Every four years, we witness the best athletes from around the world gather in one city to compete in the Olympics. The athletes have to be in peak shape. But did you know they also rely on science to optimize their performance? For example, athletes who compete in the hammer throw rely on physics to help them optimize their throwing distance as they swing a seven-kilogram iron ball attached to a steel wire before releasing it. Each competitor hopes their throw travels the farthest distance, enabling them to take the gold medal home. What these athletes do by letting go of the hammer at the precise right moment so that it zips in a straight line is a prime example of both centripetal acceleration and centripetal force. Why? The athletes enter a circle, begin to spin, and eventually release the hammer after about four or five rotations. Every rotation means the athlete is accelerating, thereby increasing velocity and optimizing the distance the hammer travels. Impressive right? Who would have thought athletes use physics?
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Jetzt kostenlos anmeldenEvery four years, we witness the best athletes from around the world gather in one city to compete in the Olympics. The athletes have to be in peak shape. But did you know they also rely on science to optimize their performance? For example, athletes who compete in the hammer throw rely on physics to help them optimize their throwing distance as they swing a seven-kilogram iron ball attached to a steel wire before releasing it. Each competitor hopes their throw travels the farthest distance, enabling them to take the gold medal home. What these athletes do by letting go of the hammer at the precise right moment so that it zips in a straight line is a prime example of both centripetal acceleration and centripetal force. Why? The athletes enter a circle, begin to spin, and eventually release the hammer after about four or five rotations. Every rotation means the athlete is accelerating, thereby increasing velocity and optimizing the distance the hammer travels. Impressive right? Who would have thought athletes use physics?
Now that we have an idea about how centripetal acceleration and force appear in everyday life, let us take a deeper look at the concepts. This article will introduce two components of rotational and circular motion: centripetal acceleration and centripetal force. We will define centripetal acceleration and centripetal force and provide a formula for each. Then, we will discuss how these concepts relate to one another before working on some examples.
In what follows, we define centripetal acceleration and centripetal force.
Centripetal acceleration is the acceleration whose direction always points radially inward toward the center of the circular path an object travels.
Acceleration is always the result of a force, which leads to the following definition of centripetal force:
Centripetal force is the radially inward external force applied to an object to keep it within a circular path.
With these definitions in mind, we now look at the formulas to calculate the centripetal acceleration and centripetal force of an object undergoing circular motion.
To calculate centripetal acceleration, we use the centripetal acceleration formula
whereis velocity measured inandis the radius of the circular path measured in.
We can obtain the centripetal force formula by following the formula for centripetal acceleration. Acceleration and force are related by Newton's Second Law of Motion, which is
.
Based on this law, we know that the force exerted on an object is the product of mass and acceleration. For centripetal force, we know its associated acceleration is also centripetal. Therefore, we must insert its formula into the above equation to obtain the formula for centripetal force. Plugging the centripetal acceleration formula into this expression, we get the centripetal force formula:
Hereis the mass of the object measured in,is the velocity measured in , and is the radius measured in.
Note that acceleration and force are vector quantities with magnitude and direction. The above formulas only give the magnitude of the vector quantities.
We just saw above how centripetal force is mathematically related to centripetal acceleration. Knowing the latter, one only has to multiply it by the object's mass to get the former. Let us now look in more detail at the conceptual relation between centripetal acceleration and centripetal force.
Centripetal acceleration and centripetal force do not have opposite directions. Remember that the acceleration of an object is always in the same direction as the net force acting on it! Circular motion is no exception to this rule. For both centripetal acceleration and centripetal force, their direction is always directed inward toward the circle's center. You can easily remember this by understanding that the word centripetal means center seeking or a tendency to move toward the center.
Since acceleration and net force always point in the same direction, and we know that centripetal means pointing towards the center, we can now appreciate why it was not necessary to write the formulas of the previous section in vector notation. Both and state explicitly that the vector form of these quantities always points toward the center of the circular path.
Centripetal forces cause centripetal acceleration. Tension, gravity, and friction are typical examples of centripetal forces responsible for circular motion. For example, if we attach an object to a string and swing it horizontally overhead, the object will travel in a circular path due to tension. The tension on the string keeps the object along its circular path and, therefore, provides the centripetal force.
Another example is the orbit of the moon around the earth. The moon is kept in its orbit by gravity, and gravity provides the centripetal force applied to the moon.
