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Calculus of a Single Variable

11th Edition
Ron Larson + 1 other
ISBN: 9781337275361

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BuyFindarrow_forward

Calculus of a Single Variable

11th Edition
Ron Larson + 1 other
ISBN: 9781337275361
Textbook Problem

Center of Mass of a Planar Lamina In Exercises 15-28, find M x , M y , and (x, y) for the lamina of uniform density p bounded by the graphs of the equations.

y = 4 y 2 , x = 0

To determine

To calculate: The Mx, My and (x¯,y¯) for the lamina bounded by the functions,

x=4y2,x=0 of uniform density ρ.

Explanation

Given:

The lamina is bounded by the functions,

x=4y2,x=0

Formula used:

The moments about x- axis and y- axis of a planar lamina are

My=ρab[f(y)+g(y)2][f(y)g(y)]dy and Mx=ρaby[f(y)g(y)]dy

The Center of mass (x¯,y¯) is x¯=Mym and y¯=Mxm

The mass of lamina is m=ρab[f(y)g(y)]dy

Calculation:

The graph of the givenequations,x=4y2,x=0 is as shown below.

Here, the lower bound is y=2 and upper bound is y=2 in the shaded region.

The moment about y-axis of this planar lamina is My=ρab[f(y)+g(y)2][f(y)g(y)]dy where,f(y)=4y2,g(y)=0,a=2,b=2.

Substitute the above values to get,

My=ρ22[4y22][4y2]dy=2ρ02(4y)22dy=ρ02(4y2)2dy=ρ02(y48y2+16)dy

On further calculation, we get

My=ρ[y558y33+16y]02=ρ2[325643+32]=256ρ15

Now, moment about x-axis of this planar lamina is Mx=ρaby[f(y)g(y)]dy where,f(y)=4y2,g(y)=0,a=2,b=2

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