Resource Lesson
A Derivation of the Formulas for Centripetal Acceleration
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An object is said to be moving in uniform circular motion when it maintains a constant speed while traveling in a circle.
Remember that since acceleration is a vector quantity comprised of both magnitude and direction, objects can accelerate in any of these three ways:
1. constant direction, changing speed (linear acceleration);
2. constant speed, changing direction (centripetal acceleration);
3. change in both speed and direction (angular acceleration).
In this lesson, we will be investigating centripetal acceleration and uniform circular motion - that is, objects moving in circular paths at constant speeds.
While moving in a circular path, an object is constantly being pulled "towards the center" of the circle away from its tangential path. Envision a stopper on the end of a string being twirled over your head in a horizontal circle. If the string were to break, the stopper would "fly off at a tangent." The tension in the string is forcing the stopper to constantly be pulled back towards the center to follow a circular, instead of a linear, path.
As shown in the diagram above, in a certain amount of time, Δt, an object traveling in a circular path would move from position A at time t
_{1}
where its velocity is labeled v
_{o}
to position B at time t
_{2}
where its velocity is labeled v
_{f}
. Note that the magnitude of v
_{f}
equals that of v
_{o}
since we are only changing the direction of the velocity, not the object's speed.
Remember that acceleration equals Δv/Δt. To diagram this acceleration, we must be able to diagram the resultant change in velocity, or Δv. Thus we must recognize the orientation of the vector -v
_{o}
. Since the vector v
_{o}
points to the right, the vector -v
_{o}
would have the exact same magnitude but point in the opposite direction.
The direction of the acceleration that an object experiences during an interval of time, Δt, is illustrated in the next diagram by showing the direction of v
_{f}
- v
_{o}
. To diagram the vector resultant v
_{f}
- v
_{o}
, we will use the head-to-tail method of vector addition where
Δv = v
_{f}
- v
_{o}
= v
_{f}
+ (- v
_{o}
)
Notice that the resultant velocity, Δv, starts at the beginning of the vector v
_{f}
and terminates at the end of the vector -v
_{o}
.
This relation can also be seen in the following diagram when we merely rearranged the vector equation
Δv = v
_{f}
- v
_{o}
to read
v
_{o}
+ Δv = v
_{f}
.
Notice that v
_{f}
is now the resultant vector since v
_{f}
starts at the beginning of the vector v
_{o}
and terminates at the end of the vector Δv.
Note that in both cases, Δv points to the center of the circle reflecting that the acceleration is also directed towards the center of the circle.
We begin out derivation of the magnitude of this centripetal acceleration by comparing two similar triangles.
The
first triangle
illustrates the relation s = rθ. In the time interval Δt, the object traveled from point A to point B, traversing through an angle of magnitude θ and along an arc of length s as shown in the following diagram. Note that as Δt → 0, the length of the arc,
s
, would approach the length of the chord,
c
**
.
Now let's look at the angles formed by v
_{f}
and v
_{o}
. Since v
_{o}
and v
_{f}
are both tangential velocities, they are perpendicular to their respective radii. Since OACB is a quadrilateral, the sum of its interior angles must equal 360º.
Note the following relationships:
m∠A = m∠B = 90º
m∠ACB + θ = 180º
m∠ACB + m∠BCD = 180
(supplementary angles)
Therefore m∠BCD = θ.
Consequently, in our
second triangle
formed by Δv = v
_{f}
- v
_{o}
, v
_{f}
and - v
_{o}
will also meet each other at an angle θ.
A comparison of corresponding parts of these two similar isosceles triangles yields
velocity triangle
displacement triangle
where
|-v
_{o}
| = |v
_{f}
| = v
In a small time interval Δt, the arc length s → c. And since we also know that distance = rate * time, we can replace the length of the chord,
c
, with the expression s = vΔt which results in the next ratio
that algebraically simplifies to
Since a = Δv / Δt,
where we have added the subscript
c
to represent the term centripetal since this formula only applies to objects moving in uniform circular motion.
For an object traveling with a constant speed we may use the relationship d = rt. For a circular path,
d
equals the circumference, C = 2πr and
t
equals the time for one revolution, or the period,
T
Substituting this expression for
c
into the equation for centripetal acceleration,
yields
Uniform circular motion requires that the object MUST move at a constant speed; therefore it can only move in a horizontal circle - that is, one in which gravity is always perpendicular to the object's tangential velocity.
When moving in vertical circles, the object’s speed is always changing and the object is not considered to be moving in uniform circular motion.
All units in these formulas are standard SI units: m, m/sec, m/sec
^{2}
, and seconds.
**
Disclaimer
: Obviously, a better derivation would let Δt approach zero, that is, involve very small time intervals and very small arc lengths. The decision was made that in order to better illustrate the vectors, we have "zoomed out" to sizes that can be more easily manipulated graphically.
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