Chapter Ten Notes:

Circular Motion

There are two types of circular motion, rotation and revolution. When an object turns about an internal axis, the motion is called rotation, or spin. When an object turns about an external axis, the motion is called revolution. Ea: The earth revolves around the sun once every 365½ days, and it rotates around it’s axis every 24 hours!

An object moving in a circle is accelerating. Accelerating objects are objects which are changing their velocity - either the speed

(i.e., magnitude of the velocity vector) or the direction. An object undergoing uniform circular motion is moving with a constant speed. Nonetheless, it is accelerating due to its change in direction. The direction of the acceleration is inwards. The animation at the right depicts this by means of a vector arrow. The final motion

characteristic for an object undergoing uniform circular motion is the net force. The net force acting upon such an object is directed towards the center of the circle. The net force is

said to be an inward or centripetal force. Without such an inward force, an object would continue in a straight line, never deviating from its direction. Yet, with the inward net force directed perpendicular to the velocity vector, the object is always changing its direction and undergoing an inward acceleration.

Types of Speed:◦ Linear Speed – A point on the outside of a

turntable moves a greater distance than a spot near the middle, in the same time. The speed of something moving along a circular path is called tangential speed because the direction of motion is always tangent to the circle.

◦ Rotational speed – (Sometimes called angular speed) is the number of rotations per unit of time. It is common to express rotational speed in revolutions per minute (RPM). Ea: phonograph records commonly rotate at 331/3 RPM

Tangential and Rotational Speed: These are related to each other. If you are on the

outside of a giant rotating platform, the faster it turns, the faster your tangential speed.

Tangential speed ~ radial distance x rotational speed

In symbol form

v ~ rω

Where v is tangential speed and ω (pronounced oh MAY guh) is rotational speed.

◦ Tangential speed depends on rotational speed and the distance you are from the

axis of rotation!

Railroad train wheels: Why does a moving freight train stay on the

tracks. Most people assume it is because of the flanges at the edge of the wheel. However, these are only for emergency situations or when they follow slots that switch the train from one set of tracks to another. They stay on the tracks because their rims are slightly tapered. See figures 10.4 and 10.5 on page 173 of your book for two of the reasons for tapered wheels. Also read pages 173 to174 for a complete discussion of this process.

FIGURE 10.6 The tapered shape of railroad train wheels (shown exaggerated here) is essential on the curves of railroad tracks.

Recall that on slides 3 & 4 when we said: “The final motion characteristic for an object undergoing uniform circular motion is the net force. The net force acting upon such an object is directed towards the center of the circle. The net force is said to be an inward or centripetal force. Without such an inward force, an object would continue in a straight line, never deviating from its direction. Yet, with the inward net force directed perpendicular to the velocity vector, the object is always changing its direction and undergoing an inward acceleration.”

Acceleration : As mentioned earlier, 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 neither decreasing nor increasing; therefore there must not be an acceleration." At the center 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 safely concluded that an object moving in a circle at constant speed is indeed accelerating. It is accelerating because the direction of the velocity vector is changing.

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 from previous chapters, 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 vi represents the initial velocity and vf represents the final velocity after some time of t. The numerator of the equation is found by subtracting one vector (vi) from a second vector (vf). 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 vi to vf. The process of subtracting vi from vf is shown in the vector diagram; this process yields the change in velocity.

Direction of the Acceleration Vector: Note in the diagram above that there is a velocity change for an object moving in a circle with a constant speed. A careful inspection of the velocity change vector in the above diagram shows that it points down and to the left.

At the midpoint along the arc connecting points A and B, the velocity change is directed towards point C - the center of the circle. The acceleration of the object is dependent upon this velocity change and is in the same direction as this velocity change. The acceleration of the object is in the same direction as the velocity change vector; the acceleration is directed towards point C as well - the center of the circle. Objects moving in circles at a constant speed accelerate towards the center of the circle.

The acceleration of an object is often measured using a device known as an accelerometer. A simple accelerometer consists of an object immersed in a fluid such as water. Consider a sealed jar which is filled with water. A cork attached to the lid by a string can serve as an accelerometer. To test the direction of acceleration for an object moving in a circle, the jar can be inverted and attached to the end of a short section of a wooden 2x4. A second accelerometer constructed in the same manner can be attached to the opposite end of the 2x4. If the 2x4 and accelerometers are clamped to a rotating platform and spun in a circle, the direction of the acceleration can be clearly seen by the direction of lean of the corks.

