Chapter 1: Problem 75

A graduated cylinder is filled to the 40.00 -mL mark with a mineral oil. The masses of the cylinder before and after the addition of the mineral oil are \(124.966 \mathrm{~g}\) and \(159.446 \mathrm{~g}\), respectively. In a separate experiment, a metal ball bearing of mass \(18.713 \mathrm{~g}\) is placed in the cylinder and the cylinder is again filled to the 40.00 -mL mark with the mineral oil. The combined mass of the ball bearing and mineral oil is \(50.952 \mathrm{~g}\). Calculate the density and radius of the ball bearing. [The volume of a sphere of radius \(r\) is \(\left.(4 / 3) \pi r^{3} .\right]\)

Short Answer

Expert verified

The density of the ball bearing is \(0.862 \mathrm{~g/cm}^3\) and its radius is \(1.50 \mathrm{~cm}\)

Step by step solution

Determine Mass and Volume of Mineral Oil

First, find the mass of the mineral oil by subtracting the mass of the empty cylinder from the mass of the cylinder with the mineral oil. This gives us \(159.446 \mathrm{~g}\) - \(124.966 \mathrm{~g}\) = \(34.480 \mathrm{~g}\). Next, we need to convert the volume of the oil from mL to cm³, as the SI unit of volume is cubic meter (m³). However, it's more convenient to use cm³ (where \(1 \mathrm{~mL}\) = \(1 \mathrm{~cm}^3\)) for this problem. Therefore, the volume of the mineral oil is \(40.00 \mathrm{~mL}\) = \(40.00 \mathrm{~cm}^3\)

Calculate the Density of Mineral Oil

Density is calculated by dividing mass by volume. Using the mass and volume from Step 1, we find that the density of the mineral oil is \(34.480 \mathrm{~g}\) / \(40.00 \mathrm{~cm}^3\) = \(0.862 \mathrm{~g/cm}^3\)

Find the Volume of Oil Displaced by Ball Bearing

When the ball bearing is added to the cylinder filled with oil, the oil level stays the same which means the ball bearing displaces an amount of oil equal to its own volume. To find this volume, we subtract the combined mass of the ball bearing and oil from the mass of the oil we found in Step 1. This gives us \(50.952 \mathrm{~g}\) - \(34.480 \mathrm{~g}\) = \(16.472 \mathrm{~g}\). Then, we divide this mass by the density of the oil to find the volume of oil displaced: \(16.472 \mathrm{~g}\) / \(0.862 \mathrm{~g/cm}^3\) = \(19.10 \mathrm{~cm}^3\)

Calculate the Radius of Ball Bearing

The volume of a sphere is given by the formula \(V = 4/3 * \pi * r^3\), where V is the volume and r is the radius. Since we have the volume of the sphere, we can rearrange this formula to solve for the radius: \(r = ((3*V)/(4*\pi))^{1/3}\). Substituting the volume of the ball bearing we found in Step 3, we find its radius: \(r = ((3 * 19.10 \mathrm{~cm}^3) /(4 * \pi))^{1/3}\) = \(1.50 \mathrm{~cm}\)

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Short Answer

Step by step solution

Determine Mass and Volume of Mineral Oil

Calculate the Density of Mineral Oil

Find the Volume of Oil Displaced by Ball Bearing

Calculate the Radius of Ball Bearing

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