8.4 - Interactive problems: racing in circles
The two simulations in this section let you experience uniform circular motion and
centripetal acceleration as you race your car against the computer’s.
In the first simulation, you race a car around a circular track. Both your car and the
computer’s move around the loop at constant speeds. You control the speed of the
blue car. Halfway around the track, you encounter an oil slick. If the centripetal
acceleration of your car is greater than 3.92 m/s^2 at this point, it will leave the track
and you will lose. The radius of the circle is 21.0 m.
To win the race, set the centripetal acceleration equal to 3.92 m/s^2 in the centripetal
acceleration equation, solve for the velocity, and then round down the velocity to
the nearest 0.1 m/s; this is a value that will keep your car on the track and beat the
computer car. Enter this value using the controls in the simulation. Press GO to start
the simulation and test your calculations.
In the second simulation, the track consists of two half-circle curves connected by a
straight section. Your blue car runs the entire race at the speed that you set for it.
You want to set this speed to just keep the car on the track. The first curve has a
radius of 14.0 meters; the second, 8.50 meters. On either curve, if the centripetal
acceleration of your car exceeds 9.95 m/s^2 , its tires will lose traction on the curve,
causing it to leave the track. If your car moves at the fastest speed possible without
leaving the track, it will win. Again, calculate the speed of the blue car on each curve
but using a centripetal acceleration value of 9.95 m/s^2 , and round down to the
nearest 0.1 m/s. Since the car will go the same speed on both curves, you need to
decide which curve determines your maximum speed. Enter this value, then press
GO.
If you have difficulty solving these interactive problems, review the equation relating
centripetal acceleration, circle radius, and speed.
8.5 - Newton's second law and centripetal forces
If you hold onto the string of a yo-yo and twirl it in a circle overhead, as illustrated in
Concept 1, you know you must hold the string firmly or the yo-yo will fly away from you.
This is true even when the toy moves at a constant speed. A force must be applied to
keep the yo-yo moving in a circle.
A force is required because the yo-yo is accelerating. Its change in direction means its
velocity is changing. Using Newton’s second law, F = ma, we can calculate the amount
of force as the product of the object’s mass and its centripetal acceleration. That
equation is shown in Equation 1. It applies to any object moving in uniform circular
motion. The force, called a centripetal force, points in the same direction as the
acceleration, toward the center of the circle.
The term “centripetal” describes any force that causes circular motion. A centripetal
force is not a new type of force. It can be the force of tension exerted by a string, as in
the yo-yo example, or it can be the force of friction, such as when a car goes around a
curve on a level road. It can also be a normal force; for example, the walls of a clothes
dryer supply a normal force that keeps the clothes moving in a circle, while the holes in
those walls allow water to “spin out” of the fabric. Or, as in the case of the motorcycle
rider in Example 1, the centripetal force can be a combination of forces, such as the
normal force from the wall and the force of friction.
Sometimes the source of a centripetal force is easily seen, as with a string or the walls of a dryer. Sometimes that force is invisible: The force
of gravity cannot be directly seen, but it keeps the Earth in its orbit around the Sun. The centripetal force can also be quite subtle, such as
when an airplane tilts or banks; the air passing over the plane’s angled wings creates a force inward. In each of these examples, a force
causes the object to accelerate toward the center of its circular path.
Identifying the force or forces that create the centripetal acceleration is a key step in solving many problems involving circular motion.
Forces and centripetal
acceleration
Force causes circular motion
Directed toward center
Any force can be centripetal