Calculating Centripetal Force: The centripetal force on an object depends on the object’s tangential speed, its mass, and the radius of its circular path. In equation form,

mass x speed2 .

Centripetal force = radius of curvature

Fc = mv2/r

Centripetal force, Fc , is measured in newtons (N) when m is expressed in kilograms (kg), v in meters/second (m/s), and r in meters (m).

Adding Force Vectors: Figure 10.11 is a sketch of a conical pendulum – a bob

held in a circular path by a string attached above. Only two forces act on the bob: mg, the force due to gravity, and tension T in the string. Both are vectors. Figure 10.12 shows vector T resolved into two perpendicular components, Tx (horizontal) and Ty (vertical).

Since the bob doesn’t accelerate vertically, the net force in the vertical direction must be zero. Therefore: Ty = -mg

Now, what do we know about Tx ? That’s the net force on the bob, centripetal force! It’s

magnitude is mv2/r. Note that this lies along the radius of the circle swept out.

Another example is shown below. There are two forces acting on the car, gravity mg and the normal force n. Gravity mg and ny balance out, and nx is the centripetal force.

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 everyday 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. It certainly might seem to you as though your body were experiencing a forwards force causing it to accelerate forwards. Yet you would once more have a difficult time identifying such a forwards force on your body. Indeed there is no physical object accelerating you forwards. 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. This is the second aspect of Newton's law of inertia - "an object in motion tends to stay in motion with the same speed and in the same direction... ." The unbalanced force acting upon the car causes the car to slow down while your body continues in its forward motion. You are once more left with the false feeling of being pushed in a direction which is opposite your acceleration.

These two driving scenarios are summarized by the following graphic.

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. That is, the car sweeps out one-quarter of a circle. The friction force acting upon the turned wheels of the car cause 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 - which 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 which is opposite your acceleration.

This apparent (fictitious) outward force on a rotating or revolving body is called centrifugal force. Centrifugal means “center-fleeing,” or “away from the center.”

Now suppose there is a ladybug inside the whirling can, as shown in figure 10.16. The can presses against the bug’s feet and provides the centripetal force that holds it in a circular path. The ladybug, in turn presses against the floor of the can.

Neglecting gravity, the only force exerted on the ladybug is the force on the can on its feet. From our outside stationary frame of reference, we see that there is no centrifugal force exerted on the ladybug. The centrifugal-force effect is attributed not to any real force but to inertia – the tendency of the moving object to follow a straight-line path.

Our view of nature depends upon the frame of reference from which we view it.

Recall the ladybug in the previous slide. We can see that there is no centrifugal force acting on her. However, we do see centripetal force acting on the can and the ladybug, producing circular motion.

But nature seen from the rotating frame of reference (the can), is different. To the ladybug, the centrifugal force appears in its own right, as real as the pull of gravity.

Centrifugal force is an effect of rotation. It is not part of an interaction and therefore it cannot be a true force.

For this reason, physicists refer to centrifugal force as a fictitious force, unlike gravitational, electromagnetic, and nuclear forces. Nevertheless, to observers who are in a rotating system, centrifugal force is very real, just as gravity is ever present at Earth’s surface, centrifugal force is ever present within a rotating system.

Even learned physics types would admit that circular motion leaves the moving person with the sensation of being thrown outward from the center of the circle. But before drawing hasty conclusions, ask yourself three probing questions:◦ Does the sensation of being thrown outward from the center of a circle

mean that there was definitely an outward force?◦ If there is such an outward force on my body as I make a left-hand turn in

an automobile, then what physical object is supplying the outward push or pull?

◦ And finally, could that sensation be explained in other ways which are more consistent with our growing understanding of Newton's laws?

If you can answer the first of these questions with "No" then you have a chance.

Key Terms:◦ Axis Rotational Speed◦ Rotation Centripetal force◦ Revolution Centrifugal force◦ Linear Speed◦ Tangential Speed