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 can use the formula:
x_cm = (1/M) * ∫(x * dm)
where x_cm is the center-of-mass coordinate, M is the total mass of the rod, x is the position along the length of the rod, and dm is an infinitesimally small mass element.
In this case, the linear density function of the rod is given by λ(x) = 111 g/m + 12.3x g/m^2. We need to find the total mass of the rod, which you have already calculated as 609 grams.
Now, let's calculate the center-of-mass coordinate step by step:
1. Express the linear density function in terms of x:
λ(x) = 111 g/m + 12.3x g/m^2
2. Integrate the linear density function with respect to x to find the mass element dm:
dm = λ(x) * dx
= (111 g/m + 12.3x g/m^2) * dx
3. Use the limits of integration (x = 0 to x = 4.41 m) and substitute the values of x and dm into the integral:
x_cm = (1/M) * ∫(x * dm)
= (1/609 g) * ∫(x * (111 g/m + 12.3x g/m^2)) dx
= (1/609) * ∫(111x + 12.3x^2) dx
4. Evaluate the integral:
x_cm = (1/609) * ((111/2) * x^2 + (12.3/3) * x^3) |_0^4.41
= (1/609) * ((111/2) * (4.41)^2 + (12.3/3) * (4.41)^3)
5. Calculate the value of x_cm using the above expression:
x_cm ≈ 1.48 m
Therefore, the center-of-mass coordinate of the rod is approximately 1.48 meters.