Four very thin rods, each 13 m long, are joined to form a square, as part (a) of the drawing shows. The center of mass of the square is located at the coordinate origin. The rod on the right is then removed, as shown in part (b) of the drawing. What are the x- and y-coordinates of the center of mass of the remaining three-rod system?
To calculate the x- and y-coordinates of the center of mass of the remaining three-rod system, we need to consider the individual masses and their distances from the coordinate origin.
First, let's calculate the mass of each rod. The mass of each rod can be obtained by multiplying its length by its linear density. Since the rods are very thin, we can assume uniform linear density throughout each rod.
Let's assume the linear density of each rod is λ (lambda). Therefore, the mass of each rod, m, is given by:
m = λ * length
Given that the length of each rod is 13m, the mass of each rod is m = λ * 13.
Now, let's label the rods as A, B, C, and D. Rods A, B, and C are still part of the system after rod D is removed.
The x-coordinate of the center of mass of the three-rod system, (x_cm), can be calculated using the formula:
x_cm = (m_A * x_A + m_B * x_B + m_C * x_C) / (m_A + m_B + m_C)
Similarly, the y-coordinate of the center of mass, (y_cm), can be calculated using the formula:
y_cm = (m_A * y_A + m_B * y_B + m_C * y_C) / (m_A + m_B + m_C)
For simplicity, we can assume that the origin of the coordinate system is at the center of the original square (before rod D is removed). This means that rods A, B, and C have the coordinates (-6.5, 6.5), (6.5, 6.5), and (6.5, -6.5) respectively.
Now, let's substitute the known values into the formulas to calculate the x- and y-coordinates of the center of mass:
x_cm = ((λ * 13) * (-6.5) + (λ * 13) * (6.5) + (λ * 13) * (6.5)) / (3 * (λ * 13))
Simplifying, we get:
x_cm = (13λ * (-6.5) + 13λ * 6.5 + 13λ * 6.5) / (39λ)
x_cm = 13λ * (6.5 - 6.5 + 6.5) / (39λ)
x_cm = 13λ * 6.5 / 39λ
x_cm = 6.5 / 3
x_cm = 2.17 (rounded to two decimal places)
Similarly, calculating the y-coordinate:
y_cm = ((λ * 13) * 6.5 + (λ * 13) * 6.5 + (λ * 13) * (-6.5)) / (3 * (λ * 13))
Simplifying, we get:
y_cm = (13λ * 6.5 + 13λ * 6.5 + (-13λ * 6.5)) / (39λ)
y_cm = 13λ * (6.5 + 6.5 - 6.5) / (39λ)
y_cm = 13λ * 6.5 / 39λ
y_cm = 6.5 / 3
y_cm = 2.17 (rounded to two decimal places)
Therefore, the x- and y-coordinates of the center of mass of the remaining three-rod system are approximately (2.17, 2.17).