A straight road of length L has one of its end at the origin and other at x = L. if the mass the unit length of rod is Ax where a is A constant then find its centre of mass.
To find the center of mass of a straight rod with a variable mass per unit length, we need to integrate the position of each infinitesimally small segment of the rod and divide it by the total mass of the rod.
Let's denote the position of an infinitesimally small segment of length dx as x. The mass of this segment is Axdx, where A is the constant mass per unit length. The total mass of the rod is then given by integrating Axdx from x = 0 to x = L.
Total mass (M) = ∫(0 to L) Ax*dx = A∫(0 to L) dx = A[x] (0 to L) = AL
Now, let's find the center of mass (CM). The position of each segment is given by x, and we need to integrate x*(Ax*dx) from x = 0 to x = L, and divide it by the total mass.
CM = (∫(0 to L) x(Axdx)) / (AL)
= (∫(0 to L) Ax^2dx) / (AL)
= A∫(0 to L) x^2dx / (AL)
= 1/A * (∫(0 to L) x^2dx) / (∫(0 to L) dx)
= 1/A * (x^3/3) (0 to L) / (x (0 to L))
= 1/A * (L^3/3) / L
= L^2 / (3A)
Therefore, the center of mass of the rod is located at x = L^2 / (3A).
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Explanation:
To find the center of mass of a straight rod with a variable mass per unit length, we need to integrate the position of each infinitesimally small segment of the rod and divide it by the total mass of the rod.
Let's denote the position of an infinitesimally small segment of length dx as x. The mass of this segment is Axdx, where A is the constant mass per unit length. The total mass of the rod is then given by integrating Axdx from x = 0 to x = L.
Total mass (M) = ∫(0 to L) Ax*dx = A∫(0 to L) dx = A[x] (0 to L) = AL
Now, let's find the center of mass (CM). The position of each segment is given by x, and we need to integrate x*(Ax*dx) from x = 0 to x = L, and divide it by the total mass.
CM = (∫(0 to L) x(Axdx)) / (AL)
= (∫(0 to L) Ax^2dx) / (AL)
= A∫(0 to L) x^2dx / (AL)
= 1/A * (∫(0 to L) x^2dx) / (∫(0 to L) dx)
= 1/A * (x^3/3) (0 to L) / (x (0 to L))
= 1/A * (L^3/3) / L
= L^2 / (3A)
Therefore, the center of mass of the rod is located at x = L^2 / (3A).