Engineering, 14.04.2020 19:15 leo4687

#198. Moment of inertia about center of a segmented bar A bar of width is formed of three uniform segments with lengths and areal densities given by: Matlab/Mathematica input: L1 = 6 rho1 = 1 L2 = 6 rho2 = 8 L3 = 4 rho3 = 5 What is the moment of inertia of the bar about the center of mass ?

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Answer from: colebash88

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The complete question is shown on the first uploaded image

The moment of inertia of the bar about the center of mass is

$I_r = 1888.80 \ kg m^2$

Explanation:

The free body diagram is shown on the second uploaded image

From the diagram we see that is

The mass of each segment is

$m_1 = \rho_1 l_1 w = 1 * 6 * 2 = 12$

$m_1 = \rho_2 l_2 w = 8 * 6 * 2 = 96$

$m_1 = \rho_2 l_2 w = 5 * 5 * 2 = 50$

The distance from the origin to the center of the segments i.e the center of masses for the individual segments

$x_2 = \frac{6}{2} + 6 = 9 m$

$x_3 = \frac{4}{2} + 12 = 14 m$

The resultant center of mass is mathematically evaluated as

$x_r = \frac{m_1 * x_1 + m_2 *x_2 + m_3 *x_3}{m_1 + m_2 + m_3}$

$= \frac{12 * 3 + 96 *9 + 50 *14}{12+ 96 + 50}$

x_r = 10.13m

The moment of Inertia of each segment of the bar is mathematically evaluated

$I_1 =\frac{m_1}{12}(l_1^2 + w^2)$ = $\frac{12}{12}(1^2 + 2^2)$

$I_1 = 4 \ kgm^2$

$I_2 =\frac{m_2}{12}(l_2^2 + w^2)$ = $\frac{96}{12}(6^2 + 2^2)$

$I_2 = 320 \ kgm^2$

$I_3 =\frac{m_3}{12}(l_3^2 + w^2)$ = $\frac{50}{12}(4^2 + 2^2)$

$I_2 = 83.334 \ kgm^2$

According to parallel axis theorem the moment of inertia about the center ( x_r ) is mathematically evaluated as

I_r = (I_1 + m_1 r_1^2) + (I_2 + m_2 r_2^2) +(I_3 + m_3 r_3^2)

I_r = (I_1 + m_1 |x_r - x_1|^2) + (I_2 + m_2 |x_r - x_2|^2) +(I_3 + m_3 |x_r - x_3|^2)

I_r = (4 + 12 |10.13 - 3|^2) + (320 + 96 |10.13 - 9|^2) +(83.334 + 50 |10.13 - 14|^2)

$I_r = 1888.80 \ kg m^2$

#198. Moment of inertia about center of a segmented bar A bar of width is formed of three uniform se

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the equation above shows how temperature f, measured in degrees fahrenheit, relates to a temperature c, measured in degrees celsius. based on the equation, which of the following must be true?

a temperature increase of 1 degree fahrenheit is equivalent to a temperature increase of

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