To solve this problem, we need to find the center of mass of a composite system consisting of two parts: a circular plate and a square plate, both having the same density and thickness.
Step 1: Understand the system
• The circular plate has a diameter aa, so its radius r=a2r = \frac{a}{2}.
• The square plate has an edge length aa.
• The density and thickness are the same for both plates, so the mass of each plate will depend only on the area.
Step 2: Center of mass of individual objects
• The center of mass of the circular plate is at its geometric center, which is at the center of the plate.
• The center of mass of the square plate is at its geometric center, which is the midpoint of the square.
Step 3: Position of the center of mass of the composite system
Since both plates have the same density and thickness, the mass of each plate is proportional to its area.
1. Area of the circular plate: Acircle=π(a2)2=πa24A_{\text{circle}} = \pi \left(\frac{a}{2}\right)^2 = \frac{\pi a^2}{4}
2. Area of the square plate: Asquare=a2A_{\text{square}} = a^2
Step 4: Formula for the center of mass of the system
The position of the center of mass xCMx_{\text{CM}} of the system is given by the weighted average of the positions of the centers of mass of the individual plates, weighted by their areas (masses):
xCM=Acirclexcircle+AsquarexsquareAcircle+Asquarex_{\text{CM}} = \frac{A_{\text{circle}} x_{\text{circle}} + A_{\text{square}} x_{\text{square}}}{A_{\text{circle}} + A_{\text{square}}}
Where:
• xcirclex_{\text{circle}} is the position of the center of mass of the circular plate (which is at the center of the circle),
• xsquarex_{\text{square}} is the position of the center of mass of the square plate (which is at the center of the square).
Since the circular plate is placed in contact with the square plate, let's assume their centers are aligned along the x-axis, with the square's center at xsquare=a2x_{\text{square}} = \frac{a}{2} and the circle’s center at xcircle=0x_{\text{circle}} = 0 (if we place the origin at the center of the circular plate).
Step 5: Calculate the center of mass position
Substituting the values into the formula for xCMx_{\text{CM}}:
xCM=(πa24)(0)+(a2)(a2)πa24+a2x_{\text{CM}} = \frac{\left(\frac{\pi a^2}{4}\right)(0) + (a^2)\left(\frac{a}{2}\right)}{\frac{\pi a^2}{4} + a^2}
Simplifying:
xCM=a3πa24+a2=a3a2(π4+1)=aπ4+1x_{\text{CM}} = \frac{a^3}{\frac{\pi a^2}{4} + a^2} = \frac{a^3}{a^2 \left(\frac{\pi}{4} + 1\right)} = \frac{a}{\frac{\pi}{4} + 1}
This value is not zero, meaning the center of mass lies somewhere between the centers of the two plates.
Conclusion:
Since the center of mass is not exactly at the point of contact and lies somewhere between the two plates, the correct answer is:
C. At the point of contact.
