Why Isn't the Centrifugal Force Real?

 Why Isn’t the Centrifugal Force Real?

By Michelle Ng

Image source: https://www.timeout.com/london/things-to-do/thorpe-park-fright-night-rides-ranked-in-order-of-fear-factor

Space mountain: a thrilling, high-speed roller coaster ride with sudden turns and sharp bends—some may love the feeling of plunging into total darkness and the sensation of weightlessness as you accelerate downwards in sudden drops, whilst others feel sick at the thought of it and prefer more forgiving rides. But whether it be a fast-moving roller coaster ride composed of a tangle of twists and turns, or just a family-friendly ride that goes round and round (such as the Slinky dog dash), you’ll feel as if you’re being thrown outwards to the side whenever the coaster curves around in a bend. Since a force is often described as a push or a pull, you might wonder: is there is a force pushing you away from the centre of the curve, causing you to feel like you’re being thrown outwards? This is where the term ‘centrifugal force’ comes from, with ‘centrifugal’ meaning ‘away from the centre’. However, this is a very common misconception when it comes to circular motion, because in fact, this force is not real—how can this be?

Circular Motion

When an object moves in a circle with no change in speed, it is still accelerating, because its direction is perpetually changing. By Newton’s second law, there must be a force acting in the same direction as acceleration; in order for this force to produce an acceleration without a change in speed, it must have no component parallel to the velocity, meaning that it must be entirely perpendicular to the velocity. But which way—outwards or inwards? If you visualise it, the direction that the object is moving in is actually bending inwards towards the centre of the circle, not outwards (figure 1): therefore, the force must be acting towards the centre of the circle. This is known as the centripetal force, or ‘centre-seeking’ force in Latin. For example, an object tied to a piece of string moving around in a circle is constantly experiencing a tension force towards the inside of the circle. A planet orbiting around a star requires the gravitational force attracting it towards the star to remain in orbit. The magnitude, or size, of the centripetal force is given by F = mv^2/r, where m is the mass of the object, v is the tangential velocity of the object and r is the radius of the circle.

Figure 1: How the velocity of an object moving in a circle changes [1]

If the force is towards the inside of the circle, you may be wondering why the car doesn’t just spiral inwards. Similarly, you might have seen a magician or a physics teacher swinging a bucket of water around above their head without the water falling out, even though both the gravitational force and the normal reaction force from the bucket acting on the water are both acting downwards when the bucket is at the highest point of the circle (I remember sitting in my classroom in year 9 watching my physics teacher doing this in astonishment.) The reason why this is the case is because we must remember forces themselves do not cause motion: a force causes acceleration. Even though the force causes the object to accelerate inwards, that does not necessarily mean the object will move inwards: since the object started off with a given tangential velocity, the object will continue moving at that tangential velocity as it continuously bends towards the inside of the circle, without ever actually reaches the middle. If the centripetal force suddenly disappeared, unless there is another force acting on it, the object would move off in a straight line at its tangential velocity with no further changes in velocity due to Newton’s first law. If the sun suddenly disappeared, the Earth would leave its orbit and start moving in a straight line. In the case of the bucket, if the teacher let go at the top of the circle, the bucket would initially move off with its original velocity but because of its weight (an external force), it will act as a projectile and fall downwards in a parabolic path.

This can be extended to non-circular curves because at any point in a given curved path, you can imagine an osculatory circle that is tangential to the velocity of the object at that point in the path (figure 2). Furthermore, even if the object’s speed is changing, although the resultant force will no longer be purely perpendicular to the tangential velocity, you can still resolve the force into its radial and tangential components, and the resultant acceleration will be a combination of the radial acceleration and tangential acceleration—the centripetal force is still required.

Figure 2: a circle that is tangential to velocity, even if the path is not entirely circular. [1]

So, the force responsible for circular motion is the centripetal force. The ‘centrifugal force’ does not exist—it is not a ‘real’ force. So why do we feel ‘flung out’?

Why do we ‘feel’ the ‘Centrifugal Force’?

