VDU_Physics_1_009

# ## Physics 1:  Mechanics: Center of Mass

Javier Montenegro Joo

## jmj@VirtualDynamics.Org

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Center of Mass.-

(1) The figure shows a laminar piece of metal that consists of 3 sheets ‘A’, ‘B’, and ‘C’, assembled one next to another.  It is known that the density of ‘B’ is double the density of ‘A’, and, that the density of ‘C’ is the triple of that of ‘A’.  Determine the center of mass of the piece.  Measurements are in cm. Solution.-

At simple sight it can be seen that for the three pieces ‘A’, ‘B’ and ‘C’  the center of mass must be along  y = 5 cm

Horizontal positions of the pieces:   A = 2 cm,    B = 7 cm,    C = 11 cm

Areas:    A = 40 cm²,      B = 48 cm²,      C = 8 cm² (2) Three compact cubical blocks of the same material, whose sides are 3L, 2L and L, respectively, are placed one next to the other, in the mentioned order and, so that their centers are aligned along the x-axis. Determine the position of the center of mass of the cubes with respect to the left side of the largest cube.

Solution.-

The sketch below shows the three blocks and the positions of their individual centers of mass; obviously, the center of mass of the trio is along the x-axis: (3) Three compact cubical blocks of the same material, whose sides are L, 2L and 3L, respectively, are placed one next to the other, in the mentioned order and, so that their centers are aligned along the x-axis. Determine the position of the center of mass of the cubes with respect to the left side of the smallest cube.

Solution.-

The sketch below shows the three blocks and the positions of their individual centers of mass; obviously the center of mass of the trio is somewhere in the x-axis: (4) Determine the center of mass of the uniform metal sheet shown in the figure, with respect to its lower left corner. Units are in cm. Solution.-

·         The sketch below shows the uniform metal sheet as three different pieces, all having the same density.

·         The center of mass of each component piece is shown as a dot and, it includes its coordinates (x,y).

·         Since the object is bi-dimensional, the computations include the surface density and the area of each piece.

·         The sought center of mass has components x and y. (5) Three compact spheres whose radiuses are R, 2R and 3R, respectively, are placed one on top of the other, so that their centers are aligned along the vertical. The density of the sphere with radius 2R is twice that of the sphere whose radius is  R and, the density of the sphere with radius 3R is thrice that of the sphere with radius R.  Determine the position of the center of mass of the group.

Solution.- (6)  In the figure, a thin metallic circular sheet of radius R has a circular cavity of radius R/2. (a) Determine the center of mass of the object. (b) Compute the center of mass for the case when the radiuses are 40 cm and 20 cm, respectively. Solution.-

In order to compute the center of mass of this hollowed object, assume that the mass of the empty region is negative. Since the object is a sheet, consider its surface density. Obviously the center of mass will be along the centers of both circles. 