8.7 - Interactive summary problem: race curves
In the simulation on the right, you are asked to race a truck on an S-shaped track
against the computer. This time, the first curve is covered with snow and you are
racing against a snowmobile. As you go around the track, the static friction between
the tires of your truck and the snow or pavement provides the centripetal force. If
you go too fast, you will exceed the maximum force of friction and your truck will
leave the track. If you go as fast as you can without sliding, you will beat the
snowmobile.
The snowmobile runs the entire race at its maximum speed. The blue truck
negotiates each curve at a constant speed, but these speeds must be different for
you to win the race. You set the speed of the blue truck on each curve.
Straightaway sections are located at the start of the race and between the two
curves. The simulation will automatically supply the acceleration you need on the
straightaway sections.
The blue truck has a mass of 1,800 kg. The first curve is icy, and the coefficient of
static friction of the truck on this curve is 0.51. (The snowmobile has a greater
coefficient thanks to its snow-happy treads.) On the second curve, the coefficient of
static friction is 0.84. The radius of the first curve is 13 m, and the second curve is 11 m. Set the speed of the blue truck on each curve as fast
as it can go without sliding off the track, and you will win.
You set the speed in increments of 0.1 m/s in the simulation. If you need to round a value after your calculations, make sure you round down to
the nearest 0.1 m/s. (If you round up, you will be exceeding the maximum safe speed.) Press GO to begin the race, and RESET if you need to
try again.
If you have difficulty with this problem, you may want to review the section on static friction in a previous chapter and the section on centripetal
acceleration in this chapter.8.8 - Gotchas
A car is moving around a circular track at a constant speed of 20 km/h. This means its velocity is constant, as well. Wrong. The car’s velocity
changes because its direction changes as it moves.
Since an object moving in uniform circular motion is constantly changing direction, it is hard at any point in time to know the direction of its
velocity and the direction of its acceleration. This is not true. The velocity vector is always tangent to the circle at the location of the object.
Centripetal acceleration always points toward the center of the circle.
No force is required for an object to move in uniform circular motion. After all, its speed is constant. Yes, but its velocity is changing due to its
change in direction, which means it is accelerating. By Newton’s second law, this means there must be a net force causing this acceleration.
Centripetal force is another type of force. No, rather it is a way to describe what a force is “doing.” The normal force, gravity, tension í each of
these forces can be a centripetal force if it is causing an object to move in uniform circular motion.8.9 - Summary
Uniform circular motion is movement in a circle at a constant speed. But while
speed is constant in this type of motion, velocity is not. Since instantaneous velocity
in uniform circular motion is always tangent to the circle, its direction changes as the
object's position changes.
The period is the time it takes an object in uniform circular motion to complete one
revolution of the circle.
Since the velocity of an object moving in uniform circular motion changes, it is
accelerating. The acceleration due to its change in direction is called centripetal
acceleration. For uniform circular motion, the acceleration vector has a constant
magnitude and always points toward the center of the circle.
Newton's second law can be applied to an object in uniform circular motion. The net
force causing centripetal acceleration is called a centripetal force. Like centripetal
acceleration, it is directed toward the center of the circle.
A centripetal force is not a new type of force; rather, it describes a role that is played by one or more forces in the situation, since there must be
some force that is changing the velocity of the object. For example, the force of gravity keeps the Moon in a roughly circular orbit around the
Earth, while the normal force of the road and the force of friction combine to keep a car in circular motion around a banked curve.(^172) Copyright 2000-2007 Kinetic Books Co. Chapter 08