Question

A rod of length 30.0 cm has linear density (mass per length) given by l 5 50.0 1 20.0x where x is the distance from one end, measured in meters, and l is in grams/meter. (a) What is the mass of the rod? (b) How far from the x 5 0 end is its center of mass?

Answer 1

Answer:

(a). The mass of the rod is 15.9 g.

(b). The center of mass is 0.153 m.

Explanation:

Given that,

Length = 30.0 cm

Linear density $\labda=50.0+20.0x$

We need to calculate the mass of rod

Using formula of mass

$M=\int{dm}$

$M=\int{(50.0+20.0x)dx}$

$M=50.0x+10x^2$

Put the value of x

$M=50.0\times0.30+10\times(0.30)^2$

$M=15.9\ g$

We need to calculate center of mass

The center of mass has an x coordinate is given by

$x_{cm}=\dfrac{\int{xdm}}{\int{dm}}$

We need to calculate the value of  $\int{xdm}$

$\int{xdm}=\int{(50.0x+20.0x^2)dx}$

$\int{xdm}=25x^2+\dfrac{20}{3}x^3$

Put the value into the formula

$\int{xdm}=25\times0.3^2+\dfrac{20}{3}\times(0.3)^3$

$\int{xdm}=2.43$

Put the value into the formula of center of mass

$x_{cm}=\dfrac{2.43}{15.9}$

$x_{cm}=0.153\ m$

Hence, (a). The mass of the rod is 15.9 g.

(b). The center of mass is 0.153 m.