Swirls and Whirls

My mom loves to make homemade ice cream. Even though we always have some in the freezer, she is constantly making more! The process is fairly simple – she first combines milk, heavy cream, sugar, and vanilla. Based on the ratios of ingredients, I am taking the mass of this mixture to be 0.765 kg. From there, she places the mixture in the cold ice cream maker. A paddle rotates and after sometime, the ice cream is ready to be eaten! I decided to help my mom make ice cream and as I waited for the ice cream to finish, I couldn’t help but think of the rotational motion it was experiencing.

The ice cream machine can be estimated as a cylinder with a radius of 8.00 cm (0.0800 m). It also rotates at a constant angular velocity. To calculate this velocity, I measured the time it took for the paddle to complete one full revolution (6.28 radians). The paddle completed the one revolution in 1.00 s, giving it an angular velocity of 6.28 radians / 1.00 s, or 6.28 rad/s.

However, we’re not making vanilla ice cream. Towards the end of the process, we added in about 252 g or 0.252 kg of Reese’s Peanut Butter Cups. Now the total mass of the ice cream is 1.017 kg. Yet the angular velocity remained the same despite the additional mass. If that’s so, the angular momentum must have changed, but by how much? To answer this we need to calculate the moment of inertia for the system before and after the addition of the PB cups.

Assuming the paddle is massless, we can use the information above to calculate the moment of inertias. For a cylinder, it is calculated by I = ½mr². Plugging in our masses and radius I_o = ½(0.765)(0.08)² = 0.00245 kgm² and I_f = ½(1.017)(0.08)² = 0.00325 kgm². Now we can calculate the change in angular momentum, where L = Iw. Using this equation, our L_o = (6.28)(0.00245) = 0.0154 and L_f = (6.28)(0.00325) = 0.020. Now to find the change we subtract L_f – L_o = 0.00506, our change in momentum. Knowing this makes the ice cream a little bit sweeter.