A circular plate of uniform thickness has a diameter of 56cm.A circular portion of diameter 42cm is removed from one edge of the plate .find the position of the center of mass of the remaining portion.
To find the position of the center of mass of the remaining portion, we need to consider the circular plate as a two-dimensional object in the x-y plane.
Let's assume that the origin (0,0) of the Cartesian coordinate system is at the center of the original circular plate.
1. First, calculate the area of the remaining portion:
- The area of the original circular plate is given by: A_plate = π * (radius_plate^2)
- Since the diameter of the plate is 56 cm, the radius (r_plate) is half of that: r_plate = 56 cm / 2 = 28 cm
- The area of the plate is: A_plate = π * (28 cm)^2
- The area of the removed circular portion is also given by: A_removed = π * (radius_removed^2)
- Since the diameter of the removed portion is 42 cm, the radius (r_removed) is half of that: r_removed = 42 cm / 2 = 21 cm
- The area of the removed portion is: A_removed = π * (21 cm)^2
- The area of the remaining portion is given by: A_remaining = A_plate - A_removed
2. Next, we need to find the x-coordinate and y-coordinate of the center of mass separately:
- The x-coordinate of the center of mass (x_com) is given by: x_com = (Σ(xi * Ai)) / A_remaining
- The y-coordinate of the center of mass (y_com) is given by: y_com = (Σ(yi * Ai)) / A_remaining
- Σ(xi * Ai) and Σ(yi * Ai) are the sums of the products of the individual coordinates (xi, yi) of every small area element with their corresponding areas (Ai).
3. We split the remaining portion into infinitesimally small concentric rings and integrate them to find the x-com and y-com.
- The x-coordinate of each small area element is the x-component of its position.
- Similarly, the y-coordinate of each small area element is the y-component of its position.
- We integrate over the remaining portion of the plate.
Once you calculate the x-com and y-com, you will have the position of the center of mass of the remaining portion.
To find the position of the center of mass of the remaining portion of the circular plate, we need to consider the geometrical properties of both the remaining portion and the portion that was removed.
1. Calculate the area of the remaining portion:
- The area of a circle is given by A = πr^2, where r is the radius.
- The radius of the remaining portion is half the diameter, which is 56 cm / 2 = 28 cm.
- Therefore, the area of the remaining portion is A_remaining = π(28 cm)^2.
2. Calculate the area of the portion that was removed:
- The radius of the removed portion is half the diameter, which is 42 cm / 2 = 21 cm.
- Therefore, the area of the removed portion is A_removed = π(21 cm)^2.
3. Calculate the position of the center of mass:
- The center of mass of an object is a point where the total mass can be considered to be concentrated.
- Since the circular plate has uniform thickness, we can assume a uniform mass distribution.
- The center of mass of a circular area (assuming uniform distribution) is at a point located 2/3 of the radius from the center.
- Therefore, the distance from the center of the remaining portion to its center of mass is (2/3) * 28 cm.
- Similarly, the distance from the center of the removed portion to its center of mass is (2/3) * 21 cm.
4. Calculate the total mass:
- Since the circular plate has uniform thickness, the mass is proportional to the area.
- The total mass of the remaining portion is proportional to the area of the remaining portion, which is A_remaining.
- The total mass of the removed portion is proportional to the area of the removed portion, which is A_removed.
- Therefore, the total mass can be considered as the ratio of A_remaining to the sum of A_remaining and A_removed, multiplied by the total mass of the circular plate.
5. Calculate the position of the center of mass of the remaining portion:
- The position of the center of mass of the remaining portion is the weighted average of the distances from the centers of each portion to their respective centers of mass, weighted by their masses.
- Using the ratio of masses calculated in step 4, the position of the center of mass of the remaining portion can be calculated as:
- (distance remaining * mass remaining + distance removed * mass removed) / (mass remaining + mass removed).
Now we can substitute the values and calculate:
A_remaining = π(28 cm)^2
A_removed = π(21 cm)^2
distance remaining = (2/3) * 28 cm
distance removed = (2/3) * 21 cm
total mass = mass remaining + mass removed
mass remaining = (A_remaining / (A_remaining + A_removed)) * mass total
mass removed = (A_removed / (A_remaining + A_removed)) * mass total
Finally, substitute these values into the formula for the position of the center of mass of the remaining portion:
position of center of mass = (distance remaining * mass remaining + distance removed * mass removed) / (mass remaining + mass removed)
The removed portion has 3/4 the diameter and 9/16 of the area and mass M of the full circular (56 cm dia) plate.
Measure CM location from the center of the larger full circle. The center of the cutout circular hole is 7 cm from the center of the large circle
[M - (9/16)M]*x = M*0 - (9/16)M*7
(7/16)x = -63/16
x = 63/7 = -9.0 cm
The CM is 9 cm from the center of the large circle, in the opposite direction from the center of the removed circle