Circular Motion

What happens to make an object move along a circular path?

We know when an object has no net force acting on it, the object travels in a straight line at constant speed.  What must happen for an object to travel along a circular path at constant speed?  Consider the situations below for a rock moving through deep space with no friction and no gravitational forces acting on it.

Copyright  © 2009-2012, by Marcus Milling

edited by Julia Pian

Situation 1,    After moving with a  constant velocity (solid red vector), the rock is pushed from behind by a constant force (blue vector).  This force causes the rock to change its velocity by speeding up, but the rock still travels along the same straight line (dashed red vector).

Example 6.1.1

A particularly cool student decides to take a break from studying and listen to John Coltrane’s Sun Ship on the original vinyl record.  He fires up his vacuum tube amplifier and turntable.

Before he puts the record on, he decides to put a coin on the turntable to see if it will move in a circle, or slide off.  He decides to calculate the centripetal force needed to keep the coin moving in a circle.  The platter spins around 33.3 times each minute and he places the penny 12 cm from the center of the platter.  With the balance he happens to have at home, he measures the mass of the coin to be 51 grams.  What is the Fc needed to keep the coin moving in the circular path?

First, we must calculate the velocity of the coin.

The turntable makes 33.3 revolutions in one minute (33.3 RPM).  This means the coin will go around the red circle 33.3 time in 60 seconds.  The distance once around the circle is the circumference.

Now we can calculate the Fc needed with the centripetal force equation.

This is the force needed to keep the coin moving along this circle at this velocity.  To have the coin actually move in this circle, something must provide the force!  What force could be present?  The frictional force!  Without enough friction between the platter and the coin, the coin will slide off.  For the coin to move in this circle, the frictional force must be 0.074 N and directed toward the center of the platter.

The coin goes around this distance 33.3 times in 60 seconds.

Situation 2,    After moving with a constant velocity (solid red vector), the rock is pushed from the front by a constant force (blue vector).  This force causes the rock to change its velocity by slowing down, but the rock still travels along the same straight line (dashed red vector).

Situation 3,    After moving with a constant velocity (solid red vector), the rock is pushed from the side by a constant force (blue vector).  This force causes the rock to change its direction and speed up.  Notice as the rock changes its direction of motion, its path becomes more parallel to the force vector, which means the direction of motion will change less in the future, but the force will be more effective at speeding the rock up.  This force causes the rock to follow a parabolic path.

Situation 4,    After moving with an constant velocity (solid red vector), the rock is pushed from the side by a constant force (blue vector).  Since the force is both from the side but also opposes the initial velocity, the force causes the rock to change its direction and slow down.  Notice as the rock changes its direction of motion, its path becomes more parallel to the force vector, which means the direction of motion will change less in the future, but the force will actually start speeding the rock up.  This force also causes the rock to follow a parabolic path.

Path the rock would have taken if

no force had acted on it

Situation 5,    After moving with an constant velocity (solid red vector), the rock is pushed from the side by a constant force (blue vector).  Since the force is both from the side but also partially in the direction of the initial velocity, the force causes the rock to change its direction and speed up.  Notice as the rock changes its direction of motion, its path becomes more parallel to the force vector, which means the direction of motion will change less in the future, but the force will be even more effective at speeding up the rock.  This force causes the rock to follow a parabolic path.

Situation 6,  Circular Motion!    After moving with an constant velocity (solid red vector), the rock is pushed from the side by a force (blue vector).  This force is constant in magnitude, but not in direction!  The force changes so that is is always pushing the rock in the direction that is perpendicular to the velocity vector.  With this ‘always perpendicular’ force situation, the force never affects the speed of the rock and the path is bent along a circular path.  (If the force suddenly ceased at a point along the circle, what path would the rock take with no force applied?  Click here for the answer.)

To establish circular motion, a force that is constant in magnitude must be applied to an object, with the object at first moving along a straight line.  The force must always be changing direction, so the force vector is always perpendicular to the velocity vector.  In other words, the force must always be applied towards the center of the circle.  This is called the centripetal force.

Also note that when a net force is applied to an object, it must accelerate. Three results of this acceleration are possible: (1) the object speeds up, (2) the object slows down, or (3) the object changes direction (or there could be some some combination of the above changes).

How much force must be applied to establish circular motion?  If you had a situation like (6) above, but the applied force was very small, what path would the rock take? The rock would take a path less curved than the circle shown.  In this case, the path would be more like a parabola.  What path would the rock take if the situation was like (6) above, but the force was very large?  The rock’s path would be much more curved than the circle in (6).

Why is this the case?

Consider a rock attached to a string and swung over your head, initially, in a circle.  The diagrams below show this from the perspective of an observer looking down from above the person swinging the rock.

If you swing the rock faster, how much force will you need to keep it going in a circle (more or less than before it sped up)?  Since the rock’s tendency is to ‘fly’ off along the tangent (green vector), the centripetal force is needed to keep it moving along the circle.  If the rock is swinging faster, you have less time to curve it along the circle.  To bend the path in a shorter time will require more centripetal force. In other words, you will need more force to keep a faster swinging rock moving in a circle.

What if you swing a rock at the same speed, but instead try to replace the original rock with a more massive rock?  With more mass, the rock has more inertia, and again, it will require more centripetal force to keep it moving along the circle.

What happens if we try to swing the original rock along a smaller circular path (smaller radius)?  The smaller path has a greater curvature (or tighter curve).  For the rock to move in the tighter circle, you will have to curve its path more, which again will require more centripetal force.

To summarize,  (1) the speed of the rock is directly proportional to the Fc needed,  (2) the mass of the rock is directly proportional to the Fc needed, and  (3)  the radius of the circular path is inversely proportional to the Fc needed.

These three conclusions would suggest the formula

F

vo

vo

F

F

F

vo

vo

F

F

F

F

vo

vo

F

F

F

F

v

v

v

v

Fc

Fc

Fc

Fc

This is the correct centripetal force equation!

Fc

v

The velocity is the total distance divided by the total time

The last point in example 6.1.1 is VERY IMPORTANT.  The equation,

only predicts the amount of force needed to get an object to move in a certain circle at a certain velocity.  Something must provide that force for the object to actually move in a circle.  Friction, a string pulling, a small rocket engine, gravity, or something entirely different must provide the force for the object to move in the desired circle!

But it turns out the Fc needed to make a rock move on a circular path actually depends on the square of the velocity (which you will confirm in your Circular Motion Lab).  So the actual equations is

We can use this equation to predict how much centripetal force is NEEDED to keep an object of a certain mass moving in a circle of certain radius, at a certain velocity.