Find the center of mass of a rectangular plate of length 10 cm and width 39 cm. The mass density varies linearly along the length. At one end, it is 8 g/cm2; at the other end, it is 18 g/cm2. (Assume the x-axis begins at the 8 g/cm2 edge and extends along the length. Assume the y-axis extends along the width.)
(??? cm, 19.5 cm)
To find the center of mass of the rectangular plate, we need to determine the position of the center of mass along the x-axis (length) and the y-axis (width).
Let's start by finding the position of the center of mass along the x-axis:
1. Convert the mass density from grams per square centimeter to grams per square millimeter to match the dimensions of the plate.
At one end, the mass density is 8 g/cm², which is equivalent to 8 g/(10 mm * 39 mm).
At the other end, the mass density is 18 g/cm², which is equivalent to 18 g/(10 mm * 39 mm).
2. Calculate the average mass density along the x-axis.
The average mass density can be calculated by taking the average of the two mass densities: (8 g/(10 mm * 39 mm) + 18 g/(10 mm * 39 mm)) / 2.
3. Determine the position of the center of mass along the x-axis.
The position of the center of mass along the x-axis can be calculated using the formula:
x_cm = (Σ(x_i * m_i)) / Σm_i
Where x_i is the distance from the x-axis (distance along the length) and m_i is the mass associated with that position.
In this case, since the mass density varies linearly along the x-axis, we can assume equal masses at equal distances.
Therefore, the x_cm can be calculated as:
x_cm = (10 cm/2) = 5 cm (since x-axis begins at the 8 g/cm² edge)
Now, let's find the position of the center of mass along the y-axis:
4. Determine the position of the center of mass along the y-axis.
Since the density variation along the y-axis is not given, we assume a uniform density distribution along the width. Therefore, the center of mass along the y-axis is simply halfway through the width:
y_cm = (39 cm) / 2 = 19.5 cm
So, the center of mass of the rectangular plate is located at (5 cm, 19.5 cm).