Centripetal and Centrifugal Force

Have you ever heard the terms "centripetal force" and "centrifugal force" used interchangeably? This wording can be confusing when trying to understand the forces that are acting on an object experiencing circular motion. In this article, we will discuss the difference between centripetal force and centrifugal force. We will also go over the equations that describe circular motion and go through some examples of objects experiencing circular motion.

Get started

Review generated flashcards

Sign up for free
You have reached the daily AI limit

Start learning or create your own AI flashcards

Contents
Contents

Jump to a key chapter

    Explanation of Centripetal Force and Centrifugal Force

    In order to understand centripetal force and centrifugal force, we need to understand the dynamics of an object moving in a circular motion. Let's consider a ball on a string moving in a circle, as shown below. If the ball is moving with a constant velocity around the circle, it is experiencing uniform circular motion.

    Uniform circular motion is the motion of an object moving with constant velocity in a circle with a fixed radius.

    Centripetal and Centrifugal Force  A ball on a string example of centripetal force StudySmarter

    A ball on a string in a uniform circular motion, StudySmarter Originals

    As shown in the image above, the velocity of the ball is always in a direction tangent to the circle and thus the direction of the velocity vector is constantly changing. This causes the ball to experience acceleration perpendicular to the velocity vector, and thus the acceleration vector is always pointing to the center of the circle, as shown in the image. This radial acceleration is known as the centripetal acceleration.

    Centripetal acceleration is the radial acceleration of an object in circular motion.

    If the ball is experiencing nonuniform circular motion in which the velocity of the ball is not constant, there will be acceleration vector components that do not point to the center of the circle. These acceleration components do not contribute to the centripetal acceleration.

    We know from Newton's second law that there must be a force acting upon an object if it has an acceleration. This law is described by the equationFnet=ma, whereFnetis the sum of the forces acting on the object,mis the mass of the object, andais the acceleration. In our case, we are considering the centripetal acceleration of a ball in uniform circular motion, so we will label our centripetal acceleration asacso thatFnet=mac.

    So, what is this force? From an observer's reference frame, if we ignore gravity there is only one force acting on the ball on the string: the tension force from the string holding it. The tension provides the radial force necessary to keep it in a circular motion. The net radial force acting on an object that keeps it moving in a circle is the centripetal force. The centripetal force vector points in the same direction as the acceleration vector according to Newton's second law.

    A centripetal force is the total radial force acting on an object in a circular motion.

    It is good to note that the centripetal force isn't an actual force, but rather we use the term centripetal force to describe the total force keeping the object in a circular motion. In the example above, the centripetal force came from the tension force from the string. Gravity is another good example of a force that keeps an object such as a satellite in orbit around the earth.

    What if we consider the ball's reference frame though? From the ball's reference frame, the ball is at rest while everything around it is in motion. If we only consider the centripetal force from the tension in the string in the ball's reference frame, the ball should be moving to its left towards the center of the circle. To fix this, we use a "fictitious force" which is the centrifugal force. The centrifugal force is directed radially outwards and has the same magnitude as the centripetal force.

    A centrifugal force is an apparent force directed radially outwards felt by an object in a rotating reference frame.

    Difference Between Centripetal and Centrifugal Force

    Centripetal force is the component of force acting on a rotating body which is directed towards the axis of rotation. Centrifugal force is a pseudo-force within the reference frame of the rotating body which acts outwards along the radius of rotation, pushing the body away from the axis of rotation.

    The difference between the centripetal force and the centrifugal force is that while the centripetal force applies to any reference frame, the centrifugal force is only applicable in a rotating reference frame. The observer's reference frame is called an inertial reference frame, which means that the law of inertia, Newton's first law, holds true. A rotating reference frame is a non-inertial reference frame because the law of inertia does not hold true.

    If we consider a ball placed on a rotating, frictionless platform, the ball will be pushed off of the platform. This makes sense for an observer watching the ball because the ball had initial velocity from the rotating platform. But from the ball's rotating reference frame, the ball begins to move radially outward from the platform without any forces acting on it. Thus in a non-inertial reference frame, Newton's laws of motion do not hold true. We can, however, use Newton's laws of motion in a non-inertial reference frame if we introduce "fictitious forces" such as centrifugal force. Fictitious forces make calculations in non-inertial reference frames easier since they allow us to use Newton's laws of motion.

    Remember that from the observer's reference frame, there is no centrifugal force. For objects in a circular motion in an inertial reference frame, there is no outward radial force pushing the object. Do not include centrifugal force in your calculations when using an inertial reference frame.

