Question

A uniform circular disk of radius R = 44 cm has a hole cut out of it with radius r = 13 cm. The edge of the hole touches the center of the circular disk. The disk has uniform area density σ.

Part (a) The vertical center of mass of the disk with hole will be located:
Part (b) The horizontal center of mass of the disk with hole will be located:
Part (c) Write a symbolic equation for the total mass of the disk with the hole.
Part (d) Write an equation for the horizontal center of mass of the disk with the hole as measured from the center of the disk.
Part (e) Calculate the numeric position of the center of mass of the disk with hole from the center of the disk in cm.

Answer 1

The vertical and horizontal centers of mass of the disk with the hole can be calculated using the properties of the original disk and the hole. The total mass of the disk with the hole can be represented symbolically. The numeric position of the center of mass can be calculated using the equation provided.

Step-by-step explanation:

Part (a): The vertical center of mass of the disk with the hole will be located at the same height as the center of mass of the original disk. This means that the vertical center of mass will be at a distance R/2 from the upper edge of the disk.

Part (b): The horizontal center of mass of the disk with the hole will be located at the same distance from the vertical axis as the original disk. This means that the horizontal center of mass will be at a distance R/2 from the vertical axis.

Part (c): The total mass of the disk with the hole can be represented symbolically as M = σ(A - A_hole), where σ is the area density, A is the area of the disk, and A_hole is the area of the hole.

Part (d): The horizontal center of mass of the disk with the hole, measured from the center of the disk, can be represented as X = (A - A_hole)/(2πR).

Part (e): To calculate the numeric position of the center of mass of the disk with the hole from the center of the disk, we can substitute the values of A, A_hole, and R into the equation X = (A - A_hole)/(2πR).

Answer 2

The vertical center of mass of the disk with a hole is located at the midpoint of the top and bottom edges of the disk. The horizontal center of mass is located at the center of the disk. The total mass of the disk can be calculated using a symbolic equation. The numeric position of the center of mass from the center of the disk in cm is 0.

Step-by-step explanation:

Part (a) The vertical center of mass of the disk with hole will be located:

The vertical center of mass of the disk with the hole will be located at the midpoint between the top and bottom edges of the disk. Since the hole is tangent to the center of the disk, the distance from the top edge of the disk to the center of mass will be half the radius of the disk, which is 22 cm.

Part (b) The horizontal center of mass of the disk with hole will be located:

The horizontal center of mass of the disk with the hole will be located at the center of the disk. This is because the disk is symmetrical in the horizontal direction, and the hole does not affect the balance horizontally.

Part (c) Write a symbolic equation for the total mass of the disk with the hole:

The total mass of the disk with the hole can be expressed symbolically as M = σπ(R² - r²), where M is the mass, σ is the area density, R is the radius of the disk, and r is the radius of the hole.

Part (d) Write an equation for the horizontal center of mass of the disk with the hole:

The horizontal center of mass of the disk with the hole is located at the center of the disk. Therefore, the equation for the horizontal center of mass is simply x = 0.

Part (e) Calculate the numeric position of the center of mass of the disk with hole from the center of the disk in cm:

Since the vertical center of mass is located at the midpoint of the disk, the distance from the center of the disk to the center of mass is 0 cm.