Consider a disk rotating on a stationary rod.
If we now view this disk from the top,
we see that when the disk rotates so that the arc length (s) equals the length of the radius of the disk (r), the subtended central angle (θ) will equal 1 radian. The resulting equation is
s = rθ
or θ = s/r
where the unit of a radian represents the dimensionless measure of the ratio of the circle's arc length to its radius.
If the disk rotates through one complete revolution, then s equals the entire circumference and θ equals 2π radians.
s = rθ
2πr = rθ
θ = 2π radians
Since one complete revolution equals 360º, we now have the conversion that
360º = 2π radians
1 radian = 180/π or approximately 57.3º.
Differentiating the basic equation s = rθ results in the next two relationships between the tangential motion of a point along the circumference and the angular behavior of the disk itself.
s = rθ
ds/dt = r(dθ/dt)
v = rω
The Greek letter ω represents the angular velocity of the disk. It is measured in the unit radians/sec. Differentiating one more time, we have
v = rω
dv/dt = r(dω/dt)
a = rα
The Greek letter α represents the angular acceleration of the disk. It is measured in the unit
radians/sec2.
Using our new variables for angular motion, we can now state analogous equations to those we have already used for linear motion. When the velocity is constant our equations are
s = vt which becomes θ = ωt.
The following chart shows the relationships between the equations for uniform linear acceleration and those that deal with uniform angular acceleration. When working with pure rotational motion, the standard unit of measurement is the radian.
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linear
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angular
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a = (vf - vo)/t
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α = (ωf - ωo)/t
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vf = vo + at
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ωf = ωo + αt
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s = ½(vf + vo)t
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θ = ½(ωf + ωo)t
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s = vot + ½at2
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θ = ωot + ½αt2
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vf2 = vo2 + 2as
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ωf2 = ωo2 + 2αθ
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Angular acceleration is considered to be constant in these types of situations:
- Any horizontal circular motion with a constant applied torque.
- Any vertical circular motion driven by a motor with constant power.
- Satellites in circular orbits.
- A yo-yo falling straight down.
Angular acceleration not considered to be constant, or uniform, in these situations:
- When a vertically rotating rod is swinging on a fixed pivot.
- In a vertical circle (pendulum), the instantaneous acceleration depends on θ.
- When there is a drag force that depends on the object’s velocity.
When an object's angular acceleration is not constant, it instantaneous final angular velocity be determined by using conservation of energy techniques. Its average velocity can be found using
θnet = ωavt
Now let's work some examples. |