I juggle, it’s a very serious hobby and I’ll always be grateful for the time I took to learn. I can’t juggle many thing, mainly balls and rings. I have been trying to juggle something new this semester: clubs. Clubs are like bowling pins with a rounded bottom and a skinnier neck.
How can one learn how to juggle such an object? Well the answer is one club at a time. I’ll start with the physics of trying to throw one of these things up in the air and catching it. The starting position of juggling a single is grasping the handle a little less than halfway with your thumb on top and your forefinger adjusted a little further up from the rest of the fingersNow to start with the physics. You need to apply a force with your thumb downward on the handle of the club. The club will rotate about in a counter clockwise direction (toward you). The goal is for the club to make one full rotation before it lands into your hand again. You'll want to move your hand out the way to allow the club to rotate fully then place it back in the original position before it ends its rotation in order to catch it nicely. If done correctly the club should rotate in place, not moving in the y direction at all. To accomplish this, you need to apply a torque that will overcome the club's moment of inertia and anticipate when the club will cover 2𝜋 radians so that you know when it's time to put your hand back to catch it.
We start with calculating the moment of inertia and we’ll need to use the parallel axis theorem because we are going to be rotating this club about a point that’s not it’s center of mass which is located near the bulge at the top of the object. We could say that our club is a long uniform rod with a point mass at the end. The equation for the total moment of inertia for the object would be the sum of the moments of inertia of the long uniform rod rotating about its center of mass and the point mass located at the end of the long uniform rod. The equation would be I= Icm + Ipoint mass. For the long uniform rod the moment of inertia would be I=1/12ML2. Let's say the mass of the rod portion of the club is 100g. Let us say that the length of the rod portion of the club is around 40 cm. Now if we plug that into this equation for the moment of inertia of just the long uniform rod we would get I= 1/12 (.100kg) (.45m)2=1.7 x 10^-3 kgm2. The equation for the moment of inertia of the point mass would be I=Mh2, where h is the distance the point mass is from the axis of rotation of the uniform rod. For that we would get I=(.200kg) (.25m)2=1.3 x10-2 kgm2. The sum of these two moments of inertia would be I= (1.3 x10-2 kgm2) +(1.7 x 10^-3 kgm2) =1.5 x 10^-2 kgm2. So, 1.5 x 10^-2 kgm2 is the resistance to the rotation of the club that we must to overcome with the torque we apply with our thumb.
Before we determine the torque, we need to apply with our thumb, let's first think about the angular momentum that we want the club to have. We basically go from having an angular velocity of 0 rad/s to about 2𝜋 radians per 0.5s or 12.6 rad/s. This change in angular velocity occurs within half a second so I'd say that your angular acceleration would be 25.2 rad/s2. The equation of the total torque equals the product of the total moment of inertia and the angular acceleration. With this equation, we can find the torque we need to apply with for thumb. So, 𝜏= (1.5 x 10^-2 kgm2) (25.2 rad/sec)=0.378Nm. It turns out that you don’t need to apply too much torque to juggle a single club which makes sense since you are applying the force with your thumb. However, this is just the beginning we haven't even gotten into the combination of rotational and translational motion as well as adding in two other clubs into the mix but, for starting out, that's enough for now.