Circular Motion and Satellite Motion - Motion Characteristics for Circular Motion

We will see the same concepts and principles used to describe and explain the motion of objects that either move in circles or can be approximated to be moving in circles. Kinematic concepts and motion principles will be applied to the motion of objects in circles and then extended to analyze the motion of such objects as roller coaster cars, a football player making a circular turn, and a planet orbiting the sun. We will see that the beauty and power of physics lies in the fact that a few simple concepts and principles can be used to explain the mechanics of the entire universe.

Suppose that you were driving a car with the steering wheel turned in such a manner that your car followed the path of a perfect circle with a constant radius. And suppose that as you drove, your speedometer maintained a constant reading of 10 mi/hr. In such a situation as this, the motion of your car could be described as experiencing uniform circular motion. Uniform circular motion is the motion of an object in a circle with a constant or uniform speed.

Calculation of the Average Speed

Uniform circular motion - circular motion at a constant speed - is one of many forms of circular motion. An object moving in uniform circular motion would cover the same linear distance in each second of time. When moving in a circle, an object traverses a distance around the perimeter of the circle. So if your car were to move in a circle with a constant speed of 5 m/s, then the car would travel 5 meters along the perimeter of the circle in each second of time. The distance of one complete cycle around the perimeter of a circle is known as the circumference. With a uniform speed of 5 m/s, a car could make a complete cycle around a circle that had a circumference of 5 meters. At this uniform speed of 5 m/s, each cycle around the 5-m circumference circle would require 1 second. At 5 m/s, a circle with a circumference of 20 meters could be made in 4 seconds; and at this uniform speed, every cycle around the 20-m circumference of the circle would take the same time period of 4 seconds. This relationship between the circumference of a circle, the time to complete one cycle around the circle, and the speed of the object is merely an extension of the average speed equation.

The circumference of any circle can be computed using from the radius according to the equation.

In fact, the average speed and the radius of the circle are directly proportional. A 2-fold increase in radius corresponds to a 2-fold increase in speed; a 3-fold increase in radius corresponds to a 3-fold increase in speed; and so on.

To illustrate, consider a strand of four LED lights positioned at various locations along the strand. The strand is held at one end and spun rapidly in a circle. Subsequently, the LEDs that are further from the center of the circle are traveling faster in order to sweep out the circumference of the larger circle in the same amount of time. This is illustrated in the diagram at the right.

The Direction of the Velocity Vector

Objects moving in uniform circular motion will have a constant speed. But does this mean that they will have a constant velocity?

Speed is a scalar quantity and velocity is a vector quantity. Velocity, being a vector, has both a magnitude and a direction. The magnitude of the velocity vector is the instantaneous speed of the object. The direction of the velocity vector is directed in the same direction that the object moves.

Since an object is moving in a circle, its direction is continuously changing. At one moment, the object is moving northward such that the velocity vector is directed northward. One quarter of a cycle later, the object would be moving eastward such that the velocity vector is directed eastward. As the object rounds the circle, the direction of the velocity vector is different than it was the instant before.

So while the magnitude of the velocity vector may be constant, the direction of the velocity vector is changing. The best word that can be used to describe the direction of the velocity vector is the word tangential. The direction of the velocity vector at any instant is in the direction of a tangent line drawn to the circle at the object's location. (A tangent line is a line that touches a circle at one point but does not intersect it.) The diagram at the right shows the direction of the velocity vector at four different points for an object moving in a clockwise direction around a circle. While the actual direction of the object (and thus, of the velocity vector) is changing, its direction is always tangent to the circle.

As mentioned earlier, an object moving in a circle is experiencing an acceleration. Even if moving around the perimeter of the circle with a constant speed, there is still a change in velocity and subsequently an acceleration. This acceleration is directed towards the center of the circle.

And in accord with Newton's second law of motion, an object which experiences an acceleration must also be experiencing a net force. The direction of the net force is in the same direction as the acceleration. So for an object moving in a circle, there must be an inward force acting upon it in order to cause its inward acceleration. This is sometimes referred to as centripetal force. The word centripetal (not to be confused with the F-word centrifugal) means center seeking. For object's moving in circular motion, there is a net force acting towards the center which causes the object to seek the center.

To understand the importance of a centripetal force, it is important to have a sturdy understanding of the Newton's first law of motion - the law of inertia. The law of inertia states

that ...

