Circular Motion, Centripetal Acceleration, and Centripetal Force
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 ball on the end of a string being
twirled over your head in a horizontal circle. If the string were to break, the ball would
"fly off at a tangent." The tension in the string is forcing the ball to constantly be pulled
back towards the center to follow a circular, instead of a linear, path.
c
c
As shown in the diagram above, the velocity, v, is tangent to the circle and the centripetal
force, Fc , and centripetal acceleration, ac, are perpendicular to this velocity and point
towards the center of the circle.
Centripetal force is the net force towards the center of the circle and is given by Newton’s
second law:
Fc = mac
In the case above, the centripetal force is equal to the tension in the string.
Centripetal acceleration is given by the formula,
For an object traveling with a constant speed we may use the relationship v=Δx/t. For a
circular path, Δx equals the circumference, C = 2πr and t equals the time for one
revolution, or the period, T
Substituting this expression for v into the equation for centripetal acceleration above,
yields
All units in these formulas are standard SI units: m, m/s, m/s2, and seconds.
Determining Centripetal Force
(Net force towards the center of the circle)
Example #1: Strings and Flat Surfaces
Suppose that a mass is tied to the end of a string and is being whirled in a circle along the
top of a frictionless table as shown in the diagram below.
A freebody diagram of the forces on the mass would show
The tension is the unbalanced central force: T = Fc = mac, it is supplying the centripetal
force necessary to keep the block moving in its circular path.
Example #2: Conical Pendulums
Our next example is also an object on the end of string but this time it is a conical
pendulum. Notice, that its path also tracks out a horizontal circle in which gravity is
always perpendicular to the object's path.
A freebody diagram of the mass on the end of the pendulum would show the following
forces.
T cos θ is balanced by the object's weight, mg. It is T sin θ that is the unbalanced central
force that is supplying the centripetal force necessary to keep the block moving in its
circular path: T sin θ = Fc = mac.
Example #3: Block on a String in a Vertical Circle
Suppose a block is being whirled on the end of a string in a vertical circle. Let's look at
the freebody diagrams of the forces acting on the block at the top of the circle and at the
bottom of the circle. Remember when drawing freebody diagrams for objects moving in
circular motion that the net force on the object is ALWAYS directed towards the center
of the circle, no matter where the object is located in its circular path.
Fc
Fc
TOP
Top of the circle:
Fc = T + mg
Bottom of the circle:
Fc = T – mg
BOTTOM
If we wanted to calculate the minimum or critical velocity needed for the block to just be
able to pass through the top of the circle without the rope sagging then we would start by
letting the tension in the rope approaches zero.
Fc = T + mg
m(v2/r) = 0 + mg
m(v2/r) = mg
v2/r = g
v2 = rg
v = √(rg)
Example #4: Roller Coaster Loop
Now, consider an example of a person riding a roller coaster through a circular section of
the track, a "loop-the-loop."
TOP
BOTTOM
Let's look at the formulas needed to calculate the normal force, N, exerted on a object
traveling on the inside surface of a vertical circle as it passes through the bottom and
through the top of the ride.
At the top:
Fc = N + mg
mv2/r = N + mg
At the bottom:
Fc = N – mg
mv2/r = N – mg
If we let the value of normal approach zero in the formula for the top of the roller coaster
we would get the same value for the critical velocity that we got when solving for the
tension in the string in our previous discussion, v = √(rg). This principle of critical
velocity is used in many places. When you watch clothes drying in a dryer, they are being
rotated in a vertical circle. But the rate of rotation does not allow the clothes to achieve
this critical value as they pass through the top of the circle. Therefore the clothes fall
away from the drum and are "fluffed" as they spin. In roller coasters, this critical velocity
is a safety threshold. The coasters MUST exceed this minimum value in order to be
certified. Obviously, no one would want the cars to fall away from the rails as the
participants experience a thrill passing through a "loop-the-loop" section of the track!
The value of the normal force at the bottom of the ride is equivalent to questions asking
about the apparent weight of a pilot as he pulls out of a vertical drive. At the bottom,
N = mv2/r + mg. The ratio of ac + g to g, (v2/r + g)/g, is often called the "g's" a pilot is
experiencing.