You will explore the effects of scaling on the strength and stability of an object.
After completing this tutorial, you will be able to complete the following:
Scaling an object bigger makes it weaker because each unit of cross-sectional area is under more force created by the increased mass of the scaled object. The cross-section of an object is the section created when a plane cuts across a solid. As an object is scaled bigger, the strength of the object depends on the amount of force its cross-sectional area can endure. To find the cross-sectional area, use the correct area formula for the object.
· Cube or rectangular solid: Area=length × width
· Cylinder: Area = ? × r2
· Square-based pyramid: Area = base2 (for largest cross-sectional area)
As an object is scaled bigger, its volume increases faster than its cross-sectional area. To calculate the volume of a solid, use the correct volume formula.
· Cube or rectangular solid: Volume = length × width × height
· Cylinder: volume = ? × r² × height
· Square-based pyramid: volume = area of the base × height × 1/3
As an object is scaled larger, its weight increases faster than its cross-sectional area. Given the mass of an object it is easy to find the weight. Multiply the mass (in kg) by the force due to gravitational acceleration. The force of gravity is 9.8 m/s2. Weight is expressed in Newtons, the amount of force needed to accelerate a one kilogram mass one meter per second per second.
Weight = mass × gravity (9.8 m/s2)
As an object is scaled larger, the density will not change. The density of an object is the amount of mass per unit of volume. To calculate density, divide the mass by the volume.
To evaluate the relative strength of a scaled object, calculate the ratio of the weight to cross-sectional area. The stronger the force acting per unit area, the weaker an object is. By comparing the ratios of the weight to cross-sectional area it can be proved that a larger scaled object will be weaker than a smaller similar object. The smaller the ratio of weight to cross-sectional area, the stronger the object will be, because there is less force (weight) acting on each unit of area. The example below compares the strength of a 10 cm cube to a 20 cm cube. The dimensions of the cube were doubled, but the ratio of the weight to cross-sectional area more than doubles. The smaller cube is significantly stronger than the larger cube.
10cm cube of water, mass = 1kg
Weight = 1kg × 9.8m/s2=9.8N
Cross-Sectional Area = 10cm × 10cm = 100cm = 1m
20cm cube of water, mass = 8kg
Weight=8kg × 9.8m/s2 = 78.4N
Area=20cm × 20cm = 200cm = 2m
|Approximate Time||20 Minutes|
|Pre-requisite Concepts||Learners should be familiar with the common properties of matter and calculating the volume of a sphere, cylinder, cube, rectangular prism, pyramid, and cone.|
|Type of Tutorial||Concept Development|
|Key Vocabulary||area, centimeter, cube|