I really don’t know much about Dua Lipa—but I actually do know something about physics. The dancing in this music video uses some cool physics for some really interesting effects. In this case, the dancers are performing on a rotating platform. This allows them to do some moves that seem impossible. One dancer lifts the other and leans back—very far. You would think the two would just tip over and fall, but they don’t.
The move happens at about 2:40 in the video.
In order to really understand this move, we need to look at some basic physics. Let’s start with an object in equilibrium. In physics, equilibrium means that an object has zero acceleration (linear equilibrium) and zero angular acceleration (rotational equilibrium). Here’s an example—a normal human standing up straight on a normal and non-rotating floor.
Yes, normal humans don’t stand around on one foot, but I wanted a fun human. Since the human has a zero acceleration, the total force must also be zero. This is straight from Newton’s second law, which states:
For this fun human, there are two forces. The gravitational force pulls straight down and seems to pull on a particular point on the human that we call the center of mass. Yes, technically, all parts of the body have mass and are therefore pulled down to the Earth. But mathematically, you can calculate the entire gravitational force as though it were acting at just one point. For a typical human, that center of mass is somewhere around your belly button. The other force is the force from the floor pushing up. Since it’s an interaction between the foot and the floor, it’s important to put the force at the contact point. In the diagram above, I labeled this as FN where the N denotes “normal.” We call this the normal force since it’s perpendicular (normal) to the floor. But the normal force and the gravitational force have to be equal in magnitude in order for the person to be in equilibrium.
Now for the other part of equilibrium, rotational equilibrium. For the human standing on one foot, this means that the fun person doesn’t rotate. Just like linear equilibrium means zero net force, rotational equilibrium means zero net torque. Torque is basically a rotational force. When you push on a door to open it, you exert a torque that causes it to go from not rotating to rotating (opening). The value of a torque depends on three things:
- The magnitude of a pushing or pulling force (like your hand pushing on the door).
- The distance from the force to the point of rotation (distance from the door hinge to your hand). We often call this the torque arm.
- The sine of the angle (θ) between the torque arm and the force. If you push perpendicular to the door, this angle would be 90 degrees.
So, as an equation the torque can be expressed as the following formula. We use the Greek letter tau (τ) for torque.
It’s pretty easy to see that the net torque for the human on one foot is zero. If you take the foot as the point of rotation, both the normal force and gravitational force have a zero torque arm and have zero torque. Since zero plus zero equals zero, the total torque is zero.
Great, now let’s use these same ideas to show why you can’t hug someone while leaning super far back (unless you are on an awesome rotating platform). Actually, just to make things easier I am going to draw the forces on a single human that is just doing a super lean back.
Even if these two forces (gravitational and normal) have the same magnitudes, the total torque will not be zero. Using the foot contact as the point of rotation, the normal force has zero torque (torque arm of zero), but the gravitational force does indeed have a non-zero torque. The total torque will cause this happy leaning human to tip over and hit the ground. Now a sad human. Sad human on the ground.
Then what the heck keeps these dancers from falling over? The answer is a fake force. Yes, a force that isn’t actually a force but instead a fake force. Oh, you’ve never heard of a fake force? Well, maybe that’s true, but I’m sure you have felt a fake force.
Imagine the following situation. You are sitting in your car at a red light (the car isn’t moving). At this moment, there are just two forces acting on you. There is the downward pulling gravitational force and the upward force from the seat. Since you aren’t accelerating, these two forces have equal and opposite magnitudes.
Oh, but wait! There is this goofy looking car in the lane next to you. The light turns green, so you hit the gas and accelerate (safely and within the posted speed limits of course). What happens next? You feel it, right? There is some force pushing you back into your seat as you accelerate. It feels like the “weight of the acceleration” or something, right? This is actually Einstein’s equivalence principle. It states that you can’t tell the difference between an acceleration and a gravitational force. So, in a sense this force you feel is as real as gravity—as far as you can tell.
The connection between forces and acceleration (Newton’s second law) only works in a non-accelerating reference frame. If you drop a ball in this accelerating car, it’s going to move as though there was some force pushing it in the opposite direction as the acceleration of the car. We can add a “fake force” that’s proportional to the acceleration of the car and boom—Newton’s second law works again. It’s really quite useful.
