Question

slender uniform rod 100 cm long is used as a meter stick. Consider twoparallel axes that are perpendicular to the rod. The first axis passes throughthe 50-cm mark and the second axis passes through the 30-cm mark. Whatis the ratio of the moment of inertia through the second axis to the momentof inertia through the first axis?

Answer 1

The ratio of the moment of inertia through the second axis to the moment of inertia through the first axis is 1.48.

Step-by-step explanation:

Given that,

Length of rod = 100 cm

The first axis passes through the 50-cm mark and the second axis passes through the 30-cm mark.

We need to calculate the moment of inertia when the axis passing through

Using formula of moment of inertia

`I_(1)=(1)/(12)ml^2`

Put the value into the formula

`I_(1)=(1)/(12)m*(1*10^(-2))^2`

We need to calculate the distance from center of mass

`x=50-30= 20 cm`

We need to calculate the moment of inertia when the axis passing through

Using formula of moment of inertia

`I_(2)=I_(1)+mx^2`

Put the value into the formula

`I_(2)=(1)/(12)m*(1*10^(-2))^2+m*(20*10^(-2))^2`

`I_(2)=m((1)/(12)*(1.00)^2+(0.2)^2)`

`I_(2)=m((1)/(12)+0.04)`

We need to calculate the ratio of the moment of inertia through the second axis to the moment of inertia through the first axis

Using formula of ration of moment of inertia

`(I_(2))/(I_(1))=(m*(1.48)/(12))/((1)/(12)m*(1*10^(-2))^2)`

`(I_(2))/(I_(1))=1.48`

Hence, The ratio of the moment of inertia through the second axis to the moment of inertia through the first axis is 1.48.