A solid aluminum cylinder with ρ=2700 kg / m^3 has a...

A solid aluminum cylinder with $\rho=2700 \mathrm{~kg} / \mathrm{m}^{3}$ has a measured mass of $67 \mathrm{~g}$ in air and $45 \mathrm{~g}$ when immersed in turpentine. Determine the density of turpentine. The $F_{B}$ acting on the immersed cylinder is $$F_{B}=(0.067-0.045)(9.81) \mathrm{N}=(0.022)(9.81) \mathrm{N}$$ This is also the weight of the displaced turpentine. The volume of the cylinder is, from $\rho=m / V$, $$ V \text { of cylinder }=\frac{m}{\rho}=\frac{0.067 \mathrm{~kg}}{2700 \mathrm{~kg} / \mathrm{m}^{3}}=2.5 \times 10^{-5} \mathrm{~m}^{3} $$ This is also the volume of the displaced turpentine. We therefore have, for the turpentine, $$ \rho=\frac{\text { Mass }}{\text { Volume }}=\frac{(\text { Weight }) / g}{\text { Volume }}=\frac{(0.022)(9.81) /(9.81)}{2.48 \times 10^{-5}} \frac{\mathrm{kg}}{\mathrm{m}^{3}}=8.9 \times 10^{2} \mathrm{~kg} / \mathrm{m}^{3} $$

A solid aluminum cylinder with $\rho=2700 \mathrm{~kg} / \mathrm{m}^{3}$ has a measured mass of $67 \mathrm{~g}$ in air and $45 \mathrm{~g}$ when immersed in turpentine. Determine the density of turpentine.
The $F_{B}$ acting on the immersed cylinder is
$$F_{B}=(0.067-0.045)(9.81) \mathrm{N}=(0.022)(9.81) \mathrm{N}$$
This is also the weight of the displaced turpentine.
The volume of the cylinder is, from $\rho=m / V$,
$$
V \text { of cylinder }=\frac{m}{\rho}=\frac{0.067 \mathrm{~kg}}{2700 \mathrm{~kg} / \mathrm{m}^{3}}=2.5 \times 10^{-5} \mathrm{~m}^{3}
$$
This is also the volume of the displaced turpentine. We therefore have, for the turpentine,
$$
\rho=\frac{\text { Mass }}{\text { Volume }}=\frac{(\text { Weight }) / g}{\text { Volume }}=\frac{(0.022)(9.81) /(9.81)}{2.48 \times 10^{-5}} \frac{\mathrm{kg}}{\mathrm{m}^{3}}=8.9 \times 10^{2} \mathrm{~kg} / \mathrm{m}^{3}
$$

Key Concepts

Volume Calculation

Calculating the volume of an object based on its mass and density is a key method in physics to establish the amount of fluid displaced when submerged. The formula V = m/? is applied, linking mass, density, and volume, which is instrumental in deriving quantities like buoyant force and fluid density.

Apparent Weight

Apparent weight is the effective weight of an object when immersed in a fluid, which is less than its true weight in air due to the buoyant force. This concept is used to determine important properties such as the density of a fluid by comparing measurements taken in air and in the fluid.

Buoyant Force

The buoyant force is the upward force experienced by an object immersed in a fluid. It is equal to the weight of the fluid displaced by the object, as described by Archimedes’ principle. This concept is essential for analyzing how objects behave in fluids, including how they float or sink.

Density

Density is defined as mass per unit volume and serves as a fundamental property of matter in physics. In problems involving fluid mechanics, density is critical for determining buoyant forces, calculating masses of displaced fluids, and comparing the properties of different substances.

Archimedes' Principle

This principle states that any object submerged in a fluid is buoyed up by a force equal to the weight of the fluid displaced by the object. It is fundamental for understanding why objects appear to weigh less when immersed in a fluid and is used to relate the loss of weight to the density of the fluid displaced.

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