Question

b. Find the center of mass if, instead of being constant, the density function is δ(x)=4/√x.

Accepted Answer

Identify the interval over which the object extends. Typically, for a density function $$ \delta(x) = \frac{4}{\sqrt{x}} $$, the domain is $$ x \u003e 0 . $$Confirm the specific interval $$[a, b]$$ for the problem, as the center of mass depends on this range. Recall the formula for the center of mass $$ \bar{x}  of a $$one-dimensional object with variable density $$ \delta(x) $$:

$$
\bar{x} = \frac{\int_a^b x \delta(x) \, dx}{\int_a^b \delta(x) \, dx}
$$

This formula represents the weighted average position, where the weights are given by the density function. Set up the numerator integral:

$$
\int_a^b x \cdot \frac{4}{\sqrt{x}} \, dx = \int_a^b 4x^{1 - \frac{1}{2}} \, dx = \int_a^b 4x^{\frac{1}{2}} \, dx
$$

This integral calculates the moment of the mass distribution about the origin. Set up the denominator integral:

$$
\int_a^b \frac{4}{\sqrt{x}} \, dx = \int_a^b 4x^{-\frac{1}{2}} \, dx
$$

This integral calculates the total mass of the object over the interval $$[a, b]. $$Evaluate both integrals using the power rule for integration:

$$
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad 	ext{for} \quad n 
eq -1
$$

After evaluating, substitute the limits $$a$$ and $$b$$ into both integrals, then divide the numerator by the denominator to find the center of mass $$ \bar{x} .$$

b. Find the center of mass if, instead of being constant, the density function is δ(x)=4/√x.

Risposta video verificata per un problema simile:

Concetti chiave

Center of Mass

Density Function

Definite Integrals in Calculus