Rotation Lecture B Centripetal Acceleration In this lesson we examine acceleration and force in circular motion. In doing so, we will learn that circular motion is unique in how the velocity and acceleration vectors are related to each other. The direction of the acceleration in circular motion may not be immediately obvious. The application of this work can range from the motion of jet planes and race cars to planets and moons moving in circular orbits to a centrifuge in a medical laboratory. We saw in the previous lesson that the velocity of an object moving in a circle is always tangent to the circle (or perpendicular to the radius). The diagram below shows an object at position X at a time t i moving in a clockwise fashion. The velocity vector at this point is drawn and labeled v i . At a later time, t f , the object has moved to a new position Y. The velocity vector at this new position is also drawn and labeled v f .
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Rotation Lecture B
Centripetal Acceleration
In this lesson we examine acceleration and force in circular motion. In doing so, we will learn that circular motion is unique in how the velocity and acceleration vectors are related to each other. The direction of the acceleration in circular motion may not be immediately obvious. The application of this work can range from the motion of jet planes and race cars to planets and moons moving in circular orbits to a centrifuge in a medical laboratory.
We saw in the previous lesson that the velocity of an object moving in a circle is always tangent to the circle (or perpendicular to the radius). The diagram below shows an object at position X at a time ti moving in a clockwise fashion. The velocity vector at this point is drawn and labeled vi. At a later time, tf, the object has moved to a new position Y. The velocity vector at this new position is also drawn and labeled vf.
Recall from our study of kinematics that the definition of acceleration is the change in velocity divided by the difference in time.
a = vf - vi or a = v/t
tf - ti
The diagram below connects the two velocity vectors to show the difference between them.
The diagram shows that the difference in velocity vectors, v, is pointing somewhere towards the interior of the circle. If the time interval between vf and vi is very small, then it would become more obvious that the direction of the difference in velocity, and therefore the acceleration, is perpendicular to the velocity vector and hence along the radius towards the center of the circle. In fact the object accelerates towards the center of the circle at every moment. This acceleration is called centripetal acceleration because the word centripetal means “center-seeking.”
It can be shown mathematically (although not discussed here) that the magnitude ac of the centripetal acceleration is given by
The centripetal acceleration vector always points toward the center of the circle and continually changes direction as the object moves.
Lastly, by combining with the centripetal acceleration can also be written as
.
Example
A 25 kg child moves with a speed of 1.93 m/s when sitting 12.5 m from the center of a merry-go-round. Calculate the centripetal acceleration.
Solution
a = (1.93 m/s)2/(12.5 m) = 0.297 m/s2 or 2.97 x 10-1 m/s2
Example
The earth orbits the sun at a distance of 1.5 x 1011 m. What is the centripetal acceleration of the earth in its orbit?
Solution
We first must calculate the earth’s period in seconds.
T = 365 days x 24 hours x 60 minutes x 60 seconds
T = 3.15 x 107 s
a = 42(1.5 x 1011 m)/( 3.15 x 107 s)2
a = 5.92 x 1012 m/9.92 x 1014 s2
a = 0.00596 m/s2 or 5.96 x 10-3 m/s2
Centripetal Force
From Newton’s second law, F = ma, we know that whenever an object accelerates, there must be a net force causing the acceleration. It follows that in uniform circular motion, there must be a net force producing the centripetal acceleration. According to the second law, the net force Fc is equal to the product of the object’s mass and acceleration. This net force is called the centripetal force and points in the same direction as the centripetal acceleration, that is, toward the center of the circle.
In the previous lesson we learned that , and if we combine this with F = ma, the centripetal force can be written as
We can describe the centripetal force in a formal way as follows:
The centripetal force is the name given to the net force required to keep an object of mass m, moving at a speed v, on a circular path of radius r that has a magnitude of
The centripetal force always points toward the center of the circle and continually changes direction as the object moves.
The term ``centripetal`` force is not a new and separate force. It is simply a label given to the net force pointing toward the center of the circular path, and this net force is the vector sum of all the force components that point along the radial direction.
Sources of Centripetal Force
(none of the following formulas will be tested in this unit, but you should know the names of all of the different sources)
TensionIn some cases, it is easy to identify the source of
centripetal force. When a model airplane on a guideline flies in a horizontal circle, the only force pulling the plane inward is the tension in the line, so this force is the centripetal force.
As a formula, one could use
FrictionWhen a car moves at a steady speed around an
unbanked curve, the centripetal force keeping the car on the curve comes from the static friction between the road and the tires. It is static rather than kinetic friction because the tires are not slipping on the road surface. If the static frictional force is not sufficient, given the radius of the turn and the speed, the car will skid off the road. For the car to make the turn, the centripetal force is equal to the force of friction.
As a formula, one could use
GravityToday there are many satellites in orbit about the
earth. The ones in circular orbits are examples of uniform circular motion. Like a model airplane on a guideline, each satellite is kept on its circular path by a centripetal force. The gravitational pull of the earth provides the centripetal force and acts like a guideline for the satellite.
As a formula, one could use
Centrifugal Force
There is a common misconception that an object moving in a circle has an outward force acting on it, a so-called centrifugal (center-fleeing) force. Consider for example a person swinging a ball on the end of a string. If you have ever done this yourself, you know that you feel a force pulling outward on your hand. This misconception arises when this pull is interpreted as an outward "centrifugal" force pulling on the ball that is transmitted along the string to the hand. But this is not what is happening at all. To keep the ball moving in a circle, the person pulls inwardly on the ball. The ball then exerts an equal and opposite force on the hand (Newton`s third law) due to the fact that a) the ball has inertia, and b) this inertia is in motion. The ball in and of itself is not pulling on your hand at all, but due to its inertia and motion, you perceive that it is. Hence, the term centrifugal force is really a misnomer.
For even more convincing evidence that a centrifugal force does not act on the ball, consider what happens when you let go of the string. If a centrifugal force were acting, the ball would fly straight out along the radius of motion. Of course this does not happen. The ball flies off tangentially in the direction of the velocity it had at the moment it was released because the inward force no longer acts.
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