Problem 77 - Fundamentals of Physics

Question

A glass ball of radius \(2.00 \mathrm{~cm}\) sits at the bottom of a container of milk that has a density of \(1.03 \mathrm{~g} / \mathrm{cm}^{3}\). The normal force on the ball from the container's lower surface has magnitude \(9.48 \times 10^{-2} \mathrm{~N}\). What is the mass of the ball?

Step-by-Step Solution

Answer

The mass of the ball is approximately 0.12 kg.

Step 1: Understand the Forces Involved

In this exercise, the glass ball is at the bottom of the liquid. The forces acting on it include its weight, the buoyant force due to the liquid, and the normal force from the container's surface.

Step 2: Apply the Buoyant Force Formula

The buoyant force is given by Archimedes' principle, which states that it equals the weight of the displaced liquid. The volume of milk displaced by the submerged ball is the same as the ball's volume.\(V = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (2)^3 = \frac{32 \pi}{3} \text{ cm}^3\).

Step 3: Calculate the Buoyant Force

Convert the volume from \(\text{cm}^3\) to \(\text{m}^3\) for consistency (because the result from step 2 is in terms of centimeters):\(V = \frac{32 \pi}{3} \times 10^{-6} \text{ m}^3\).The buoyant force, \(F_b\), is given by \(F_b = \text{density of milk} \times V \times g = 1.03 \times 10^3 \text{ kg/m}^3 \times \frac{32 \pi}{3} \times 10^{-6} \times 9.8\).

Step 4: Calculate the Ball's Weight

The weight \(W\) of the ball is the combination of the net downward force it has to support itself with, which includes the buoyant force and the normal force:\(W = F_b + F_n\).Given that \(F_n = 9.48 \times 10^{-2} \text{ N}\).

Step 5: Solve for the Mass of the Ball

The weight of the ball also equals the mass times gravity, i.e., \(W = m \times g\).Substitute \(W = F_b + F_n\) to solve for mass \(m\):\(m = \frac{F_b + F_n}{g}\).

Step 6: Find the Numeric Solution

Firstly, compute \(F_b\) from step 3. Afterward, substitute \(F_b\) and \(F_n\) into the equation from step 5. Solve for \(m\). When solved, \(m\) equals approximately \(0.12 \text{ kg}\).

Key Concepts

Archimedes' principle

Archimedes' principle is a fundamental concept in fluid mechanics. It states that a body submerged in a fluid experiences a buoyant force equal to the weight of the fluid that the body displaces. This principle helps explain why objects either float or sink in fluids.
When we submerge an object in a liquid, two forces act on it: the object's weight and the buoyant force. Archimedes' principle allows us to determine the buoyant force by calculating the weight of the displaced fluid.
For instance, in the case of the glass ball in milk, Archimedes' principle tells us that the buoyant force is equivalent to the weight of the milk displaced by the ball. This displaced volume is identical to the volume of the ball since it is completely submerged.

Buoyant force

Buoyant force is the upward force exerted by a fluid on a submerged object. It arises because pressure within a fluid increases with depth, creating a net upward force on objects within the fluid.
The magnitude of the buoyant force can be calculated using the formula:

• \[ F_b = \rho_f \times V \times g \]

where \( F_b \) is the buoyant force, \( \rho_f \) is the fluid's density, \( V \) is the volume of the fluid displaced, and \( g \) is the acceleration due to gravity.
This concept is used extensively in various applications, from designing ships to understanding how hot air balloons lift off. In our exercise, the buoyant force helps reduce the total weight supported by the bottom of the container, affecting the normal force observed.

Fluid density

Fluid density is a measure of how much mass is contained in a given volume of fluid. It is often denoted by the symbol \( \rho_f \) and its unit is typically \( \text{kg/m}^3 \).
Density plays a crucial role in determining buoyant force. Higher density fluids exert a stronger buoyant force on a submerged object compared to lower density fluids of the same volume.
In this exercise, the milk's density is provided as \( 1.03 \text{ g/cm}^3 \), which is converted to \( 1030 \text{ kg/m}^3 \) for consistency in calculations involving the buoyant force. The density value enables us to calculate how much the milk supports the weight of the submerged glass ball.

Normal force

Normal force is the contact force exerted by a surface perpendicular to an object resting on it. It counteracts other forces acting on the object, such as gravity.
In scenarios involving a submerged object, like the glass ball at the bottom of the milk container, the normal force balances the combined effect of the weight of the ball and the buoyant force exerted by the fluid.
Mathematically, the normal force \( F_n \) can be expressed as:

• \[ F_n = W - F_b \]

where \( W \) is the weight of the object and \( F_b \) is the buoyant force. In the exercise, the calculated normal force helped determine the mass of the ball by considering the forces working together to keep the ball stationary at the container's bottom.