In the figure the L- shaped machine part is composed of two homogeneous bars . Bar 1 is tungsten alloy with mass density 14,000 kg/m3 , and bar 2 is steel with mass density 7800 kg/m3 . Determine its moment of inertia about the x axis.
In the figure the L- shaped machine part is composed of two homogeneous bars . Bar 1 is tungsten alloy with mass density 14,000 kg/m3 , and bar 2 is steel with mass density 7800 kg/m3 . Determine its moment of inertia about the x axis.
The Correct Answer and Explanation is :
It seems that I cannot access the figure directly from the provided link. However, I can guide you on how to approach calculating the moment of inertia of the L-shaped machine part about the x-axis based on the given materials.
Step-by-Step Explanation:
- Moment of inertia of a uniform bar about its center of mass:
The moment of inertia for a bar rotating about its center of mass is given by:
[
I = \frac{1}{12} m L^2
]
where (m) is the mass of the bar, and (L) is its length. - Density and mass of each bar:
The mass of each bar can be found using its density ((\rho)) and the volume ((V)) of the bar:
[
m = \rho \times V = \rho \times A \times L
]
where (A) is the cross-sectional area, and (L) is the length of the bar. - Parallel axis theorem:
If the rotation axis is not at the center of mass of the bar, we use the parallel axis theorem to adjust the moment of inertia:
[
I = I_{\text{cm}} + m d^2
]
where (d) is the distance between the axis of rotation and the center of mass. - Sum of moments of inertia:
For the L-shaped part, calculate the moment of inertia of each bar (using the center of mass formula and adjusting with the parallel axis theorem), then add them together for the total moment of inertia about the x-axis.
Example Calculations (Hypothetical):
For bar 1 (tungsten alloy):
- Density ((\rho_1)) = 14,000 kg/m³
- Length ((L_1)) = 2 m
- Cross-sectional area ((A_1)) = 0.01 m²
For bar 2 (steel):
- Density ((\rho_2)) = 7,800 kg/m³
- Length ((L_2)) = 1.5 m
- Cross-sectional area ((A_2)) = 0.01 m²
Now, using the formulas above, calculate the mass of each bar, then the moment of inertia for each and apply the parallel axis theorem if needed based on the positions of the bars relative to the x-axis.
If you could provide the dimensions or any further specifics about the arrangement of the bars, I can help refine the answer.