[amp_mcq option1=”4.5 cm” option2=”5.0 cm” option3=”5.25 cm” option4=”5.5 cm” correct=”option1″]<\/p>\n
The correct answer is $\\boxed{\\text{B)} 5.0 \\text{ cm}}$.<\/p>\n
The center of mass (COM) of a solid cylinder is at its geometric center, which is at a height of $4 \\times \\frac{1}{2} = 2$ cm from the base. The COM of a right circular cone is at its apex, which is at a height of $4$ cm from the base.<\/p>\n
When the cone is scooped out of the cylinder, the COM of the remainder will be located somewhere between the COM of the cylinder and the COM of the cone, closer to the COM of the cylinder. This is because the cone has a smaller mass than the cylinder, so its COM will have a smaller effect on the overall COM of the remainder.<\/p>\n
To calculate the exact height of the COM of the remainder, we can use the following equation:<\/p>\n
$$h = \\frac{m_1 h_1 + m_2 h_2}{m_1 + m_2}$$<\/p>\n
where $h$ is the height of the COM of the remainder, $m_1$ is the mass of the cylinder, $h_1$ is the height of the cylinder, $m_2$ is the mass of the cone, and $h_2$ is the height of the cone.<\/p>\n
We know that $m_1 = \\pi r^2 h$, $m_2 = \\frac{1}{3} \\pi r^2 h$, $h_1 = 8$ cm, and $h_2 = 4$ cm. Substituting these values into the equation, we get:<\/p>\n
$$h = \\frac{\\pi (4)^2 (8) + \\frac{1}{3} \\pi (4)^2 (4)}{\\pi (4)^2 + \\frac{1}{3} \\pi (4)^2} = \\frac{32 + 16}{16 + 4} = 5.0 \\text{ cm}$$<\/p>\n
Therefore, the COM of the remainder is at a height of $5.0$ cm from the base.<\/p>\n","protected":false},"excerpt":{"rendered":"
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