There is a caveat to what has been described above—in considering the centripetal force acting on an object moving in a circle, it has been assumed that the observer—the frame of reference—is at rest. This is because Newton’s laws only apply in an inertial frame of reference: this is a frame of reference that is not accelerating. [2] When considering the moving car, we have assumed the position of someone who is standing next to the roller coaster ride, not someone who is in the roller coaster ride. So the confusion occurs when you try to consider the forces from the viewpoint of someone inside the roller coaster, but since the person inside the rollercoaster is accelerating, Newton’s laws as we know them will not apply. 

The apparent feeling of being thrown outwards is because of your body’s tendency to move in a straight line if no external forces act on you (Newton’s first law). Consider the roller coaster turning around a bend again: initially, the roller coaster experiences a driving force towards the inside of the circle and starts to bend. However, if the passenger is sat in the middle of the car, at this particular moment, the passenger is not touching the side walls of the car—the walls are not exerting a reaction force on them. If the friction between the seat and the car is not great enough to provide the passenger with the centripetal force required to keep them in circular motion, according to a stationary observer, the passenger will move off away from the path of the circle. [1] (If friction was 0, the passenger would move off in a straight line.) From the viewpoint of the rollercoaster, this creates the illusion that the passenger is being thrown outwards—the centrifugal force is only ‘real’ in a rotating reference frame.

Figure 3: Inside a car, a coffee cup resting on a surface will appear to slide outwards because it is moving away from the circular path (as friction is not great enough to keep it within the path), creating the illusion of a ‘centrifugal force’.

But since the car itself is turning whilst the passenger continues to move at its original velocity, this causes the passenger to appear to move towards the outside wall of the car until the passenger becomes in contact with the outside wall. At this point, there will be a reaction force acting on the passenger towards the inside of the circle and the passenger will now have the same acceleration as the car and bend with it. Because of this, from the viewpoint of the rollercoaster, the passenger is not longer moving outwards. From the viewpoint of the passenger, it seems like they are pushing against the wall of the car, which is the feeling we get when we ride a rollercoaster.

When is the ‘Centrifugal Force’ Useful to Consider?

If the centrifugal force is not ‘real’, why are there machines called centrifuges, and why is it sometimes used by industries to explain certain mechanisms?

Actually, some industries may find it useful to look at a situation relative to the rotating reference frame. [3] A centrifuge is a machine used to separate different particles in a solution according to their different densities by spinning them around. The spinning container itself experiences a centripetal force towards the centre of rotation. But if the frictional force and upthrust force experienced by the particle is not great enough to give them the same acceleration as the motor and the solution (as the force required is the centripetal force, mv^2/r), the particles will appear to be ‘flung out’ to the sides of the container, causing them to sediment and collect on the walls. [4] As the velocity of the machine is increased, the centripetal force that is needed, so particles of different masses can be separated because the forces they require will be different.

Hence it may be useful for industries that use this machine to visualise the situation from the point of view of the spinning container and to study the behaviour of the particles relative to the rotating container rather than to a stationary observer. From this situation it will seem like the particles are experiencing a ‘centrifugal force’ pushing them outwards due to the rotation of the solution itself, and if this is not ‘balanced’ by the friction and upthrust, then the particles will sediment. 

However, in normal usage of Newton’s laws, the centrifugal force is merely fiction: it is a common misconception about circular motion. So the next time you ride a roller coaster and feel flung outwards, you now know that circular motion is in fact due to a centripetal force, viewed from an inertial frame of reference—the ‘centrifugal force’ is only an apparent force you experience. 

BIBLIOGRAPHY

[1] Hyperphysics.Phy-Astr.Gsu.Edu, 2021, http://hyperphysics.phy-astr.gsu.edu/.

[2] Janet Jagger, and Kevin Lord. “What Is Centrifugal Force?” The Mathematical Gazette, vol. 79, no. 486, 1995, pp. 484–488. JSTOR, www.jstor.org/stable/3618074. Accessed 10 Apr. 2021.

[3] "Centrifugal Force | Physics". Encyclopedia Britannica, 2021, https://www.britannica.com/science/centrifugal-force.

[4] "Centrifugation Theory". Fishersci.Co.Uk, 2021, https://www.fishersci.co.uk/gb/en/scientific-products/centrifuge-guide/centrifugation-theory.html#:~:text=A%20centrifuge%20is%20a%20device,through%20use%20of%20a%20rotor.&text=As%20a%20rotor%20spins%20in,centrifugal%20force%20applied%20to%20it.