Center Of Mass

Step 1: The position of the centre of mass of a uniform plate can be calculated by using the formula for the centre of mass of a rectangular body. For a uniform rectangular plate of dimensions $a%20%5Ctimes%20b$ , the centre of mass lies at the intersection of the diagonals.

Step 2: For a uniform plate, the coordinates of the centre of mass are given by:

$%5Cleft(%20%5Cfrac%7Ba%7D%7B2%7D%2C%20%5Cfrac%7Bb%7D%7B2%7D%20%5Cright)$

where $a$ and $b$ are the length and width of the plate, respectively, and the origin is taken at one corner of the plate.

Step 3: If the plate is located such that the origin is at a point shifted from the center of the plate, the position of the centre of mass will be shifted accordingly.

Step 4: The position of the centre of mass relative to the given origin in the figure (assuming a symmetric uniform plate) will be:

$%5Cleft(%20-%5Cfrac%7Ba%7D%7B6%7D%2C%20-%5Cfrac%7Bb%7D%7B6%7D%20%5Cright)$

This represents the correct coordinates for the centre of mass based on the dimensions and position of the plate in the figure.

Step 1: Understanding the Problem The mass of a small element of the rod at a distance

x

from one end is given by:

dm%20%3D%20%5Crho(x)%20%5C%2C%20dx%20%3D%20%5Crho_0%20%5Cfrac%7Bx%5E2%7D%7BL%5E2%7D%20%5C%2C%20dx

The position of the center of mass is given by the formula:

x_%7B%5Ctext%7Bcm%7D%7D%20%3D%20%5Cfrac%7B%5Cint_0%5EL%20x%20%5C%2C%20dm%7D%7B%5Cint_0%5EL%20dm%7D

Where: - The numerator is the first moment of mass, - The denominator is the total mass of the rod. Step 2: Total Mass of the Rod The total mass

M

of the rod is obtained by integrating the mass element

dm

over the length of the rod:

M%20%3D%20%5Cint_0%5EL%20%5Crho(x)%20%5C%2C%20dx%20%3D%20%5Cint_0%5EL%20%5Crho_0%20%5Cfrac%7Bx%5E2%7D%7BL%5E2%7D%20%5C%2C%20dx%20%3D%20%5Crho_0%20%5Cfrac%7B1%7D%7BL%5E2%7D%20%5Cint_0%5EL%20x%5E2%20%5C%2C%20dx

The integral of

x%5E2

from 0 to

L

is:

%5Cint_0%5EL%20x%5E2%20%5C%2C%20dx%20%3D%20%5Cfrac%7BL%5E3%7D%7B3%7D

Thus:

M%20%3D%20%5Crho_0%20%5Cfrac%7B1%7D%7BL%5E2%7D%20%5Ccdot%20%5Cfrac%7BL%5E3%7D%7B3%7D%20%3D%20%5Cfrac%7B%5Crho_0%20L%7D%7B3%7D

Step 3: First Moment of Mass The first moment of mass is:

%5Cint_0%5EL%20x%20%5C%2C%20dm%20%3D%20%5Cint_0%5EL%20x%20%5Crho_0%20%5Cfrac%7Bx%5E2%7D%7BL%5E2%7D%20%5C%2C%20dx%20%3D%20%5Crho_0%20%5Cfrac%7B1%7D%7BL%5E2%7D%20%5Cint_0%5EL%20x%5E3%20%5C%2C%20dx

The integral of

x%5E3

from 0 to

L

is:

%5Cint_0%5EL%20x%5E3%20%5C%2C%20dx%20%3D%20%5Cfrac%7BL%5E4%7D%7B4%7D

Thus:

%5Cint_0%5EL%20x%20%5C%2C%20dm%20%3D%20%5Crho_0%20%5Cfrac%7B1%7D%7BL%5E2%7D%20%5Ccdot%20%5Cfrac%7BL%5E4%7D%7B4%7D%20%3D%20%5Cfrac%7B%5Crho_0%20L%5E2%7D%7B4%7D

Step 4: Position of the Center of Mass Now we can calculate the position of the center of mass:

x_%7B%5Ctext%7Bcm%7D%7D%20%3D%20%5Cfrac%7B%5Cfrac%7B%5Crho_0%20L%5E2%7D%7B4%7D%7D%7B%5Cfrac%7B%5Crho_0%20L%7D%7B3%7D%7D%20%3D%20%5Cfrac%7B3L%7D%7B4%7D

Step 5: Conclusion The position of the center of mass from one end is

%5Cfrac%7B3L%7D%7B4%7D

. Thus, the correct answer is:

%5Cboxed%7B(D)%7D%20%5C%2C%20%5Cfrac%7B3L%7D%7B4%7D

High-Yield Trend

Chapter Questions
3 MCQs

Official Solution

Official Solution

Official Solution

Center Of Mass

High-Yield Trend

Chapter Questions 3 MCQs

Official Solution

Official Solution

Official Solution

Chapter Questions
3 MCQs