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Only an inward force is required for uniform circular motion.
The figures on the previous page showed the string pulling in straight
along a radius of the circle, but many people believe that when they are
doing this they must be “leading” the rock a little to keep it moving along.
That is, they believe that the force required to produce uniform circular
motion is not directly inward but at a slight angle to the radius of the circle.
This intuition is incorrect, which you can easily verify for yourself now if
you have some string handy. It is only while you are getting the object going
that your force needs to be at an angle to the radius. During this initial
period of speeding up, the motion is not uniform. Once you settle down
into uniform circular motion, you only apply an inward force.
If you have not done the experiment for yourself, here is a theoretical
argument to convince you of this fact. We have discussed in chapter 6 the
principle that forces have no perpendicular effects. To keep the rock from
speeding up or slowing down, we only need to make sure that our force is
perpendicular to its direction of motion. We are then guaranteed that its
forward motion will remain unaffected: our force can have no perpendicular
effect, and there is no other force acting on the rock which could slow it
down. The rock requires no forward force to maintain its forward motion,
any more than a projectile needs a horizontal force to “help it over the top”
of its arc.
Why, then, does a car driving in circles in a parking lot stop executing
uniform circular motion if you take your foot off the gas. The source of
confusion here is that Newton’s laws predict an object’s motion based on the
total force acting on it. A car driving in circles has three forces on it
(1) an inward force from the asphalt, controlled with the steering wheel;
(2) a forward force from the asphalt, controlled with the gas pedal; and
(3) backward forces from air resistance and rolling resistance.
You need to make sure there is a forward force on the car so that the
backward forces will be exactly canceled out, creating a vector sum that
points directly inward.
In uniform circular motion, the acceleration vector is inward
Since experiments show that the force vector points directly inward,
Newton’s second law implies that the acceleration vector points inward as
well. This fact can also be proven on purely kinematical grounds, and we
will do so in the next section.
To make the brick go in a circle, I had
to exert an inward force on the rope.
A series of three hammer taps makes
the rolling ball trace a triangle, seven
hammers a heptagon. If the number
of hammers was large enough, the ball
would essentially be experiencing a
steady inward force, and it would go
in a circle. In no case is any forward
force necessary.
When a car is going straight at con-
stant speed, the forward and backward
forces on it are canceling out, produc-
ing a total force of zero. When it moves
in a circle at constant speed, there are
three forces on it, but the forward and
backward forces cancel out, so the
vector sum is an inward force.
Chapter 9Circular Motion