/r
centripetal force is F = ma = rmω² = mv²/r
momentum of the body is p = mv = rmω
angular momentum is L = rmv = r²mω = I·ω
kinetic energy is E = ½mv² = ½r²mω² = p²/2m =
   ½Iω = L²/2I

Power (watts) = torque (N-m) x 2π x rotational speed
   (rev/sec)



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