Squirrels are all right. They’re better than your average rodent, and they jump around and stuff. But if you have a bird feeder, you might hate them. These animals just don’t get that some food is set aside for birds. They don’t respect boundaries, and they’re not above wrecking your feeder to get the goods.
That’s why some people employ anti-squirrel technology. A company called Droll Yankees makes dispensers with names like the Tipper, the Whipper, and the Flipper. That latter one has a motor on the bottom and a weight-activated spinning perch. Birds aren’t heavy enough to trip the switch, but a squirrel is.
Now usually, a squirrel will jump off a bird feeder that starts to spin—but not the one in this viral video. You have to admire his spirit, actually. He hangs on till the bitter end, but it’s not enough and he catches some major air.
You know what’ I’m thinking? This is a perfect example of the forces involved in circular motion. Let’s take a look at some of the interesting physics questions here.
Why Does the Squirrel Fly Off?
So you have this furby attached to a spinning contraption. Clearly it’s not easy to hold on—but why? Is this all about centrifugal force?
Yes, it’s true that this deals with centrifugal force. It’s also true that most physics teachers hate using centrifugal force, because it’s conceptually dangerous for beginning students. Let me first describe the idea, and then I’ll tell you why it’s not included in introductory physics courses.
You know about centrifugal force, right? When you’re sitting in a car that’s turning left, you feel something pushing you to the right—away from the center of the circle that the car is moving in. (A turn is temporarily part of a circular motion.) That’s what centrifugal means—to flee (fugere) the center. It’s a force that pushes away from the center of a circle. The faster the car goes, the greater the force. The tighter the turn (i.e., the smaller the radius of the circle), the greater the force.
That’s what happens to the squirrel. As the rate of rotation increases, he gets pulled and stretched outward, away from the center, until his little paws can’t hold on and he loses contact with the bird feeder.
But wait! Centrifugal forces are different from the usual physics forces. We typically describe forces as an interaction between two objects. If you hold out an apple and let go, it will fall. That falling motion is due to a gravitational interaction between the Earth and the apple. But what is the force-paired object pushing on the squirrel? There isn’t one.
Another way at this is to think about what it is that forces do. A force acting on an object changes its momentum—where momentum is the product of mass and velocity. When you drop that apple, the gravitational force increases its speed as it falls, thus increasing its momentum.
So here’s a little thought experiment: Let’s say this apple starts 1 meter above the ground. If you drop it with zero initial velocity, it will move down with an acceleration of 9.8 m/s2, and it will take 0.45 seconds to hit the ground.
Now drop the apple again, but this time, do it inside an elevator that is just starting to go up. (You know the elevator is accelerating upward because you feel “heavier.”) If you measure the falling time, you’ll see that it now takes less than 0.45 seconds to hit the floor.
Why is that? There is still only the same gravitational force acting on the apple, so it seems like the normal force-motion laws don’t work—the apple hits the floor too soon. Well, the reason is that it didn’t fall as far. Because the elevator is accelerating upward, the distance from the starting point to the ending point is less than 1 meter. (If you find an elevator with a glass window you can see this quite well.)
Motion is always relative. We can only measure how things move relative to something else. That “something else” is called a reference frame. So this is a good example of how you can get confused when the reference frame itself is accelerating. Those physics laws only work in an inertial (i.e., non-accelerating) reference frame.
To make the apple in the elevator follow the normal physics laws, we have to add another force pushing it down. This is an example of what I like to call a “fake force.” A fake force needs to be added to an accelerating reference frame in order to make the physics work again. In general, a fake force takes the following form:
This says that the fake force you add to your accelerating system is just the mass of the object multiplied by the acceleration of the reference frame (aframe)—but in the opposite direction.
Imagine you’re in a car that’s accelerating forward. You feel yourself being pushed back into the seat, right? Since you’re in the car, you automatically make that your reference frame, and you think there’s a force pushing you back. But there is no force; there’s no object acting on you. But to make our normal physics work, you can add a fake force pushing backward, in the opposite direction from the motion of the car.
That’s exactly what happens to the squirrel. For an object moving in a circle, that object must have an acceleration pointing toward the center of that circle. But if you were the one getting spun around in a circle, you would add a fake centrifugal force that points in the opposite direction of the real acceleration.
And now we can talk about centripetal, or “center-pointing,” acceleration. The force that causes this circular acceleration is then called the centripetal force. For the squirrel, this (real) force is applied from the perch that he’s holding on to, and it’s yanking him toward the center. Once this force gets too high, the squirrel can’t hang on anymore. It’s as if the handle is ripped out of his grip.
