Math Problem Statement

Two physics problems: 1) Determine the densities of three objects submerged in water, and 2) Calculate the force needed to operate a hydraulic jack based on Pascal's principle.

Solution

Let's delve into solving these two problems from the image:

Question 13:

We are dealing with floating objects in a liquid with a density of 1000kg/m31000 \, \text{kg/m}^3. The key here is Archimedes' principle, which says:

  • Objects less dense than water (less than 1000kg/m31000 \, \text{kg/m}^3) will float.
  • Objects with equal density will be submerged but not sink.
  • Objects denser than water (more than 1000kg/m31000 \, \text{kg/m}^3) will sink.

From the diagram:

  • Object 44 floats at the surface, so its density is less than 1000kg/m31000 \, \text{kg/m}^3.
  • Object 22 is submerged but does not sink, so its density is equal to 1000kg/m31000 \, \text{kg/m}^3.
  • Object 33 sinks, so its density is greater than 1000kg/m31000 \, \text{kg/m}^3.

Thus, the answer is: A. 700kg/m3;1000kg/m3;1200kg/m3700 \, \text{kg/m}^3; 1000 \, \text{kg/m}^3; 1200 \, \text{kg/m}^3.


Question 14:

This is a hydraulic jack problem, where Pascal's principle applies: F1/A1=F2/A2F_1 / A_1 = F_2 / A_2 Here:

  • F2=4000NF_2 = 4000 \, \text{N}
  • A2=200cm2=0.02m2A_2 = 200 \, \text{cm}^2 = 0.02 \, \text{m}^2
  • A1=5cm2=0.0005m2A_1 = 5 \, \text{cm}^2 = 0.0005 \, \text{m}^2

Rearranging for F1F_1: F1=F2×(A1/A2)F_1 = F_2 \times (A_1 / A_2) Substitute values: F1=4000×0.00050.02=100NF_1 = 4000 \times \frac{0.0005}{0.02} = 100 \, \text{N}

Thus, the answer is: A. 100N100 \, \text{N}.


Do you want further explanations or clarifications? Here are 5 related questions for practice:

  1. How does Archimedes' principle apply to irregularly shaped objects?
  2. Can you derive Pascal's principle using fluid mechanics laws?
  3. What is the impact of temperature on fluid density in buoyancy problems?
  4. How does the area ratio affect force in hydraulic systems?
  5. What are practical applications of hydraulic jacks in engineering?

Tip: Always check the units when working with density, force, or area in fluid dynamics problems!

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Math Problem Analysis

Mathematical Concepts

Archimedes' principle
Pascal's principle
Density calculations
Hydraulics and pressure

Formulas

Buoyancy: Density = Mass / Volume
Hydraulics: F1 / A1 = F2 / A2

Theorems

Archimedes' Principle
Pascal's Principle

Suitable Grade Level

Grades 10-12