A thin rectangular piece of aluminum (density is 2700 kg/m3) is designed to pivot about a pin that smoothly slides into a small hole. Where should the hole be drilled in order to minimize the period of harmonic motion? The aluminum has dimensions of 2 cm x 40 cm x 0.1 cm
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To determine the optimal position for the hole, we need to consider the principles of simple harmonic motion (SHM) and the equation for the period of a simple pendulum.
The period (T) of a simple pendulum is given by the equation:
T = 2π √(L/g)
where L is the length of the pendulum and g is the acceleration due to gravity.
In this case, we will consider the aluminum piece as a simple pendulum when it pivots about the pin. The length of the pendulum is the distance between the pin and the center of mass of the aluminum piece.
To find the center of mass, we can calculate the centroid of the rectangular piece. For a rectangle, the center of mass coincides with the centroid, which is given by:
x_bar = (x1 + x2)/2
y_bar = (y1 + y2)/2
z_bar = (z1 + z2)/2
where (x1, y1, z1) and (x2, y2, z2) are the coordinates of opposite corners of the rectangle.
In this case, the dimensions of the aluminum piece are:
x1 = 0 cm, x2 = 40 cm
y1 = 0 cm, y2 = 2 cm
z1 = 0 cm, z2 = 0.1 cm
Using the centroid formula, we find:
x_bar = (0 + 40)/2 = 20 cm
y_bar = (0 + 2)/2 = 1 cm
z_bar = (0 + 0.1)/2 = 0.05 cm
Therefore, the center of mass is located at (20 cm, 1 cm, 0.05 cm).
To minimize the period of harmonic motion, we want to minimize the length of the pendulum. In this case, the length is the distance between the pin and the center of mass.
If we assume that the small hole is drilled along the longer side of the rectangular piece, the distance between the pin and the center of mass would be 20 cm. This would result in a longer pendulum and a longer period.
Therefore, to minimize the period of harmonic motion, the hole should be drilled along the shorter side of the rectangular piece, which is 2 cm. This would result in a shorter distance between the pin and the center of mass and, consequently, a shorter pendulum and a shorter period of harmonic motion.