Consider the following mass distribution, where x- and y- coordinates are given in meters: 5.0 kg at (0.0, 0.0) m, 4.0 kg at (0.0, 7.0) m, and 7.0 kg at (5.0, 0.0) m. Where should a fourth object of 8.0 kg be placed so that the center of gravity of the four- object arrangement will be at (0.0, 0.0) m? Enter the x-coordinate.
To find the x-coordinate of the fourth object, we need to calculate the total moment produced by the three existing objects and set it equal to zero. By doing so, we ensure that the center of gravity of the four-object arrangement will be at (0.0, 0.0) m.
First, let's calculate the total moment produced by the existing objects. The moment (M) is the product of the mass (m) and the distance (r) from the center of gravity. The equation for the total moment (Mtotal) is given by:
Mtotal = (m1 * r1) + (m2 * r2) + (m3 * r3)
Plugging in the given values:
m1 = 5.0 kg, r1 = 0.0 m (as it is placed at the origin)
m2 = 4.0 kg, r2 = 7.0 m (distance to the y-axis)
m3 = 7.0 kg, r3 = 5.0 m (distance to the x-axis)
Mtotal = (5.0 kg * 0.0 m) + (4.0 kg * 7.0 m) + (7.0 kg * 5.0 m)
Mtotal = 0 + 28.0 + 35.0
Mtotal = 63.0 kg*m
Since we want the center of gravity to be at (0.0, 0.0) m, the total moment must be zero. Therefore:
Mtotal = 0
0 = 63.0 kg*m + (8.0 kg * x)
Rearranging the equation to solve for x:
8.0 kg * x = -63.0 kg*m
x = (-63.0 kg*m) / 8.0 kg
x ≈ -7.88 m
Hence, the x-coordinate of the fourth object should be approximately -7.88 m to position it at the center of gravity at (0.0, 0.0) m.