Out of My Element

Recently, a friend shared a meme about how COVID has made the idea of bowling particularly unappealing: Sticking your fingers into a ball handled by many people, which has also been rolling on the floor, then eating greasy food with those same fingers. After getting that image out of my head, I started thinking about the physics of bowling. In particular, I was interested in the curve that skilled bowlers are able to give to the ball:

Typically in physics, we make the assumption of "rolling without slipping," meaning that an object rotates at the correct speed for the contact between the two surfaces not to slide, i.e.

where v is the velocity along the floor and ω and r are the angular velocity and radius of the rotating object. If you're mathematically minded, you may notice this is the derivative of the relation between angle and arc length for circles. Bowling balls though have significant slip, as illustrated in this animation:

Wikipedia

When the player throws the ball, they give it some initial linear velocity and angular velocity, or spin. When the ball makes contact with the floor, friction begins pulling on the ball, which applies a torque. Since the ball is slipping, we need to know the relative velocity between the surface of the ball and the floor:

Note that the angular velocity is given according to the right-hand rule: If you point your thumb in the direction (ωx, ωy), your fingers curl in the direction of rotation. Using this, we can get the force and torque on the ball:

where μ is the coefficient of friction, m is the mass of the ball, g is the acceleration from gravity, and X is the cross product.

Since I was on vacation this week, I spent some time thinking about this, and decided to put together another interactive demo for the problem. The inputs are the initial speed and spin, and the friction between the ball and floor. To get a nice arc, turn down the friction and increase the x angular velocity. The coordinates are arranged so +x is forward and +y is right.

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	x	y
Lin. Vel.
Ang. Vel.
Friction Coef.