How Is the Center of Mass Calculated for a Plate with a Circular Hole?

Click For Summary
SUMMARY

The calculation of the center of mass (CM) for a rectangular plate with a circular hole involves using specific formulas for both the x and y coordinates. The plate has an areal density of σ = 3.06 kg/m², dimensions of 37.0 cm by 25.0 cm, and a circular hole with a radius of 6.50 cm centered at (12.50, 10.50) cm. The x-coordinate of the CM was correctly calculated as 0.194 m, while the y-coordinate was initially miscalculated but later confirmed to be 0.0555 m. The distance from the origin to the center of mass was ultimately determined using the Pythagorean theorem, yielding a correct distance of approximately 0.195 m.

PREREQUISITES
  • Understanding of center of mass calculations
  • Familiarity with geometric shapes and their properties
  • Knowledge of basic calculus and integration for area calculations
  • Proficiency in using the Pythagorean theorem
NEXT STEPS
  • Study the derivation of center of mass formulas for composite shapes
  • Learn about the impact of varying densities on center of mass calculations
  • Explore applications of center of mass in physics problems
  • Investigate numerical methods for calculating center of mass in irregular shapes
USEFUL FOR

Students in physics, engineering, or mathematics who are learning about center of mass calculations, particularly those dealing with composite shapes and geometric properties.

djgh
Messages
2
Reaction score
0

Homework Statement



A thin rectangular plate of uniform areal density σ = 3.06 kg/m2 has length of 37.0 cm and width of 25.0 cm. The lower left hand corner is located at the origin, (x,y)= (0,0) and the length is along the x-axis.

There is a circular hole of radius 6.50 cm with center at (x,y) = (12.50,10.50) cm in the plate. Calculate the mass of plate.

Homework Equations


The Attempt at a Solution



I need to:

Calculate the distance of the plate's CM from the origin.

I've calculated the x-coordinate of the center of mass with the equation:

x= [ ((L/2)*(l*w)) - ((2r)*pi*r^2) ] / [ (l*w) - (pi*r^2) ]

basically x= (x-component center of mass of rectangle)*area - (center of mass of circle)*area divided by total area with L = 0.37m (length of rectangular plate), l*w= area of plate, r= 0.65m, radius of circle cut out of plate. This comes out to be 0.194m and I know is correct.

However, I use the same equation, but replacing L with H, the height of the rectangle or 0.25m to find the y component, which I've calculated to be 0.124m. Then when I find that I used Pythagorean theorem or sqrt(x^2 + y^2) to find distance from origin to center of mass but I get a wrong answer, or answer that is supposedly wrong, which I calculate to be 0.230m. I think I'm doing it correctly, but supposedly I'm wrong, any ideas?

EDIT: SOLVED

Homework Statement


Homework Equations


The Attempt at a Solution

 
Last edited:
Physics news on Phys.org
The method looks good. I got .195 for the x and .0555 for the y when I ran it through.
 
nevermind, figured out what was wrong
 

Similar threads

Calculate the speed and angle of the center of mass before a collision
  • · Replies 3 ·
Replies
3
Views
1K
Kinetic energy in center of mass reference frame
  • · Replies 3 ·
Replies
3
Views
2K
Center of mass and momentum- A question about systems
  • · Replies 25 ·
Replies
25
Views
2K
Finding the center of mass of a simple 2D shape
  • · Replies 9 ·
Replies
9
Views
3K
Finding the center of mass of a solid hemisphere
  • · Replies 19 ·
Replies
19
Views
4K
Find the center of mass of a plate
  • · Replies 57 ·
2
Replies
57
Views
6K
Calculating System Work and Displacement of Center of Mass
  • · Replies 15 ·
Replies
15
Views
2K
Help Me Solve My Diagram Problem: Calculating Center of Mass
  • · Replies 8 ·
Replies
8
Views
3K
Finding the POSITION of the center of mass
  • · Replies 2 ·
Replies
2
Views
2K
The center of mass of a semicircular arc of non-negligible width
  • · Replies 3 ·
Replies
3
Views
2K