Chapter 19
Rotational Motion Variables, Tangential Acceleration, Constant Angular Acceleration
117
19
Rotational Motion Variables, Tangential Acceleration,
Constant Angular Acceleration
Because so much of the effort that we devote to dealing with angles involves acute angles,
when we go to the opposite extreme, e.g. to angles of thousands of degrees, as we often
do in the case of objects spinning with a constant angular acceleration, one of the most
common mistakes we humans tend to make is simply not to recognize that when someone
asks us; starting from time zero, how many revolutions, or equivalently how many turns
or rotations an object makes; that someone is asking for the value of the angular
displacement
Δ
θ
.
To be sure, we typically calculate
Δ
θ
in radians, so we have to convert
the result to revolutions before reporting the final answer, but the number of revolutions
is simply the value of
Δ
θ
.
In the last chapter we found that a particle in uniform circular motion has centripetal acceleration
given by equations 185 and 186:
r
v
2
c
=
a
2
c
w
r
=
a
It is important to note that any particle undergoing circular motion has centripetal acceleration,
not just those in uniform (constant speed) circular motion.
If the speed of the particle (the value
of
v
in
r
v
2
c
=
a
) is changing, then the value of the centripetal acceleration is clearly changing.
One can still calculate it at any instant at which one knows the speed of the particle.
If, besides the acceleration that the particle has just because it is moving in a circle, the speed of
the particle is changing, then the particle also has some acceleration directed along (or in the
exact opposite direction to) the velocity of the particle.
Since the velocity is always tangent to
the circle on which the particle is moving, this component of the acceleration is referred to as the
tangential acceleration of the particle.
The magnitude of the tangential acceleration of a particle
in circular motion is simply the absolute value of the
rate of change of the speed of the particle
dt
d
a
v
=
t
.
The direction of the tangential acceleration is the same as that of the velocity if the
particle is speeding up, and in the direction opposite that of the velocity if the particle is slowing
down.
Recall that, starting with our equation relating the position
s
of the particle along the circle to the
angular position
θ
of a particle,
θ
r
=
s
, we took the derivative with respect to time to get the
relation
w
r
v
=
.
If we take a second derivative with respect to time we get
dt
d
dt
d
w
r
v
=
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Chapter 19
Rotational Motion Variables, Tangential Acceleration, Constant Angular Acceleration
118
On the left we have the tangential acceleration
t
a
of the particle.
The
dt
d
w
on the right is the
time rate of change of the angular velocity of the object.
The angular velocity is the spin rate, so
a nonzero value of
dt
d
w
means that the imaginary line segment that extends from the center of
the circle to the particle is spinning faster or slower as time goes by.
In fact,
dt
d
w
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 Fall '08
 RABE
 Physics, Derivative, Acceleration, Circular Motion, Rotation, tan θ

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