Moment of Inertia (M.I.) of four bodies having same mass ‘M’ and radius ‘2R’ are as follows:

I₂ = M.I. of solid sphere about its diameter

I₂ = M.I. of solid cylinder about its axis

I₃ = M.I. of solid circular disc about its diameter

I₄ = M.I. of thin circular ring about its diameter

if 2(I₂ + I₃) + I₄ = x. I₁ then the value of x will be…………

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alok kumar singh | Contributor-Level 10

2 months ago

l₁ = M. l of solid sphere about its diameter

$= \frac{2}{5} M R'^{2} = \frac{2}{5} M {(2 R)}^{2} = \frac{8}{5} M R^{2}$

l₂ = M. I of solid cylinder about its axis

$= \frac{M R^{12}}{2} = \frac{M {(2 R)}^{2}}{2} = 2 M R^{2}$

I₃ = M. I of solid circular disc about its diameter

$= \frac{M R^{12}}{4} = \frac{M {(2 R)}^{2}}{2} = M R^{2}$

I₄ = M. I of this circular ring about its diameter

$= \frac{M R^{12}}{2} = \frac{M {(2 R)}^{2}}{2} = 2 M R^{2}$

$\Rightarrow 6 M R^{2} + 2 M R^{2} = \frac{x 8 M R^{2}}{5}$

x = 5

l1 = M. l of solid sphere about its diameter <math> <mrow> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mn>5</mn> </mrow> </mfrac> <mi>M</mi> <mtext> </mtext> <mtext> </mtext> <mi>R</mi> <msup> <mrow> <mo>'</mo> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mn>5</mn> </mrow> </mfrac> <mi>M</mi> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>R</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mfrac> <mrow> <mn>8</mn> </mrow> <mrow> <mn>5</mn> </mrow> </mfrac> <mi>M</mi> <msup> <mrow> <mi>R</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </math>               l2 = M. I of solid cylinder about its axis <math> <mrow> <mo>=</mo> <mfrac> <mrow> <mi>M</mi> <msup> <mrow> <mi>R</mi> </mrow> <mrow> <mn>1</mn> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mi>M</mi> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>R</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>=</mo> <mn>2</mn> <mi>M</mi> <msup> <mrow> <mi>R</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </math>               I3 = M. I of solid circular disc about its diameter <math> <mrow> <mo>=</mo> <mfrac> <mrow> <mi>M</mi> <msup> <mrow> <mi>R</mi> </mrow> <mrow> <mn>1</mn> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mi>M</mi> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>R</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>=</mo> <mi>M</mi> <msup> <mrow> <mi>R</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </math>               I4 = M. I of this circular ring about its diameter  <math> <mrow> <mo>=</mo> <mfrac> <mrow> <mi>M</mi> <mtext> </mtext> <mtext> </mtext> <msup> <mrow> <mi>R</mi> </mrow> <mrow> <mn>1</mn> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mi>M</mi> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>R</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>=</mo> <mn>2</mn> <mi>M</mi> <msup> <mrow> <mi>R</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </math>               <math> <mrow> <mo>⇒</mo> <mn>6</mn> <mi>M</mi> <msup> <mrow> <mi>R</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>M</mi> <msup> <mrow> <mi>R</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mfrac> <mrow> <mi>x</mi> <mn>8</mn> <mi>M</mi> <msup> <mrow> <mi>R</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>5</mn> </mrow> </mfrac> </mrow> </math>               x = 5

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