Peoples Physics Book Version-2

10.2. Formalizing Rotational Motion http://www.ck12.org

10.2 Formalizing Rotational Motion

FIGURE 10.1

Illustration of Rotational Motion

Key Concepts

• To determine therotation axis,wrap your right hand’s fingers in the direction of rotation and your thumb will
  point along the axis (see figure).


• When something rotates in a circle, it moves through aposition angleθthat runs from 0 to 2πradians and
  starts over again at 0. The physical distance it moves is called thepath length. If the radius of the circle
  is larger, the path length traveled is longer. According to the arc length formulas=rθ, the path length∆s
  traveled by something at radiusrthrough an angleθis:
  ∆s=r∆θ[1]


• Just like the linear velocity is the rate of change of distance, angular velocity, usually calledω, is the rate of
  change ofθ. The direction of angular velocity is either clockwise or counterclockwise. Analogously, the rate
  of change ofωis the angular accelerationα.


• The linear velocity and linear acceleration of rotating object also depend on the radius of rotation, which is
  called themoment arm(See figure) If something is rotating at a constant angular velocity, it moves more
  quickly if it is farther from the center of rotation. For instance, people at the Earth’s equator are moving faster
  than people at northern latitudes, even though their day is still 24 hours long – this is because they have a
  greater circumference to travel in the same amount of time. According to [1],

ω=

∆θ

∆t

=

∆s

rt

=

v

r

orv=ωr[2]

• Alternatively, we could derive [2] by setting the time to travel a path length equal to the circumference, 2πrat
  speedvequal to the time it takes to travel one full angular revolution, 2πat angular velocityω.


• In exactly the same fashion we can derive the fact that angular accelerationαis related to linear acceleration
  ain the following way:

a=αr[3]

Note: The above two relations hold for the situations where it is a single object (like the Earth, merry go round, etc.)
or if a rolling object is not slipping with respect to the ground or if a pulley is not slipping with respect to the rope.