Question

For a material to float on the surface of water, the material must have a density less than that of water \((1.0 \mathrm{~g} / \mathrm{mL})\) and must not react with the water or dissolve in it. A spherical ball has a radius of 0.50 \(\mathrm{cm}\) and weighs \(2.0 \mathrm{~g}\). Will this ball float or sink when placed in water? (Note: Volume of a sphere \(\left.=\frac{4}{3} \pi r^{3} .\right)\)

Short answer

The sphere has a density of approximately \(3.85 \mathrm{~g/cm}^3\), which is greater than the density of water (\(1.0 \mathrm{~g/cm}^3\)). Therefore, the sphere will sink when placed in water.

Step-by-step solution

Calculate the volume of the sphere

Using the given radius (0.50 cm), we can find the volume of the sphere using the volume formula: \[ V = \frac{4}{3} \pi r^3 \] where V is the volume and r is the radius of the sphere. Let's plug the given radius into the formula: \[ V = \frac{4}{3} \times \pi \times (0.50 \mathrm{~cm})^3 \]

Compute the volume

Now, we can compute the volume of the sphere: \[ V = \frac{4}{3} \times \pi \times (0.125 \mathrm{~cm}^3) \approx 0.52 \mathrm{~cm}^3 \] The volume of the sphere is approximately 0.52 cm³.

Calculate the density of the sphere

Next, we will find the density of the sphere using its mass and volume. The formula for density is: \[ \rho = \frac{m}{V} \] where ρ is the density, m is the mass, and V is the volume of the sphere. We know the mass of the sphere is 2.0 g, and we've calculated the volume as 0.52 cm³. Now, we can plug these values into the formula: \[ \rho = \frac{2.0 \mathrm{~g}}{0.52 \mathrm{~cm}^3} \]

Compute the density

Now, we can compute the density of the sphere: \[ \rho \approx 3.85 \mathrm{~g} / \mathrm{cm}^3 \] The density of the sphere is approximately 3.85 g/cm³.

Determine if the sphere will float or sink

To determine if the sphere will float or sink in water, we need to compare its density to the density of water, which is 1.0 g/cm³. Since the density of the sphere (3.85 g/cm³) is greater than the density of water (1.0 g/cm³), the sphere will sink when placed in water.

Key concepts

Buoyancy

Buoyancy refers to the upward force that allows objects to float in a fluid, like water. When an object is submerged in a fluid, the fluid exerts an upward force called buoyant force. This force is equal to the weight of the fluid displaced by the object.

For an object to float, its density has to be less than the density of the fluid. In simple terms, if an object can displace enough fluid to counteract its own weight, it floats. Otherwise, it sinks.

• Buoyant force: The force that pushes objects up in a fluid.
• Density: A key factor to determine whether an object will float or sink.

Understanding buoyancy helps us predict the floating or sinking behavior of objects in various fluids.

Sphere Volume

The volume of a sphere is a measure of the space it occupies. Calculating the volume is particularly useful when dealing with spherical objects, as it allows us to understand their capacity and properties.

The formula to find the volume of a sphere is: \[ V = \frac{4}{3} \pi r^3 \]where:

• \( V \) is the volume,
• \( r \) is the radius of the sphere.

This mathematical formula involves the value of \( \pi \), which is a constant approximated as 3.14159. By using the radius of the sphere, you can easily determine its volume.

This calculation is useful in various scientific and engineering applications.

Density Calculation

Density is a physical property that describes the mass of an object relative to its volume. It's an essential characteristic that helps in identifying substances and predicting their interactions with other materials.

The formula for calculating density is \[ \rho = \frac{m}{V} \]where:

• \( \rho \) represents the density,
• \( m \) is the mass of the object,
• \( V \) is the volume of the object.

This formula tells us how much mass is packed in a given volume. By knowing both the mass and volume, you can accurately calculate an object's density.

Understanding density is crucial for determining whether an object will float or sink in a fluid, as in the case of the spherical ball detailed in the exercise.