Conceptual Physics: Chapter 12 Solids
Weight depends on volume, and strength comes from the area of the cross-section of limbs -tree limbs or animal limbs
A 1 cm cube has a cross-section of 1 cm² and its volume is 1 cm³. A cube of the same material that has double the linear dimensions has a cross-sectional area of 4 cm² and a volume of 8 cm³
A tree branch is found to obey Hooke's law. When a 20 kg load is hung from the end of it, the branch sags 10 cm. If a 40 kg load is hung from the same place, how much will the branch sag? What would you find if a 60 kg load were hung from the same place?
A 40 kg load has twice the weight of a 20 kg load. In accord with Hooke's law, F ~ Δx, The branch should sag 20 cm. The weight of the 60 kg load will make the branch sag 30 cm.
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A body's elasticity describes how much it changes shape when a deforming force acts on it, and how well it returns to its original shape when the deforming force is removed
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A horizontal beam supported at one or both ends is under stress from the load it supports, including its own weight. It in undergoes a stress of both compression and tension (stretching)
Elastic
A material that returns to its original shape after it has been stretched or compressed
Density
A measure of how much matter occupies a given space; it is the amount of mass per unit of volume Density = mass/volume
Specific gravity
A standard measure of density; the ratio of the mass of a substance to the mass of an equal volume of water (A substance that weighs five times as much as an equal volume of water has a specific gravity of five. Specific gravity is a ratio of the density of a material to the density of water. Specific gravity has no units.)
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Air resistance depends on the surface area of the moving object. If you fell off of a cliff even with air resistance, your speed would increase at the rate of very nearly 1 g-unless you wore a parachute. Small animals need no parachute. They have plenty of surface area relative to their small weights.
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As an object grows proportionally in all directions, there is a greater increase in volume than in surface area
Neutral layer
Between the top and the bottom there is a region that is neither stretch more compressed
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Density is a property of a material; density varies somewhat with temperature and pressure
The density of gold is 19.3 g/cm³. What is it specific gravity?
Density of gold/density of water = 19.3/1.0 = 19.3 Specific gravity of gold is 19.3
Weight density
Expressed by the amount of weight a body has per unit of volume - weight density is commonly used when discussing liquid pressure Weight density = weight/volume
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Hooke's law holds only as long as the force does not stretch or compress the material beyond its elastic limit
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If an elastic material is stretched or compressed more than a certain amount it will not return to its original state
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If the linear dimensions of an object are multiplied by some number, the area will grow by the square of the number, and the volume will grow by the cube of the number
Inelastic
Materials that do not resume their original shape after being distorted (clay, putty, dough)
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Not all crystals are evident to the naked eye - their existence in many solids was not discovered until x-rays became a tool of research
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Not all materials return to their original shape when a deforming force is applied and then removed
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One of the properties of solids (as well as liquids and even gases), is the measure of how tightly the material is packed together
Crystals
Regular geometric shapes whose components are arranged in an orderly, repeating pattern
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Smaller objects have more surface area per kilogram. (Cooling occurs at the surface of objects, so crushed icd will cool a drink faster than cubed icd. Crushed ice presents more surface area to the beverage
Hooke's law
The amount of stretch (or compression), x, is directly proportional to the applied force F. (F ~ Δx) (Double the force and you double the stretch; triple the force and you get three times the stretch, and so on)
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The bottom part of the beam is compressed. Atoms there are pushed toward one another, so the bottom part is slightly shorter
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The density of a material depends upon the masses of the individual atoms that make it up, and the spacing between those atoms
Which has greater density-1 kg of water or 10 kg of water? 5 kg of lead or 10 kg of aluminum?
The density of any amount of water is 1.00 g/cm³. Any amount of lead always has a greater density than any amount of aluminum.
Elastic limit
The distance at which permanent distortion occurs
If you make a hole horizontally through a tree branch, what location weakens it the least -top, the middle, or the bottom?
The middle. Fibers in the top part of the branch are stretched and fibers in the lower part are compressed. In the neutral later the hole will not affect the strength of the branch
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The proportion of things in nature are in accord with their size
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The rusting of iron is also a surface phenomenon. The greater amount of surface exposed to the air, the faster rusting takes place.
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The shape of a crystal mirrors the geometric arrangement of atoms with in the crystal
A force of 10 N stretches a certain spring 4 cm, how much stretch will occur it for an applied force of 15 N?
The spring will stretch 6 cm. By ratio and proportion: 10 N/4 cm = 15 N/x Then X= (15 N) x (4 cm)/(10 N) = 6 cm
Scaling
The study of how size affects the relationship between weight, strength, and surface area (as the size of a thing increases, it grows heavier much faster then it grows stronger)
Consider a beam that is supported at both ends, and carries a load in the middle
The top part of the beam is in compression and the bottom is in tension. Again, there is a neutral layer along the middle portion of the length of the beam.
Consider a beam with a weight on the end (a cantilever beam)
The top part of the beam is stretched and the bottom part is being compressed. The middle portion is neither stretched or compressed
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The top part of the horizontal beam is stretched. Atoms are tugged away from one another and the top part is slightly longer.
How does surface area compare with volume?
Volume grows as the cube of the enlargement, and both cross-sectional area and total surface area grow as the square of the enlargement. As an object grows, the ratio of surface area to volume decreases
Suppose a cube 1 cm long on each side were scaled up to a cube 10 cm long. What would be the volume of the scaled up cube? What would be its cross-sectional surface area? It's total surface area?
Volume of the scaled up cube is 1000 cm³. It's cross-sectional surface area is 100 cm². It's total surface area is 600 cm².
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Weight grows as the cube of linear enlargement, while strength grows as the square of linear enlargement
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When linear dimensions are enlarged, the cross-sectional area (as well as the total surface area) grows as the square of the enlargement, whereas volume and weight grow as the cube of the enlargement. As the linear size of an object increases, the volume grows faster than the total surface area
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When you hang a weight on a spring, the weight applies a force to the spring and it stretches in direct proportion to the applied force