To sum up: Centrifugal force is a fake force that is added to an accelerating reference frame, and centripetal force is the force required in an inertial reference frame to make an object move in a circle. Because the centrifugal force is fake, most physics instructors don’t want students to use it—they have enough problems with real forces.
Now for some other important physics questions (with answers)!
How Hard Is It to Hang On?
Let’s start with some data. I put this squirrel video into the Tracker video analysis app and found that it takes 0.5 seconds for the feeder to make one complete rotation. This gives it an angular velocity (ω) of 12.6 radians per second. The approximate radius (r) of the squirrel’s “orbit” is about 0.15 meter (6 inches). This means the centripetal acceleration is:
Oh, in case you’re wondering, that’s 2.4 g’s. But what about the force? For that, I need to guess the mass of the squirrel. Let’s go with 0.45 kilograms. That puts the magnitude of the centrifugal force at 10.7 newtons—a pretty big force for a small squirrel.
That’s good enough for garden-variety math. For simplicity, I used as a radius the distance from the center of the squirrel to the axis of rotation. But in fact, since different parts of the squirrel move in circles with different radii, each part has a different acceleration. So if you wanted a more exact estimate, you’d have to use calculus and integrate the differential acceleration over the length of the squirrel. Now that would be a nice real-world math problem for you.
Is Angular Momentum Conserved?
I’m just adding this question since I noticed quite a few internet comments about angular momentum. So what the heck is angular momentum? In short, angular momentum is a quantity we can calculate that is sometimes conserved. For a single particle (not quite true for a squirrel), the angular momentum can be calculated as:
In this expression, L is the angular momentum, r is the vector distance from some point (it could be the center of the circle) to the object, and p is the linear momentum of the object (mass times velocity). Oh, that “×” isn’t for multiplication; that’s the vector cross product.
Angular momentum is useful because it’s a quantity that remains constant in some situations. For a closed system with zero torque (torque is like a twisting force), angular momentum is conserved. But for the system consisting of the squirrel, there is indeed an external torque. The motor in the feeder twists the rotating perch so as to increase angular momentum. It’s not conserved.
Now, if the perch was freely rotating without an electric motor, then angular momentum would be conserved. As the squirrel moved farther away from the axis of rotation, the angular speed would decrease but the angular momentum would be constant. This is exactly what happens when a spinning figure skater moves from an “arms in” to an “arms out” position to slow down their rotation rate.
Can the Squirrel Get Completely Horizontal?
No—at least not for a full, complete rotation. It might look like the squirrel is horizontal if you look at just one frame of the video, but that position is just temporary. Let’s imagine that this animal is in a stable rotation. At one point, it might have the following force diagram.
There are really just two forces on this squirrel (in the real, inertial reference frame): (1) the downward-pulling gravitational force (mg), and (2) the force the squirrel has to exert to hold on to the spinning feeder (Fs). If he is spinning in a flat horizontal plane, then the total force in the y direction must be zero. Since there are only these two forces, the squirrel can’t just pull horizontally. He also needs to pull upward some in order to bring the net vertical force to zero. Yes, it’s true that the faster the squirrel spins the more horizontal he will get. But he’ll never be completely horizontal.
What Path Will He Take When He Lets Go?
This is actually a classic physics question that is often used in classes. It goes like this: Suppose you view the rotating squirrel from above. Once he lets go of the bird feeder, which path will he likely take: A, B, C, or D?
Go ahead, pick one and write it down, along with some type of justification for your choice. You could probably make a reasonable case for each of these paths. But only one of them is correct.
So the key question is, what forces are acting on the squirrel after he lets go? There’s still the downward gravitational force, but that wouldn’t change the motion as seen from above. But that’s it; there are no other forces. With zero forces in the horizontal plane, there is zero change in horizontal motion. Remember that forces only change the motion of an object. With no change in motion, the object will just continue along in a straight line. That means it can’t be A.
Really, to choose between paths B, C, and D, you just need to think about what direction the squirrel is traveling at the point of release. If he’s moving in a circle, his velocity will be in a direction tangent to the circle. So the only possible path for the released squirrel is B. He isn’t flung “outward,” as you might be tempted to say—there is no “centrifugal force”!—he’s flung forward.
Of course, from the squirrel’s reference frame, all that matters is that none of these paths lead to bird food.
