Center of Mass of a Composite Object The L-shaped machine part in Fig. 7.35 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 the center of mass of the machine part.

Question

Center of Mass of a Composite Object

The L-shaped machine part in Fig. 7.35 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 the center of mass of the machine part.

Accepted Answer

The volume of bar 1 is

[latex](80)(240)(40) = 7.68 imes 10^{5} mm^{3} = 7.68 imes 10^{- 4} m^{3} ,[/latex]

so its mass is [latex] \left(7.68 imes 10^{- 4} ight)\left(1.4 imes 10^{4} ight) = 10 .75 kg [/latex] The center of mass of bar 1 coincides with the centroid of its volume: [latex]\overline{x}_{1} = 40 mm , \overline{y}_{1} = 120 mm ,\overline{z}_{1} = 0 .[/latex]

Bar 2 has the same volume as bar 1, so its mass is [latex] (7.68 imes 10^{- 4} ) (7.8 imes 10^{3} ) = 5.99 kg .[/latex] The coordinates of its center of mass are [latex]\overline{x}_{2} = 200 mm ,\overline{y}_{2} = 40 mm , \overline{z}_{2} = 0 .[/latex] Using the information summarized in Table 7.7, we obtain the x coordinate of the center of mass,

[latex]\overline{x} = \frac{\overline{x} _{1}m_{1} + \overline{x} _{2}m_{2} }{m_{1} + m_{2} } = \frac{(40)(10.75) + (200)(5.99)}{10.75 + 5.99} = 97.2 mm,[/latex]

and the v coordinate,

[latex]\overline{y} = \frac{\overline{y} _{1}m_{1} + \overline{y} _{2}m_{2} }{m_{1} + m_{2} } = \frac{(120)(10.75) + (40)(5.99)}{10.75 + 5.99} = 91.4 mm,[/latex]

Because of the symmetry of the object, [latex]\overline{z} =0 .[/latex]

Table 7.7 Information for determining the center of mass

	[latex]\pmb{m_{1}(kg)}[/latex]	[latex]\pmb{\overline{x}_{i}(mm)}[/latex]	[latex]\pmb{\overline{x}_{i}m_{i}(mm-kg)}[/latex]	[latex]\pmb{\overline{y}_{i}(mm)}[/latex]	[latex]\pmb{\overline{y}_{i}m_{i} (mm-Kg)}[/latex]
Bar 1	10.75	40	(40)(10.75)	120	(120)(10.75)
Bar 2	5.99	200	(200)(5.99)	40	(40)(5.99)

Question 7.18: Center of Mass of a Composite Object The L-shaped machine pa...

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