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"(c) 2016-2017 Colin B. Macdonald, CC-BY licensed. Supplementary notes for UBC's Math 253, to follow Section 13.4 \"Center of Mass\" of our text APEX Calculus 3, version 3.0.\n",
"\n",
"Moments of Inertia\n",
"==================\n",
"\n",
"We've previously seen _moments_ when calculating centre of mass of a lamina.\n",
"This involved two double integrals:\n",
"\n",
" $$ M_y = \\int\\int_D x \\rho(x,y)dA $$\n",
" $$ M_x = \\int\\int_D y \\rho(x,y)dA $$\n",
" \n",
"These can also be called the \"first moments\"; here we look at the \"second moments\"\n",
"or \"moments of inertia\"."
]
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"Kinetic Energy of a spinning lamina\n",
"-----------------------------------\n",
"\n",
"Suppose our lamina (which lies in the $x$-$y$ plane) is rotating around the $z$-axis\n",
"(note this is orthogonal to the lamina) at a constant angular rotational speed\n",
"$\\omega$ radians/s. (E.g., 60 rpm = 1 rev/s = $2\\pi$ rad/s).\n",
"Find the *Kinetic Energy* of the lamina. [Draw diagram!]\n",
"\n",
"\n",
"Riemann sum idea: as before, we consider a small rectangular piece $R_{ij}$ with area\n",
"$\\Delta x \\Delta y$. The kinetic energy of a point mass is $\\frac{1}{2} m v^2$.\n",
"Its going to be small in the limit so we use this to get:\n",
"\n",
" $$ \\frac{1}{2} \\rho(x_i, y_j) \\Delta x \\Delta y |\\vec{v}_{ij}|^2. $$\n",
" \n",
"The piece $R_{ij}$ moves faster the further it is from the axis of rotation ($z$-axis, $(x,y)=(0,0)$).\n",
"Different pieces move at different speeds. Our piece has kinetic energy:\n",
"\n",
" $$ \\frac{1}{2} \\rho(x_i, y_j) \\Delta x \\Delta y \\omega^2 \\left(x_i^2 + y_j^2)\\right). $$\n",
" \n",
"So take the Riemann sum over all pieces of the lamina and we get:\n",
"\n",
" $$K = \\frac{1}{2} \\omega^2 \\int\\int_D (x^2+y^2) \\rho(x,y) dA. $$\n",
" \n",
"We define $I_0$ the **moment of inertia** about the $z$-axis as just the integral part:\n",
"\n",
" $$ I_0 = \\int\\int_D (x^2+y^2) \\rho(x,y) dA. $$\n",
"\n",
"Larger $I_0$ means more energy (work) to rotate the lamina about the $z$-axis."
]
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"About some other axis?\n",
"----------------------\n",
"\n",
"A similar argument shows how to compute the moment of inertia about some other axis parallel to the $z$-axis,\n",
"centred at $(x,y) = (a,b)$:\n",
"\n",
" $$ I_0 = \\int\\int_D ((x-a)^2+(y-b)^2) \\rho(x,y) dA. $$\n",
" \n",
"And in particular about the centre of mass $(x,y) = (\\bar{x}, \\bar{y})$, this would be:\n",
"\n",
" $$I_{0,c} = \\hspace{30ex} $$"
]
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"Rotation around $x$ or $y$ axes\n",
"----------------------------\n",
"\n",
"What about rotating around the $x$-axis and $y$-axis? This gives the moment of inertia about the $y$-axis denoted $I_y$ and the moment of inertia about the $x$-axis denoted $I_x$. [Draw diagrams]\n",
" \n",
" $$ I_y = \\hspace{30ex} $$\n",
"\n",
" $$ \\phantom{x} $$\n",
"\n",
" $$ I_x = \\hspace{30ex} $$\n",
" \n",
"Note relationship to previous,\n",
"\n",
" $$I_0 = \\hspace{30ex} $$"
]
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"Changing the axis of rotation\n",
"-----------------------------\n",
"\n",
"Suppose we have $I_{0,c}$ and want rotation around $z$-axis? Let $M$ be overall mass of lamina.\n",
"We get:\n",
"\n",
"$$I_0 = \\hspace{30ex} $$\n",
"\n",
"Examples\n",
"--------\n",
"\n",
"1. Find moment of inertia about the $z$-axis of a uniform circular disc of radius $R$ and total mass $M$, centred at the origin.\n",
"\n",
"2. Find same, but with disc centred at point $(a,b)$.\n",
"\n",
"3. Find same, for a uniform rectangular plate, mass $M$, axis through centre, size $a \\times b$.\n"
]
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