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 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 that 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.
As the cork-water combination spins in a circle, the cork leans towards the center of the circle. The least massive of the two objects always leans in the direction of the acceleration.
In the case of the cork and the water, the cork is less massive on a per mL basis and thus it experiences the greater acceleration. Having less inertia owing to its smaller mass on a per mL basis , the cork resists the acceleration the least and thus leans to the inside of the jar towards the center of the circle.
This is observable evidence that an object moving in circular motion at constant speed experiences an acceleration that is directed towards the center of the circle. Another simple homemade accelerometer involves a lit candle centered vertically in the middle of an 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 you hold the glass-candle system with an outstretched arm and 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 can be explained by asserting that the hot gases of the flame are less massive on a per mL basis and thus have less inertia than the cooler gases that surround it.
Subsequently, the hotter and lighter gases of the flame experience the greater acceleration and will lean in 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. This is one more piece of observable evidence that indicates that objects moving in a circle at a constant speed experience an acceleration that is 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 to not believe. This leaves us with the following work-energy equation. Remember that the work is zero and the disk starts at position 1 from rest and not rotating. Now I can add to this two ideas. First, I know the expression for the moment of inertia of disk. Second, the disk is rolling and not sliding.
Since the disk is rolling, the speed of the center of mass of the disk is equal to the angular speed times the radius of the disk. Putting this all together, I can solve for the velocity at the bottom. Ok, but what about the acceleration? I will assume that the object rolls down the incline with a constant acceleration.
In this case, it starts from rest and ends with the final speed all the time while moving a distance s down the incline. In the direction along the incline, I can find the acceleration:. Remember, the initial velocity was zero - that's why the v 1 term drops out. But what about the time interval?
Here I can use the definition of average velocity. Oh wait. Couldn't I have just used that kinematic equation? You know, the one that looks like this:. Yes, I could easily have used that equation instead. Also, I could make waffles in the morning using one of those box mixes.
Personally, I prefer to make my waffles from scratch. I should put in the value for the final velocity from the rolling part. This gives the disk an acceleration of:. The sine of this angle will be the opposite side h divided by the hypotenuse s. That means I can rewrite the equation as:.
Can we get the acceleration of the disk without using the work-energy principle? Let's start with a force diagram of the disk as it rolls down the incline. Three forces, this should be simple - right? The disk only accelerates along the x-direction along the plane so this should be a simple problem. But no. It's not that simple. The problem is the friction force. This frictional force is what prevents the disk from slipping.
Since the disk rolls without slipping, the frictional force will be a static friction force. We can model the magnitude of this force with the following equation. It depends on the two types of materials interacting. I know that seems crazy, but imagine a super-rough surface for the disk and plane.
For a case like this, it's possible the frictional force is quite large. What if the frictional force was larger than the component of the gravitational force in the direction of the plane?
This would make the disk accelerate UP the plane. That would be crazy. The static frictional force is called a constraint force. It exerts whatever force it needs such that the disk rolls instead of slides - up to some maximum value.
But what is that value? Who knows. Should that stop us? Here is the equation for the net forces in the x-direction I am calling down the incline as the positive direction :. If I could only find this frictional force, I would have an answer for the acceleration. Yet acceleration has nothing to do with going fast.
A person can be moving very fast and still not be accelerating. Acceleration has to do with changing how fast an object is moving. If an object is not changing its velocity, then the object is not accelerating.
The data at the right are representative of a northward-moving accelerating object. The velocity is changing over the course of time. Anytime an object's velocity is changing, the object is said to be accelerating; it has an acceleration. Sometimes an accelerating object will change its velocity by the same amount each second. This is referred to as a constant acceleration since the velocity is changing by a constant amount each second.
An object with a constant acceleration should not be confused with an object with a constant velocity. Don't be fooled! If an object is changing its velocity -whether by a constant amount or a varying amount - then it is an accelerating object. And an object with a constant velocity is not accelerating. The data tables below depict motions of objects with a constant acceleration and a changing acceleration.
Note that each object has a changing velocity. A falling object for instance usually accelerates as it falls. Our free-falling object would be constantly accelerating. Given these average velocity values during each consecutive 1-second time interval, we could say that the object would fall 5 meters in the first second, 15 meters in the second second for a total distance of 20 meters , 25 meters in the third second for a total distance of 45 meters , 35 meters in the fourth second for a total distance of 80 meters after four seconds.
These numbers are summarized in the table below. This discussion illustrates that a free-falling object that is accelerating at a constant rate will cover different distances in each consecutive second. Further analysis of the first and last columns of the data above reveal that there is a square relationship between the total distance traveled and the time of travel for an object starting from rest and moving with a constant acceleration.
The total distance traveled is directly proportional to the square of the time. For objects with a constant acceleration, the distance of travel is directly proportional to the square of the time of travel.
The average acceleration a of any object over a given interval of time t can be calculated using the equation.
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