91Ó°ÊÓ

A hydrometer is a device used to measure the density of a liquid. It is a cylindrical tube weighted at one end, so that it floats with the heavier end downward. The tube is contained inside a large "medicine dropper," into which the liquid is drawn using the squeeze bulb (see the drawing). For use with your car, marks are put on the tube so that the level at which it floats indicates whether the liquid is battery acid (more dense) or antifreeze (less dense). The hydrometer has a weight of \(W=5.88 \times 10^{-2} \mathrm{N}\) and a cross-sectional area of \(A=7.85 \times 10^{-5} \mathrm{m}^{2}\) How far from the bottom of the tube should the mark be put that denotes (a) battery acid \(\left(\rho=1280 \mathrm{kg} / \mathrm{m}^{3}\right)\) and (b) antifreeze \(\left(\rho=1073 \mathrm{kg} / \mathrm{m}^{3}\right) ?\)

Step by step solution

01

Understanding Archimedes' Principle

To solve this problem, we use Archimedes' principle which states that the buoyant force on the immersed part of the hydrometer is equal to the weight of the liquid displaced. This principle is given by \( F_b = \rho V g \), where \( \rho \) is the density of the liquid, \( V \) is the volume displaced, and \( g \) is the acceleration due to gravity.

02

Calculating Displaced Volume

Given that the hydrometer is in equilibrium, the buoyant force equals the weight of the hydrometer. Therefore, \( \rho V g = W \), where \( W = 5.88 \times 10^{-2} \mathrm{N} \). To find the volume \( V \), rearrange the equation to \( V = \frac{W}{\rho g} \).

03

Find the Submerged Length for Battery Acid

For battery acid with \( \rho = 1280 \mathrm{kg/m^3} \), substitute \( \rho \) into the volume equation: \( V = \frac{5.88 \times 10^{-2}}{1280 \times 9.81} \). Calculate this to find \( V \). Then use \( V = A h \) where \( h \) is the height, to find \( h = \frac{V}{A} \).

04

Find the Submerged Length for Antifreeze

For antifreeze with \( \rho = 1073 \mathrm{kg/m^3} \), use the same steps as with battery acid: \( V = \frac{5.88 \times 10^{-2}}{1073 \times 9.81} \). Compute \( V \) and then find \( h = \frac{V}{A} \) using the cross-sectional area \( A = 7.85 \times 10^{-5} \mathrm{m}^{2} \).

05

Calculate Final Values

Compute the exact values for \( h \) for both battery acid and antifreeze using the equations derived. This will give the distances from the bottom of the tube at which the marks should be placed.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Buoyant Force

When objects are submerged in a fluid, they experience a force known as the buoyant force. This force acts upwards, opposing the force of gravity, and is responsible for making objects float. According to Archimedes' Principle, the buoyant force on an object is equal to the weight of the fluid that the object displaces. This can be expressed with the formula:

• \( F_b = \rho V g \)

Here, \( F_b \) is the buoyant force, \( \rho \) represents the density of the fluid, \( V \) is the volume of fluid displaced, and \( g \) is the acceleration due to gravity. This principle helps us understand why objects of different densities float or sink.
When the buoyant force equals the weight of the object, it floats. If it's less, the object sinks.

Density Measurement

Density measures how much mass is contained in a given volume. It is expressed as mass per unit volume (e.g., kg/m³). High-density liquids, like battery acid, have more mass packed in a given volume compared to low-density liquids, like antifreeze. Understanding density is key to many applications, such as determining whether a liquid will support a floating object or not.
In the context of a hydrometer, the tool measures the density of liquids to identify substances. By knowing the density, one can predict the liquid’s behavior and its potential uses. For instance, with a hydrometer, you could identify a denser liquid as battery acid by observing how much of the hydrometer sinks.

Hydrometer

A hydrometer is a simple yet effective tool used for measuring liquid density. Typically, it consists of a weighted tube that floats upright in a liquid. The level at which the hydrometer floats depends on the liquid's density. When placed in denser liquids, less of the hydrometer is submerged, as it doesn't need to displace much liquid to equal its weight.
To determine the density using a hydrometer:

• Place it in the liquid.
• Observe the depth to which it sinks.

For a vehicle, hydrometers can indicate the type of liquid, such as differentiating between battery acid and antifreeze, ensuring safe vehicle operation.

Volume Displacement

Volume displacement is a concept related to the amount of fluid that moves when an object is submerged. The displaced volume is directly proportional to how much of the object is underwater. In the formula \( V = A h \), \( V \) represents the displaced volume, \( A \) is the cross-sectional area, and \( h \) is the submerged height.
This concept is critical for calculating how deep a hydrometer will sink in a given liquid. By knowing the displaced volume, you can determine the point where the buoyant force balances the weight of the hydrometer. For instance, in denser liquids, a smaller volume is needed to match the hydrometer’s weight, resulting in a shallower subsurface depth.