Guess what? A rotating platform accelerates. In fact, any object moving in a circle accelerates. Acceleration is defined as the rate of change of velocity (in calculus, this would be the derivative of velocity with respect to time). But velocity is a vector. That means that moving to the left is different than moving to the right with the same speed. In fact an object moving at a constant speed but changing direction is a changing velocity. So, turning in a circle is indeed an acceleration. We call this “centripetal” acceleration—which literally means “center pointing” acceleration. Yes, the acceleration for an object moving in a circle points toward the center of that circle.
The magnitude of this acceleration depends on two things: the speed of the object (the magnitude of the velocity) and the radius of the circular motion. Sometimes it’s useful to write the centripetal acceleration in terms of the angular velocity (ω) instead, since all points on a rotating platform have the same angular velocity but not the same speed (points farther from the center have to move faster).
We are ready. Ready for the impossible seeming physics of a dancer on a rotating platform. Let’s start with a diagram.
There’s a lot going on here. But really, there are only two new forces. First, there is the fake force. At this instant, the center of the circular motion is to the right. That means the centripetal acceleration is also toward the right. So, if we want to consider the rotating dancer as our reference frame, there will need to be a fake force pushing to the left (opposite the acceleration). But wait! Did you notice that I put a new green dot for the fake force? Yes, that’s legit. Technically, all parts of the human are accelerating. But just like the gravitational force can be calculated as though it were acting at one point (the center of mass), the same is true for the fake force—it feels the same as gravity according to Einstein.
However, the Earth’s gravitational force is pretty much constant. It doesn’t noticeably change as you move up or down. This is not true for the fake rotational force. As you get closer to the center of the rotating platform, the acceleration (and thus the fake force) decrease down to zero at the exact center. So, the single point that acts as the “center of acceleration” would be a little bit farther away from the axis of rotation. I will let you calculate the exact location of this center of acceleration as a homework problem. (It depends on the density distribution of the human, the angular velocity of the platform, and the location of the human.)
So, then why doesn’t the dancer fall over? In the rotating reference frame, you can see that there is also a torque produced by the fake force. Using the foot contact as the pivot point, the gravitational force causes a clockwise torque, but the fake force produces a counterclockwise torque. With these two torques, it’s possible that they add up to zero torque so that the human stays at that lean angle. Of course, if the platform rotates too fast, the torque from the fake force will cause the person to rotate out and away from the platform. If the human leans too far, the gravitational torque will be greater—then they will end up falling down.
But wait! There’s another force in that diagram—friction. Since there is a fake force pushing sideways now, there must be a frictional force pushing back to make the net force zero. Without that frictional force, the dancer would just slide right off the rotating platform. Our basic model of the frictional force has the magnitude proportional to the normal force using the following relationship.
In this expression, μs is the coefficient of friction that depends on the two materials interacting (like rubber and wood). This frictional force is whatever value it needs to be in order to prevent the person’s foot from sliding—up to some maximum value. That’s why there is a less than or equal to sign in there. But now we can use this to get a rough estimate of the value of this frictional force (and coefficient) needed to prevent the dancer from slipping. Really, I just need a value for the angular velocity and the rotational distance.
Looking at the video, the dancers make a quarter rotation in about 0.8 seconds. (I used Tracker Video Analysis to get the time.) From this, I get an angular velocity of 0.98 radians per second. For the radius of rotation, I am going to approximate the center of acceleration at around 1 meter. This gives me the following two equations for the net force in the x and y directions (in the rotating frame).
Using these two equations, I can get the following expression for the coefficient.
Notice that the mass cancels—that just makes things easier. If I put in my estimations for the radius and the angular velocity (and use a gravitational constant of g = 9.8 m/s2), I get a coefficient of static friction value of around 0.1. Remember, this is for the maximum frictional force that can occur between the dancer’s shoe and platform. The coefficient could be greater than this value, but if it’s less there will be a slip and fall. But if he is wearing rubber shoes, the dancer can easily get a coefficient of static friction over 0.5 to prevent a slip. So, it looks like you don’t even need rubber shoes, but you still need some awesome physics for this dance move.
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