📘 Numerical Problems (Mechanical Properties of Matter)
Practice problems covering Hooke's law, density, pressure, hydraulic systems, and atmospheric pressure
Related Resources:
Important Formulas
Density
\[\rho = \frac{m}{V}\]Where \( \rho \) is density, \( m \) is mass, and \( V \) is volume
Pressure
\[P = \frac{F}{A}\]Where \( P \) is pressure, \( F \) is force, and \( A \) is area
Spring Constant (Hooke's Law)
\[k = \frac{F}{x}\]Where \( k \) is spring constant, \( F \) is force, and \( x \) is extension
Hydraulic Press Equation
\[\frac{F_1}{A_1} = \frac{F_2}{A_2}\]Where \( F_1 \) and \( F_2 \) are forces, \( A_1 \) and \( A_2 \) are areas
Pressure at Depth in Liquid
\[P = \rho g h\]Where \( P \) is pressure, \( \rho \) is density, \( g \) is gravity, and \( h \) is depth
Problem Categories
Hooke's Law and Spring Constant Problems
These problems involve calculating spring constant, force, or extension using Hooke's law:
- Finding spring constant from force and extension
- Calculating force for a given extension
- Determining extension for a given force
Density Problems
These problems involve calculating density, mass, or volume:
- Finding density from mass and volume
- Calculating mass from density and volume
- Determining volume from mass and density
- Using displacement method for irregular solids
Pressure Problems
These problems involve calculating pressure, force, or area:
- Finding pressure from force and area
- Calculating force from pressure and area
- Determining area from force and pressure
- Maximum and minimum pressure for objects with different orientations
Hydraulic Systems Problems
These problems involve Pascal's law and hydraulic systems:
- Calculating force multiplication in hydraulic presses
- Finding pressure in hydraulic brake systems
- Determining forces on different pistons
Atmospheric Pressure Problems
These problems involve calculations related to atmospheric pressure:
- Finding height of liquid columns in barometers
- Calculating pressure at different depths in liquids
- Pressure variations with altitude
Problem-Solving Strategies
Follow these steps for solving numerical problems effectively:
1. Understand the Problem
- Read the problem carefully and identify what is being asked
- Note down all given quantities with their units
- Identify the unknown quantity to be found
2. Select the Appropriate Formula
- Choose the formula that relates the given and unknown quantities
- Ensure you understand what each variable in the formula represents
- Check if any rearrangements of the formula are needed
3. Convert Units
- Convert all quantities to SI units before calculations
- Pay special attention to:
- Length: cm to m (1 cm = 0.01 m)
- Area: cm² to m² (1 cm² = 0.0001 m²)
- Volume: cm³ to m³ (1 cm³ = 0.000001 m³)
- Mass: g to kg (1 g = 0.001 kg)
4. Perform Calculations
- Substitute the values into the formula
- Perform the calculations step by step
- Keep track of units throughout the calculation
5. Check Your Answer
- Ensure the answer has the correct units
- Verify that the magnitude of the answer is reasonable
- Check if the answer makes physical sense
Unit Conversion Guide
Practice Tips
Effective Practice Strategies
- Start with basic problems: Begin with simple calculations to build confidence
- Focus on unit conversions: Many errors occur due to incorrect unit conversions
- Practice regularly: Consistent practice helps reinforce concepts
- Review mistakes: Analyze errors to understand where you went wrong
- Time yourself: Practice solving problems within time limits
- Create formula sheets: Make personalized formula summaries for quick reference
Created by Hira Science Academy | Aligned with PECTA 2025 Syllabus