Question

a rod with density δ(x)=2 sin(x) (in mass per unit length) lies on the x-axis between x=0 and x=2π/3 . find the center of mass of the rod.

Answer 1

The center of mass of the rod with density δ(x) = 2 sin(x) along the x-axis between x=0 and x=2π/3 is located at x = π/3.

Step-by-step explanation:

The center of mass for a continuous rod along an axis is calculated using the formula:
`\(\bar{x} = (\int_(a)^(b) x \cdot \delta(x) \,dx)/(\int_(a)^(b) \delta(x) \,dx)\)` , where
`\(\delta(x)\)` represents the

density function. For this rod, the density is given as δ(x) = 2 sin (x) over the interval x=0 to x=2π/3.

To find the center of mass, integrate
`\(x \cdot \delta(x)\)` and
`\(\delta(x)\)` separately over the given interval. For the numerator, integrate
`\(x \cdot \delta(x)\)` from 0 to 2π/3, which gives
`\(\int_(0)^(2π/3) x \cdot 2 \sin(x) \,dx\)` . For the denominator, integrate δ(x) from 0 to 2π/3, resulting in
`\(\int_(0)^(2π/3) 2 \sin(x) \,dx\)` .

Solving these integrals, we get the values of the numerator and denominator. After dividing the numerator by the denominator, the center of mass
`(\(\bar{x}\))` is computed to be π/3, indicating that the center of mass of the rod lies at x = π/3 along the x-axis. This means that the balancing point of the rod is at this position considering its varying density along its length.