The same applies to friction. If a car is driving on a banked curve, friction acts as an external force causing the car to remain along its path.
From the above examples, we can appreciate that every time an object undergoes circular motion, it has a centripetal acceleration because the direction of its velocity at any point along the path constantly changes. This acceleration is caused by a centripetal force constraining the object's motion to a circle.
In this section, we provide a geometric derivation of centripetal acceleration. The accurate derivation would require calculus to derive the equation for centripetal acceleration. However, it's possible to work around this using informal visual proof. Let us begin by considering a particle moving on a circle with constant speed and draw the position vector and the corresponding velocity vector at different times.
When an object is in a circular motion, its position vector and velocity vector constantly change. The position vector changes due to the velocity vector. Therefore, if the particle moves with a speed, , along a circle with radius, , we can determine the time it requires for the particle to make one complete revolution
.
For a circle, we know that the distance will be equal to , which is the circumference of a circle. Therefore, we find the time around the circumference of this first circle via:
.
Now, we also know that the velocity direction is changing due to centripetal acceleration.
As the particle moves along the green circle of radius,, the velocity vector changes due to the acceleration vector. Now, if we rearrange these velocity and acceleration vectors while keeping them perpendicular to each other, we get the following:
The changes in the velocity vector describe a second circle. This circle will have a radiusand requires the same amount of time that the particle takes to move along the circular path of radiusfrom the first image. However, for this second circle, we know that centripetal acceleration is responsible for the particle's speed. Therefore, if the circle has a radiusand a speedwe can determine the time it takes for the particle to make one complete revolution:
.
Both circles must complete a full revolution in the same amount of time because they are describing the same case of circular motion. Therefore, if we set both time equations equal to each other, we can solve for centripetal acceleration as follows:
.
Cross multiplying, one gets
which, upon dividing both sides by , yields
.
This is the equation for centripetal acceleration we were looking for.
In this section, we'll work out two calculations involving examples of centripetal acceleration and centripetal force. Before beginning a problem, always remember first to read and identify all variables provided to you before applying the necessary formulas to answer the question.
A mass is attached to a rope,in length, and swung horizontally overhead. If the mass has a velocity of , determine the centripetal acceleration and centripetal force of the mass.
From the information given, the mass is , the radius is , and the velocity is . So we have that
We can now calculate the centripetal acceleration and centripetal force for this situation using their respective formulas.
The centripetal acceleration is
We now use Newton's Second Law to calculate the centripetal force:
A skater skating on a circular track whose radius is , has a speed of Determine the centripetal acceleration and the centripetal force of the skater.
From the information given, the mass is, the radius is , and the velocity is So, we have that
We can now calculate the centripetal acceleration and centripetal force for this situation using their respective formulas.
The centripetal acceleration is
We now use Newton's Second Law to calculate the centripetal force
Centripetal acceleration is an acceleration that is specific to objects undergoing rotational or circular motion.
Centripetal force is the force applied to an object that keeps the object within a circular path.
Centripetal force is the radially inward external force applied to an object to keep it within a circular path. Centripetal acceleration is the direction pointing inward toward the center of the circular path objects subjected to a centripetal force follow.
Centripetal acceleration is caused by centripetal forces. Centripetal forces are in turn provided by some external force such as tension, gravity, and friction.
Centripetal acceleration and centripetal force do not have opposite directions. For both, their direction is always directed inward toward the center of the circle
The centripetal acceleration equation is used to calculate an object's centripetal acceleration. Multiplying it by the object's mass yields the centripetal force acting on it.
An accurate derivation of centripetal acceleration involves taking the time derivative of an object undergoing circular motion at a constant angular velocity. Without using calculus, it's possible to arrive at the same result geometrically through a comparison of the position and acceleration vectors of an object as it traces out a full revolution.
What is the definition of centripetal acceleration?
Centripetal Acceleration is the acceleration of an object undergoing rotational motion whose direction is always inward
What is the definition of centripetal force?
A centripetal force is a force applied to an object to keep it within a circular path, whose direction is always inward.
The word centripetal is defined by what phrase
Center-seeking
Centripetal acceleration is caused by a centripetal force.
True
What three external forces cause a centripetal force?
Friction
Tension
Gravity
The direction of centripetal acceleration is
inward, toward the center of the circle
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