    Developing Formulas for Centripetal Force and Centrifugal Force

    To find the formulas for centripetal force and centrifugal force, let's take a closer look at the equations that describe the centripetal acceleration. Consider the ball on a string at two different points,P1andP2, on a circle of radiusR. At these points, the ball has corresponding velocities that we will callv1andv2. We will also call the distance that the ball traveledxand the angleθ. The time it took for the ball to travel fromP1toP2ist.

    Centripetal and Centrifugal Force A ball on a string example of centripetal force different positions StudySmarter

    A ball at two points on the circle, StudySmarter Originals

    From the image shown above, we can draw two similar triangles for the change in velocityvand the change in distancex. Since the triangles are similar triangles, the ratio of similar sides of the triangles is equal and gives us:

    xR=vv1

    vv1=xRv=xRv1

    Now let's think about the average acceleration. Acceleration is defined as the change in velocity divided by the change in time. Thus, we can write:

    a =vt =xRv1t =v1Rxt

    If we consider very small changes in distance and time, we can take the limit as the change in time approaches zero. As the change in time approaches zero, the change in distance over the change in time approachesv1.

    limt0xtv1

    Now we can write the acceleration as:

    a=v1Rlimt0xt =v1Rv1

    Since we have taken the limit as the change in time approaches zero, we can drop the subscripts. We now arrive at the equation for the centripetal acceleration:

    ac=v2R

    It is important to remember that the equation above for centripetal acceleration is considering the acceleration of the center of mass of the system. The variables used above for the position, velocity and acceleration are all quantities based on the center of mass of the object.

    The acceleration can also be described by the period of the motion. Let's call the time it takes the ball to complete one revolution,T. The average velocity is given by the circumference of the circle divided by the period: v=2πRT. Using our equation for centripetal acceleration, we can then write the centripetal acceleration as: ac=v2R=4π2RT2.

    We find the equation for the magnitude of the centripetal force,Fc

    , from Newton's second law as mentioned above:

    Fc=mac =mv2R

    The centripetal force points in the same direction as the centripetal acceleration, to the center of the circle.

    The centrifugal force,Fcf, used only in a non-inertial reference frame, has the same magnitude as the centripetal force:

    Fcf=mv2R

    Though the centripetal force and the centrifugal force are equal in magnitude, they point in opposite directions. The centrifugal force is directed radially outwards.

    Example of Centripetal Force and Centrifugal Force

    Let's consider a ball on a string again, but this time we'll make it a pendulum, as shown below.

    Centripetal and Centrifugal Force Pendulum example of centripetal force StudySmarter

    Forces acting on a pendulum, StudySmarter Originals

    We will consider the pendulum in an inertial reference frame for now. The only forces acting on the ball are gravity and the tension force from the string. As discussed above, the centripetal acceleration points toward the center of the circle drawn in the image, and thus the centripetal force must also point in that direction. To calculate the centripetal force, we would need to find the x component of the tension force. Let's draw some triangles to help us!

    Centripetal and Centrifugal Force A ball on a string example of centripetal force triangle components StudySmarter

    Triangle components of pendulum problem, StudySmarter Originals

    Our first triangle shows that the ball is at an angleθwith respect to the normal. The hypotenuse of the triangle is given by the length of the string; we'll call itLfor now. The radius of the circle,R, also defines one side of the triangle. We can make our second triangle from the forces working on the ball. We have the same angleθwith respect to the normal, the tension forceTis the hypotenuse and the force of gravityFgis the adjacent side. Our last triangle divides the tension force into its x and y components. Using trigonometry, we obtain these equations:

    sinθ=RLcosθ=FgT sinθ=TxT

    Since we're looking for the centripetal force, we need to solve for the x component of the tension force as that is the component that is pointing to the center of the circle. Solving forTxin terms of known variables gives us:

    θ=sin-1RL

    T=Fgcosθ =mgcosθ

    Tx=Tsinθ =mgcosθsinθ =mgtanθ

    A ball on a pendulum is attached to a10 cmstring moving in a circle with a radius of5 cm. The mass of the ball is200 g. First, find the centripetal force acting on the ball in an inertial reference frame. Then, consider the rotating reference frame of the ball, and find the centrifugal force.

    We can use the equation we found for the x-component of the tension force to find the centripetal force:

    Tx=(0.2 kg)9.8 ms2tansin-10.05 m0.1 m =1.132 N

    Thus the centripetal force is1.132 Ndirected towards the center of the circle.

    The centrifugal force is equal in magnitude to the centripetal force and is thus1.132 Npointing in an opposite direction, radially outwards.

