MI of rectangular and triangular lamina

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The discussion centers on calculating the moment of inertia (MI) for rectangular and triangular lamina, specifically seeking links and formulas for these calculations. The formula I = ∫ r² dm is highlighted, focusing on two-dimensional area density for lamina. A user inquires about the limits of integration when deriving the MI for a rectangular lamina with specified dimensions. Additionally, there is confusion regarding the concept of moment of inertia about a point, as it is noted that such a definition is problematic in one dimension. The conversation emphasizes the need for clarity in understanding how MI is defined across different dimensions.
Mansi Khanna
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Homework Statement




can anyone please send me the link to calculate the moment of inertia of cuboid,rectangular and triangular lamina ??(along with figure)

Homework Equations





The Attempt at a Solution

 
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Welcome to PF.

Just use the formula:

I = ∫ r2 dm

Taken over all the mass elements.

Since you are interested in lamina then you're only interested in two dimensional area density.
 
thank you so much for your help..
while deriving the formula for rectangular lamina,of sides 2a and 2b(i.e.,length 2a and breadth 2b),when i consider a thin horizontal strip parallel to x-axis at distance y from x-axis and infinitesimally small thickness dy,
after using the formula you have given..what limits of the integral should i take??please help..
one last thing i need to know is that in the link which you have sent,under extended explanation someone's written
WARNING: there is no such thing as moment of inertia about a point
and then the definition of moment of inertia about a point is mentioned...if there's no such thing as moment of inertia about a point then how do we define MI in 1 dimension??
like MI in 2-d is about a line and MI in 3-d is about coordinate axis..then in 1-d is it not supposed to be about a point??
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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