Question

Define moment of inertia and radius of gyration. Prove that I=2E. Derive the M.I. of a spherical shell about its diameter.​

Answer 1

Answer:

Mass per unit volume of the shell =ρ" role="presentation" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: 0px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(0, 0, 0); font-family: Lato, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; background-color: rgb(224, 234, 255); text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial; position: relative;">=ρ=ρ

ρ=M(43)π(R3-r3)" role="presentation" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: 0px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(0, 0, 0); font-family: Lato, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; background-color: rgb(224, 234, 255); text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial; position: relative;">ρ=M(43)π(R3−r3)ρ=M(43)π(R3-r3)

=3M4π(R3-r3)" role="presentation" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: 0px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(0, 0, 0); font-family: Lato, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; background-color: rgb(224, 234, 255); text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial; position: relative;">=3M4π(R3−r3)=3M4π(R3-r3)

M.I. of the solid sphere of radius R about a diameter

=23×mass×(radius)2" role="presentation" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: 0px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(0, 0, 0); font-family: Lato, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; background-color: rgb(224, 234, 255); text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial; position: relative;">=23×mass×(radius)2=23×mass×(radius)2

=25×43πR3ρR2" role="presentation" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: 0px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(0, 0, 0); font-family: Lato, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; background-color: rgb(224, 234, 255); text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial; position: relative;">=25×43πR3ρR2=25×43πR3ρR2

=815πR5ρ" role="presentation" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: 0px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px;