10.3. Analogies Between Linear and Rotational Motion http://www.ck12.org
10.3 Analogies Between Linear and Rotational Motion
Linear
Quantity Units
~x m
~v m/s
m kg
~F=∆∆~pt N
~a=FNet~m m/s^2
~p=m~v kg m/s
K=^12 mv^2 JRotational
Quantity Units
~θ Radians
ω Radians/s
~I kg m^2
τ=
∆~L
∆t N m
α=τNet~I Radians/s^2
~L=~Iω kg m^2 /s
K=^12 Iω^2 JIn addition to [1], [2], and [3], there are other important relationships for rotational motion. These are summarized
in table 1.1.
TABLE10.1:
Equation Explanation
ac=v^2 /r=rω^2 the centripetal acceleration of an object.
ω= 2 π/T= 2 πf Relationship between period and frequency.
θ(t) =θ 0 +ω 0 t+ 1 / 2 αt^2 The ’Big Three’ equations work for rotational motion
too!
ω(t) =ω 0 +αt Rotational equivalent ofvf=vi+at
ω^2 =ω^20 + 2 α(∆θ) Rotational equivalent ofv^2 f=v^2 i+ 2 ad
α=τnet/I Angular accelerations are produced by net torques,with
inertia opposing acceleration; this is the rotational ana-
log ofa=Fnet/m
τnet=Στi=Iα The net torque is the vector sum of all the torques acting
on the object. When adding torques it is necessary to
subtract CW from CCW torques.
~τ=~r×~F=r⊥F=rF⊥ Individual torques are determined by multiplying the
force applied by theperpendicularcomponent of the
moment arm
L=Iω Angular momentum is the product of moment of inertia
and angular velocity.
τ= 4 L/ 4 t Torques produce changes in angular momentum; this is
the rotational analog ofF= 4 p/ 4 t
K= 1 / 2 Iω^2 Rotational motion contributes to kinetic energy as well!