Question

Consider a rod of length L, whose mass density is given by λ = C(1 + ax^2), where x is the distance from the light end and C is a constant with a dimension of mass per unit length. Calculate the CM of the rod.

Answer 1

To calculate the center of mass of the rod with varying mass density, you can use the formula: CM = (1/total mass) * ∫(x * dm) = (1/total mass) * ∫(x * λ(x) * dx), where x is the distance from the light end, dm is the mass of an infinitesimally small element of the rod, and λ(x) = C(1 + ax^2) is the mass density function.

To calculate the center of mass (CM) of the rod with varying mass density, we can use the formula:

CM = (1/total mass) * ∫(x * dm) = (1/total mass) * ∫(x * λ(x) * dx)

where x is the distance from the light end, dm is the mass of an infinitesimally small element of the rod, and λ(x) = C(1 + ax^2) is the mass density function.

We can integrate this equation over the entire length of the rod (0 to L).

The total mass of the rod can be found by integrating the mass density function over the length of the rod:

Total mass = ∫λ(x) * dx = ∫C(1 + ax^2) * dx

After finding the total mass and evaluating the integrals, we can substitute the values into the CM equation to calculate the center of mass of the rod.