Question

You have two solid steel spheres; sphere 2 has twice the radius of sphere 1. what is the ratio of the moments of inertia, i2 : i1?

Answer 1

Answer:

$\dfrac{I_2}{I_1}=\dfrac{32}{1}$

Explanation:

The momentum of inertia of the solid sphere is :

$I_1=\dfrac{2}{5}mr^2$

m is the mass of solid sphere

r is the radius of sphere

Since, $m=\rho V$

$m=\dfrac{4\pi r^3\rho }{3}$

$I_1=\dfrac{2}{5}r^2\dfrac{4\pi r^3\rho }{3}$........................(1)

Let r' is the radius of second sphere such that, r' = 2r. New moment of inertia is given by :

$I_2=\dfrac{2}{5}mr'^2$

$I_2=\dfrac{2}{5}m(2r)'^2$

Similarly,

$I_2=\dfrac{2}{5}r'^2\dfrac{4\pi r'^3\rho }{3}$.

$I_2=\dfrac{2}{5}(2r)^2\dfrac{4\pi (2r)^3\rho }{3}$........................(2)

Dividing equation (1) and (2) as :

$\dfrac{I_2}{I_1}=\dfrac{2^5}{1}$

$\dfrac{I_2}{I_1}=\dfrac{32}{1}$

So, the ratio of their moments off inertia is 32:1

Answer 2

The ratio of the moments of inertia of the two solid steel spheres,

I₂ : I₁ = 32 : 1

$\texttt{ }$

Further explanation

Given:

radius of sphere 1 = R₁ = R

radius of sphere 2 = R₂ = 2R

Asked:

ratio of the moments of inertia = I₂ : I₁ = ?

Solution:

We will use moment of inertia for sphere as follows:

$I_2 : I_1 = \frac{2}{5} m_2 (R_2)^2 : \frac{2}{5} m_1 (R_1)^2$

$I_2 : I_1 = \frac{2}{5} \rho V_2 (R_2)^2 : \frac{2}{5} \rho V_1 (R_1)^2$

$I_2 : I_1 = V_2 (R_2)^2 : V_1 (R_1)^2$

$I_2 : I_1 = [\frac{4}{3} \pi (R_2)^3] (R_2)^2 : [\frac{4}{3} \pi (R_1)^3] (R_1)^2$

$I_2 : I_1 = (R_2)^5 : (R_1)^5$

$I_2 : I_1 = (2R)^5 : (R)^5$

$I_2 : I_1 = 32R^5 : R^5$

$I_2 : I_1 = 32 : 1$

$\texttt{ }$

Conclusion:

The ratio of the moments of inertia, I₂ : I₁ = 32 : 1

$\texttt{ }$

Learn more

• Impacts of Gravity : brainly.com/question/5330244
• Effect of Earth’s Gravity on Objects : brainly.com/question/8844454
• The Acceleration Due To Gravity : brainly.com/question/4189441
• Moment of Inertia of Four Small Spheres : brainly.com/question/6760068

$\texttt{ }$

Answer details

Grade: High School

Subject: Physics

Chapter: Circular Motion