Lesson 1: Motion Characteristics for Circular Motion

Any moving object can be described using the kinematic concepts . The motion of a moving object can be explained using either Newton's Laws  and vector principles  or by means of the Work-Energy Theorem . The same concepts and principles used to describe and explain the motion of an object can be used to describe and explain the parabolic motion of a projectile. In this unit, we will see that these same concepts and principles can also be used to describe and explain the motion of objects which 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. Lesson 1 of this study will begin with the development of kinematic and dynamic ideas can be used to describe and explain the motion of objects in circles.

 

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 would be described to be experiencing uniform circular motion. Uniform circular motion is the motion of an object in a circle with a constant or uniform 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. At a uniform speed of 5 m/s, if the circle had a circumference of 5 meters, then it would take the car 1 second to make a complete cycle around the circle. 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

Combining these two equations above will lead to a new equation relating the speed of an object moving in uniform circular motion to the radius of the circle and the time to make one cycle around the circle (period).

where R represents the radius of the circle and T represents the period. This equation, like all equations, can be used as a algebraic recipe for problem solving. Yet it also can be used to guide our thinking about the variables in the equation relate to each other. For instance, the equation suggests that for objects moving around circles of different radius in the same period, the object traversing the circle of larger radius must be traveling with the greatest speed. In fact, the average speed and the radius of the circle are directly proportional. A twofold increase in radius corresponds to a twofold increase in speed; a threefold increase in radius corresponds to a three--fold increase in speed; and so on. This principle was best illustrated in an classroom demonstration using a series of LCD lights. The LCD lights were positioned along an electrical wire at varying locations from the end. The end of the wire was held and spun rapidly in a circle. Each LCD light traversed a circle of different radius. Yet since they were connected to the same wire, their period of rotation was the same. Subsequently, the LCDs which were further from the center of the circle were traveling faster in order to sweep out the circumference of the larger circle in the same amount of time. With the room lights turned off, the LCDs created an arc which could be perceived to be longer for those LCDs which were traveling faster - the LCDs with the greatest radius. This is illustrated in the diagram above.

Objects moving in uniform circular motion will have a constant speed. But does this mean that they will have a constant velocity? Recall  that speed and velocity refer to two distinctly different quantities. Speed is a scalar and velocity is a vector quantity. Velocity, being a vector, has both a magnitude and a direction. The magnitude of the velocity vector is merely the instantaneous speed of the object; the direction of the velocity vector is directed in the same direction which 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 which touches the circle at one point but does not intersect it.) The diagram at the right shows the direction of the velocity vector at four different point 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, it's direction is always tangent to the circle.

 

To summarize, an object moving in uniform circular motion is moving around the perimeter of the circle with a constant speed. While the speed of the object is constant, its velocity is changing. Velocity, being a vector, has a constant magnitude but a changing direction. The direction is always directed tangent to the circle and as the object turns the circle, the tangent line is always pointing in a new direction. As we proceed through this unit, we will see that these same principles will have a similar extension to noncircular motion.

 

As mentioned , an object moving in uniform circular motion is moving in a circle with a uniform or constant speed. The velocity vector is constant in magnitude but changing in direction. Because the speed is constant for such a motion, many students have the misconception that there is no acceleration. "After all," they might say, "if I were driving a car in a circle at a constant speed of 20 mi/hr, then the speed is not decreasing or increasing; therefore there must not be an acceleration." At the heat of this common student misconception is the wrong belief that acceleration has to do with speed and not with velocity. But the fact is that an accelerating object is an object which is changing its velocity. And since velocity is a vector which has both magnitude and direction, a change in either the magnitude or the direction constitutes a change in the velocity. For this reason, it can be boldly declared that an object moving in a circle at constant speed is indeed accelerating. It is accelerating because its velocity is changing its directions.

To understand this at a deeper level, we will have to combine the definition of acceleration with a review of some basic vector principles. Recall that acceleration as a quantity was defined as the rate at which the velocity of an object changes. As such, it is calculated using the following equation:

where v_i represents the initial velocity and v_f represents the final velocity after some time of t. The numerator of the equation is found by subtracting one vector (v_i) from a second vector (v_f). But the addition and subtraction of vectors from each other is done in a manner much different than the addition and subtraction of scalar quantities. Consider the case of an object moving in a circle about point C as shown in the diagram below. In a time of t seconds, the object has moved from point A to point B. In this time, the velocity has changed from v_i to v_f. The process of subtracting v_i from v_f is shown in the vector diagram; this process yields the change in velocity.

Note in the diagram below that there is a velocity change for an object moving in a circle with a constant speed. Furthermore, note that this velocity change vector is directed towards the center. An object moving in a circle at a constant speed from A to B experiences a velocity change and therefore an acceleration; this acceleration is directed towards point C - the center of the circle.