    Applications of Centripetal Force and Centrifugal Force

    A common application that involves the centripetal force and centrifugal force is a car driving around a flat curve at a constant speed. In an inertial reference frame, the forces acting on the car are gravity, the normal force, and friction, which keeps the car from sliding. The centripetal force is supplied by the friction, keeping the car moving in a circle. A passenger in the car could argue that they feel an outward force pushing them into the door as the car goes around the curve, but in an inertial reference frame, there is no centrifugal force pushing outward. What the passenger is feeling comes from the law of inertia. The passenger has an initial velocity from the car in motion and wants to move in a straight line, but since the car is rounding a curve, the side of the car keeps the passenger from moving in a straight line. If we consider a non-inertial reference frame from the passenger's perspective, the passenger and car are at rest while everything else is moving. The outward force the passenger feels can be called the centrifugal force in their reference frame, but we need to remember that it is not a real force since there is no object applying it.

    A1000 kgcar is going around a curve of radius20 mwith a velocity of15 m/s. What is the friction coefficient? From the car's non-inertial reference frame, what is the magnitude of the centrifugal force?

    We learned above that the centripetal force can be found byFc=mv2R. Our total force in this problem is given by frictionFc=μFn, whereμis the friction coefficient andFnis the normal force. As stated above, the normal force is equal to the force of gravity so thatFn=mg. Substituting these into our equation for the centripetal force gives us:

    Fc=μFn mv2R=μmg

    μ=v2gR =(15 ms)2(9.8 ms2)(20 m) =1.15

    The magnitude of the centrifugal force is given by:

    Fcf=mv2R

    Thus it isFcf=(1000 kg)(15 m/s)220 m=11250 N.

    Centripetal and Centrifugal Force - Key takeaways

    • Centripetal acceleration is the radial acceleration of an object moving in a circular motion.
    • Centripetal force is the net radial force acting on an object in a uniform circular motion.
    • Centrifugal force is an apparent force directed radially outwards felt by an object in a non-inertial reference frame.
    • Centrifugal forces are only applicable in a non-inertial reference frame. There are no centrifugal forces in inertial reference frames.
    • Using a fictitious centrifugal force in a non-inertial reference frame allows us to use Newton's laws of motion in that reference frame.
    • The centripetal force can be found using Newton's second law, and it is proportional to the centripetal acceleration.
    • Centripetal force and centrifugal force are equal in magnitude and point in opposite directions.
    • The centripetal force vector and the centripetal acceleration vector always point to the center of the circle.
    Learn faster with the 1 flashcards about Centripetal and Centrifugal Force

    Sign up for free to gain access to all our flashcards.

    Centripetal and Centrifugal Force
    Frequently Asked Questions about Centripetal and Centrifugal Force

    What are centripetal force and centrifugal force? 

    Centripetal force is the net radial force acting on an object in a uniform circular motion. Centrifugal force is an apparent force directed radially outwards felt by an object in a noninertial reference frame. 

    What is the difference between centripetal force and centrifugal force? 

    Centripetal force is the real radial force acting on an object causing it to move in a circular motion. Centrifugal force is a fictitious force that we use for objects in a noninertial reference frame so that we can use Newton’s laws of motion for calculations in that frame of reference. 

    Are centripetal and centrifugal forces equal? 

    The centripetal and centrifugal forces are equal in magnitude, but point in opposite directions. 

    How are centripetal and centrifugal forces related? 

    Centripetal force and centrifugal force both refer to forces on objects experiencing circular motion, but centrifugal force is only used in a noninertial reference frame. Centrifugal force does not exist in an inertial reference frame. 

    Are centripetal force and centrifugal force equal in magnitude? 

    Centripetal force and centrifugal force are equal in magnitude and opposite in direction. 

    Save Article

    Discover learning materials with the free StudySmarter app

    Sign up for free
    1
    About StudySmarter

    StudySmarter is a globally recognized educational technology company, offering a holistic learning platform designed for students of all ages and educational levels. Our platform provides learning support for a wide range of subjects, including STEM, Social Sciences, and Languages and also helps students to successfully master various tests and exams worldwide, such as GCSE, A Level, SAT, ACT, Abitur, and more. We offer an extensive library of learning materials, including interactive flashcards, comprehensive textbook solutions, and detailed explanations. The cutting-edge technology and tools we provide help students create their own learning materials. StudySmarter’s content is not only expert-verified but also regularly updated to ensure accuracy and relevance.

    Learn more
    StudySmarter Editorial Team

    Team Physics Teachers

    • 13 minutes reading time
    • Checked by StudySmarter Editorial Team
    Save Explanation Save Explanation

    Study anywhere. Anytime.Across all devices.

    Sign-up for free

    Sign up to highlight and take notes. It’s 100% free.

    Join over 22 million students in learning with our StudySmarter App

    The first learning app that truly has everything you need to ace your exams in one place

    • Flashcards & Quizzes
    • AI Study Assistant
    • Study Planner
    • Mock-Exams
    • Smart Note-Taking
    Join over 22 million students in learning with our StudySmarter App
    Sign up with Email