Question

Where is the center of mass of a slice of pizza that was cut into eight equal slices? Assume the origin is at the apex of the slice and measure angles with respect to an edge of the slice. The radius of the pizza is R.

a) ((0, 3R/8))
b) ((0, R/4))
c) ((0, R/8))
d) ((0, R/16))

Answer 1

The center of mass for a slice of pizza cut into eight equal slices is at the y-coordinate R/4 from the apex of the slice, along the y-axis.

Step-by-step explanation:

The center of mass of a uniform semicircular object is located at a distance 4R/(3π) from the center of the flat side along the symmetry axis of the semicircle. For a pizza cut into 8 equal slices, each slice can be looked at as a quarter of a semicircle. Therefore, since the distance from the center (apex) to the crust is R, and each slice makes up π/4 rad of the circle, the center of mass of each slice would be 4R/(3π) × (2/π) = 8R/(3π²). However, we need the distance along the y-axis, so we consider only the y-component. The y-coordinate is this distance times the sine of 45°, or √2/2. Therefore, the y-coordinate of the center of mass for the pizza slice is (8R/(3π²)) × (√2/2) which simplifies to approximately R/4. Consequently, the correct answer is (b) ((0, R/4)).