The density of a 4.41-m long rod can be described by the linear density function λ(x) = 111 g/m + 12.3x g/m2. One end of the rod is positioned at x = 0 and the other at x = 4.41 m.
found total mass of rod to be:
609 grams
need help finding :
the center-of-mass coordinate.
Thank You so much!!
To find the center-of-mass coordinate of the rod, you need to calculate the average position of all the individual mass elements that make up the rod.
The linear density function λ(x) gives you the mass per unit length of the rod at any point x along its length. To find the total mass of the rod, you need to integrate this linear density function over the entire length of the rod.
To do this, you can use the formula for calculating the total mass:
M = ∫λ(x)dx
where M is the total mass and λ(x) is the linear density function.
In this case, the linear density function is given as:
λ(x) = 111 g/m + 12.3x g/m^2
To find the total mass, integrate λ(x) with respect to x from 0 to 4.41:
M = ∫(111 + 12.3x)dx from 0 to 4.41
M = [111x + 6.15x^2] from 0 to 4.41
M = 111(4.41) + 6.15(4.41)^2 - (111(0) + 6.15(0)^2)
M = 614.51 + 108.64 - 0
M = 723.15 grams
Therefore, the total mass of the rod is 723.15 grams.
Now, to find the center-of-mass coordinate, you need to consider the distribution of mass along the length of the rod. Since the rod is uniform in shape and mass distribution, the center-of-mass coordinate will be located at the midpoint of the rod's length.
The midpoint can be found by taking half of the total length of the rod:
Center-of-mass coordinate = (0 + 4.41) / 2
Center-of-mass coordinate = 2.2055 meters
Therefore, the center-of-mass coordinate of the rod is located at x = 2.2055 meters.
To find the center-of-mass coordinate, you need to calculate the average of all the individual masses multiplied by their respective positions along the rod.
The formula for the center of mass is given by:
x_cm = (∫x * λ(x) dx) / (∫λ(x) dx)
First, let's find the total mass of the rod:
m_total = ∫λ(x) dx
m_total = ∫(111 g/m + 12.3x g/m^2) dx
To find the integral of λ(x), integrate each term separately:
m_total = ∫111 dx + ∫12.3x dx
m_total = 111x | from 0 to 4.41 + 6.15x^2 / 2 | from 0 to 4.41
Evaluate the integral:
m_total = 111(4.41) - 111(0) + (6.15(4.41)^2) / 2 - (6.15(0)^2) / 2
m_total = 489.51 + 279.490695 / 2
m_total = 489.51 + 347.352
m_total = 836.862 g
Now, let's find the integral of x * λ(x):
∫x * λ(x) dx = ∫x * (111 g/m + 12.3x g/m^2) dx
∫x * λ(x) dx = ∫111x dx + ∫12.3x^2 dx
Evaluate each integral:
∫111x dx = 111 * (x^2/2) | from 0 to 4.41
∫111x dx = 111 * (4.41^2 / 2) - 111 * (0^2 / 2)
∫111x dx = 111 * (97.4889 / 2)
∫111x dx = 5356.725 / 2
∫111x dx = 2678.3625
∫12.3x^2 dx = 12.3 * (x^3/3) | from 0 to 4.41
∫12.3x^2 dx = 12.3 * (4.41^3 / 3) - 12.3 * (0^3 / 3)
∫12.3x^2 dx = 12.3 * (88.9342937 / 3)
∫12.3x^2 dx = 289.9399251 / 3
∫12.3x^2 dx = 96.6466417
Now, substitute the values into the center of mass formula:
x_cm = (∫x * λ(x) dx) / (∫λ(x) dx)
x_cm = (2678.3625 + 96.6466417) / 836.862
x_cm = 2775.0091417 / 836.862
x_cm ≈ 3.313 m
Therefore, the center-of-mass coordinate of the rod is approximately 3.313 m.