... objects in motion tend to stay in motion with the same speed and the same direction unless acted upon by an unbalanced force.

According to Newton's first law of motion, it is the natural tendency of all moving objects to continue in motion in the same direction that they are moving ... unless some form of unbalanced force acts upon the object to deviate its motion from its straight-line path. Moving objects will tend to naturally travel in straight lines; an unbalanced force is only required to cause it to turn. Thus, the presence of an unbalanced force is required for objects to move in circles.

Inertia, Force and Acceleration for an Automobile Passenger

The idea expressed by Newton's law of inertia should not be surprising to us. We experience this phenomenon of inertia nearly every day when we drive our automobile. For example, imagine that you are a passenger in a car at a traffic light. The light turns green and the driver accelerates from rest. The car begins to accelerate forward, yet relative to the seat which you are on your body begins to lean backwards. Your body being at rest tends to stay at rest. This is one aspect of the law of inertia - "objects at rest tend to stay at rest." As the wheels of the car spin to generate a forward force upon the car and cause a forward acceleration, your body tends to stay in place. It certainly might seem to you as though your body were experiencing a backwards force causing it to accelerate backwards. Yet you would have a difficult time identifying such a backwards force on your body. Indeed there isn't one. The feeling of being thrown backwards is merely the tendency of your body to resist the acceleration and to remain in its state of rest. The car is accelerating out from under your body, leaving you with the false feeling of being pushed backwards.

Now imagine that you are in the same car moving along at a constant speed approaching a stoplight. The driver applies the brakes, the wheels of the car lock, and the car begins to skid to a stop. There is a backwards force upon the forward moving car and subsequently a backwards acceleration on the car. However, your body, being in motion, tends to continue in motion while the car is skidding to a stop. The feeling of being thrown forwards is merely the tendency of your body to resist the deceleration and to remain in its state of forward motion.

Suppose that on the next part of your travels the driver of the car makes a sharp turn to the left at constant speed. During the turn, the car travels in a circular-type path.

The friction force acting upon the turned wheels of the car causes an unbalanced force upon the car and a subsequent acceleration. The unbalanced force and the acceleration are both directed towards the center of the circle about which the car is turning.

Your body however is in motion and tends to stay in motion. It is the inertia of your body - the tendency to resist acceleration - that causes it to continue in its forward motion.

While the car is accelerating inward, you continue in a straight line. If you are sitting on the passenger side of the car, then eventually the outside door of the car will hit you as the car turns inward.

This phenomenon might cause you to think that you are being accelerated outwards away from the center of the circle. In reality, you are continuing in your straight-line inertial path tangent to the circle while the car is accelerating out from under you.

The sensation of an outward force and an outward acceleration is a false sensation. There is no physical object capable of pushing you outwards. You are merely experiencing the tendency of your body to continue in its path tangent to the circular path along which the car is turning. You are once more left with the false feeling of being pushed in a direction that is opposite your acceleration.

The Centripetal Force and Direction Change

Any object moving in a circle (or along a circular path) experiences a centripetal force. That is, there is some physical force pushing or pulling the object towards the center of the circle. This is the centripetal force requirement. The word centripetal is merely an adjective used to describe the direction of the force. We are not introducing a new type of force but rather describing the direction of the net force acting upon the object that moves in the circle.

Whatever the object, if it moves in a circle, there is some force acting upon it to cause it to deviate from its straight-line path, accelerate inwards and move along a circular path. Three such examples of centripetal force are shown below.

As a car makes a turn, the force of friction acting upon the turned wheels of the car provides centripetal force required for circular motion.

As a bucket of water is tied to a string and spun in a circle, the tension force acting upon the bucket provides the centripetal force required for circular motion.

As the moon orbits the Earth, the force of gravity acting upon the moon provides the centripetal force required for circular motion.

The centripetal force for uniform circular motion alters the direction of the object without altering its speed. The idea that an unbalanced force can change the direction of the velocity vector but not its magnitude may seem a bit strange. How could that be? There are a number of ways to approach this question. One approach involves to analyze the motion from a work-energy standpoint. We will discuss work-energy in more detail later.

As the centripetal force acts upon an object moving in a circle at constant speed, the force always acts inward as the velocity of the object is directed tangent to the circle. This would mean that the force is always directed perpendicular to the direction that the object is being displaced.