You have a one-dimensional slice of cheese with total mass \(M\) and total length \(L\text{.}\) You know the linear mass density is \(\lambda(x) = k \sqrt{x}\text{.}\) Find the constant of proportionality \(k\) (including its units) and the center of mass of the slice of cheese.
Tip1.
Analyze and Represent: Sketch and label a physical diagram showing at least one small piece of the cheese slice.
Tip2.
Sensemake: For your numerical sensemaking (part 3b), sketch a graph of the mass density vs. \(x\) and use it to support a qualitative explanation for why the center of mass is located where it is relative to the geometric center of the cheese slice.
A*R*C*S13.9.2.The Cube I.
A wooden cube with side length \(L\) and total mass \(M\) is rotating about an axis passing through one of its edges. Use an integral to calculate the moment of inertia.
Explanation13.9.3.The Cube II.
The cube from the previous activity is instead rotating about an axis passing through its center (and through the center of one of its faces). The parallel axis theorem can be used to demonstrate that the moment of inertia about this axis is smaller than the moment of inertia about the axis in the previous activity. Give a qualitative explanation accounting for this result.
