Physics 1 Topic 6: Torque changes Rotation (spin)
. . . things spin around an axis ... tend to spin at constant rate.
Even if the Sum of all Forces (applied to an object) is zero, the object's motion might be changing rotationally.
. . . a thing's spinning motion tends to stay the same <=> analogous to linear motion
. . . angular location, or angular position, angular orientation traditionally called θ or φ, in [radians] = (arc length)/r <=> x, y, or z .
(right-hand rule : curl right fingers along arc's arrow and thumb points (perp.to fingers) in direction of θ )
. . . counter-clockwise arc has θ out of page .
. . . angular velocity ω [radian/sec] = Δθ / Δt = v/r is analogous to linear velocity
. . . angular acceleration α [radian/s²] = Δω / Δt = a/r is analogous to linear acceleration .
The analogy to any kinematic formula is obtained by directly replacing each linear quantity by its angular sibling :
θf = θi + ωavg Δt ;
ωf = ωi + αavg Δt ;
Δθ = ωi Δt + ½ αavg (Δt)² ;
ωf² = ωi² + 2 αavg·Δθ ; etc.
Rotational Dynamics statements are almost a straight-forward analogy to the familiar linear (translational) statements .
. . . the rotational analogy to Force is Torque , abbreviated "tau" τ = r × F , with units [m×N] .
Torque around a rotation axis depends on the distance from the axis (tail of r) to the point the Force is applied
. . . and only the component of the Force that is | to r provides torque
. . . or, use the entire Force at its lever-arm distance : the distance from the axis | to the Force's line of action
generally, τ [m N] = r × F = r F sin(θ) ; here, θ is the angle from the r-vector direction to the F-vector direction
. . . the general right-hand rule has : right rist along r , fingers sweep through angle θ until fingers point along F ; Thumb points ( | ) along Torque .
. . . the result from any cross product like torque) must always be perpendicular to both of the factors ...
=> so torque is always perpendicular to the Force that generates it, and is perpendicular to the radius vector (to the F application place)
. . . if the total Torque is zero, the conserved rotational quantity is angular momentum , abbreviated L = r x p , with units [m N s]
of course the anguLar momentum is also around the rotation axis ... L = r | p = r p_|_ = r p sin(θ) ... use the same right-hand rule as for Torque .
The "story-board" statement of how Nature behaves is just as valid in rotational form as in linear form :
"the starting anguLar momentum is changed by the total Torque thru a duration, into the later anguLar momentum."
. . . Σ Li + Σ τ Δt = Σ Lf .
If you try to isolate the more easily-measured kinematic quantities , the analogies become more complicated.
. . . the analogy to mass is rotational Inertia or "angular Inertia" or "(second) Moment of Inertia" , abbreviated I = Σ m r² , with units [kg m²]
... it is the second mass moment (notice the square!) ... it depends not only on the mass , but strongly on how far from the axis that mass is ... which can easily change
... because its velocity is v = − r × ω , more distant mass has more momentum
. . . we will approximate real shapes by similar geometries that have known "formulas" for their moment of Inertia (page 291)
using the parallel-axis theorem if we need the rotation axis in a different place (just add "M ΔR²" to move the axis by ΔR)
. . . once the Inertia is known, the angular momentum is (yes, as you expect from then linear analogy):
=> => L = I ω
If some mass's distance (from the axis) changes , it's essentially a "soft rotational collision" between the moving masses and the others.
. . . If you know the initial and final locations, you may compute the initial rotational Inertia and the different final Rotational Inertia
Ii ωi + Σ τ = If ωf
If the object does NOT change shape, and the rotation axis does not change,
then the moment of Inertia (around that axis) does NOT change ;
. . . then any change in angular momentum comes from change in angular velocity ω
. . . so the total external torque causes the object's rotational Inertia to have angular acceleration α :
=> ΣτA = I ΔωA / Δt = I αA ... where "A" is the axis , which the Inertia will rotate around ...
... a mass has tangential acceleration component aφ = − r × α , but radial acceleration component ar = − rω² .
because the different parts of the object do not have the same speed, there is an additional Kinetic Energy term :
=> KErotation = ½ I ω² calso be written KErot = ½ L ω , or KErot = L²/2I .
. . . this KE of mass around the rotation axis is added to the linear KE as if it was moving at the speed that the axis has .
this rotational KE is changed by rotational Work :
=> Wrot = τ ·Δθ . . . parallel components of Torque and angular displacement .
an implication of this is that an object's moment of inertia is smallest around its center-of-mass,
. . . more rotational inertia the farther the axis is from the c.o.m. ; if axis is distance d from c.o.m. :
=> Iaxis = Ic.o.m. + M d² .
You STILL get to choose which objects in the scene are included INSIDE your system
. . . use "bigger" systems if you intend to use angular momentum during a "rotational collision"
so that internal Forces do not produce external torques to Sum ... but the Total Inertia inside does.