The acceleration of an object is often measured using a device known as an accelerometer. A simple home-made accelerometer involves a lit candle centered vertically in the middle of a open-air glass. If the glass is held level and at rest (such that there is no acceleration), then the candle flame extends in an upward direction. However, if the glass is held at the end of an outstretched arm as you spin in a circle at a constant rate (such that the flame experiences an acceleration), then the candle flame will no longer extend vertically upwards. Instead the flame deflects from its upright position. This signifies that there is an acceleration when the flame moves in a circular path at constant speed. The deflection of the flame will be in the direction of the acceleration. This is because the hot gases of the flame are less massive (on a per mL basis) and thus have less inertia than the cooler gases which surround. Subsequently, the hotter and lighter gases of the flame experience the greater acceleration and will lurch towards the direction of the acceleration. A careful examination of the flame reveals that the flame will point towards the center of the circle, thus indicating that not only is there an acceleration; but that there is an inward acceleration. Objects moving in a circle at a constant speed experience an acceleration which is directed towards the center of the circle.

A further demonstrations of this principle was performed in class using a cork accelerometer. A cork was submerged in a sealed flask of water. The flask was then held in an outstretched arm and moved in a circle at a constant rate of turning. Thus, the flask with both the water and the cork were moving in uniform circular motion. Again, the least massive of the two objects will lean in the direction of the acceleration. In the case of the cork and the water, the cork is least massive (on a per mL basis) and thus it experiences the greater acceleration. As the cork-water combination spun in the circle, the cork leaned towards the center of the circle. Once more, there is proof that an object moving in circular motion at constant speed experiences an acceleration which directed towards the center of the circle.

So thus far, we have seen a geometric proof and two real-world demonstrations of this inward acceleration. At this point it becomes the decision of the student to believe or not to believe. Is it sensible that an object moving in a circle experiences an acceleration which is directed towards the center of the circle? Can you think of a logical reason to believe in say no acceleration or even an outward acceleration experienced by an object moving in uniform circular motion? later, additional logical evidence will be presented to support the notion of an inward force for an object moving in circular motion.

 

Check Your Understanding

1. The initial and final speed of a ball at two different points in time is shown below. The direction of the ball is indicated by the arrow. For each case, indicate if there is an acceleration. Explain why or why not. Indicate the direction of the acceleration.

a.
Acceleration: Yes or No? Explain.	If there is an acceleration, then what direction is it?
Depress mouse.Since the velocitydid not change,there is no acceleration.	Depress mouse to view answer.There is no acceleration.

b.
Acceleration: Yes or No? Explain.	If there is an acceleration, then what direction is it?
Depress mouse.Since the velocity changed(the object increased itsspeed), there is anacceleration.	Depress mouse to view answer.If a rightward-moving object speeds up,then it experiences a rightward acceleration.

c.
Acceleration: Yes or No? Explain.	If there is an acceleration, then what direction is it?
Depress mouse.Since the velocity changed(the object decreased itsspeed), there is anacceleration.	Depress mouse to view answer.If a rightward-moving object slows down,then it experiences a leftward acceleration.

d.
Acceleration: Yes or No? Explain.	If there is an acceleration, then what direction is it?
Depress mouse.Since the velocity changed(the object decreased itsspeed), there is anacceleration.	Depress mouse to view answer.If a rightward-moving object slows down,then it experiences a leftward acceleration.

e.
Acceleration: Yes or No? Explain.	If there is an acceleration, then what direction is it?
Depress mouse.Yes. Even though theinitial and final speedsare the same, there hasbeen a change in directionfor the object. Thus,there is an acceleration.	Depress mouse to view answer.The object was moving rightward and sloweddown to 0 m/s before changing directionsand speeding up while traveling leftward.This constitutes a leftward acceleration.

2. Explain the connection between your answers to the above questions and the reasoning used to explain why an object moving in a circle at constant speed can be said to experience an acceleration.

 

3. Dizzy Smith and Hector Vector are still discussing #1e. Dizzy says that the ball is not accelerating because its velocity is not changing. Hector says that since the ball has changed its direction, there is an acceleration. Who do you agree with? Argue a position by explaining the discrepancy in the other student's argument.

 

4. Identify the three controls on an automobile which allow the car to be accelerated.

For questions #5-#8: An object is moving in a clockwise direction around a circle at constant speed. Use your understanding of the concepts of velocity and acceleration to answer the next four questions. Use the diagram shown at the right.

 

5. Which vector below represents the direction of the velocity vector when the object is located at point B on the circle?

6. Which vector below represents the direction of the acceleration vector when the object is located at point C on the circle?

7. Which vector below represents the direction of the velocity vector when the object is located at point C on the circle?

8. Which vector below represents the direction of the acceleration vector when the object is located at point A on the circle?