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PhET Hooke’s Law Simulation

Use this simulation to measure the deflection of a spring when a force is applied to it. You can also see what the effects of springs connected in series and in parallel have on the deflection of the springs.

Video Transcript

Hello there! Welcome to lecture 12: solids!

Solids are one of the four most common states of matter. This video will explain the properties of solid objects, and the effects those have on our daily lives. This knowledge will help explain topics from the stretching of a slinkey, to the design of bridges and buildings, to why cells divide.

Each of the following concepts will be discussed in this video: states of matter, the structure of solids, density, elasticity, forces in solids, and scaling.

States of Matter

In our everyday lives, we experience four states of matter: solid, liquid, gas, and plasma. We’ll be covering these phases of matter in this and the next two lectures.

A solid is a material where the molecules or atoms are very tightly bound together. This gives a solid a very rigid volume and shape. Solid objects do not change their shape to fit into a container, as a liquid does. Solids also do not change their volume to take up all available space, as a gas does.

The Structure of Solids

The atoms in a solid can be arranged in one of a few ways. When the atoms are arranged in an orderly, repeating manner, that solid is called a crystalline solid. In a crystal, knowing the location of one set of atoms allows us to determine the location of all other atoms in the solid, because that atomic pattern, known as a unit cell, will repeat itself regularly. Examples of crystalline solids include: diamonds, quartz and snowflakes.

Some crystalline solids have special properties. Silicon, for example, is grown into large crystals called boules in labs. These silicon boules are sliced into wafers. The crystalline nature of silicon makes it a high-quality semiconductor, and it is used predominantly in computer chip manufacturing.

Other solids have atomic arrangements that are disordered, or that otherwise lack an orderly arrangement. These solids are known as amorphous solids. Examples of amorphous solids include: plastics, wax, and glass.

In between amorphous and crystalline materials is a class of material called polycrystalline. Polycrystalline materials have some order to them, but are not made of a single crystal. You can think of polycrystals as being composed of many individual crystals. Most metals and ceramics are polycrystalline.

The properties of crystalline, polycrystalline, and amorphous solids are a subject of interest to materials scientists. Materials scientists determine the properties of materials, and design new materials, for use in everything from industry and manufacturing to biomaterials and building materials.

Density

Density defines how compact the matter in an object is. The symbol for density is the capital letter D. The equation for density is mass divided by volume. Because the units of mass are kilograms and the units of volume are meters-cubed, the units of density are kilograms per cubic meter.

Sometimes we may use units of grams per cubic centimeter for volume. How can we convert from one to the other? To convert from grams per cubic centimeter to kilograms per cubic meter, multiply by one thousand. To convert from kilograms per cubic meter to grams per cubic centimeter, divide by one thousand.

The density of gold is 19.3 grams per cubic centimeter. Multiplying by one thousand, we see that is equal to 19,300 kilograms per cubic meter.

The density of water is 1,000 kilograms per cubic meter. Dividing by one thousand, we see that is equal to 1 gram per cubic centimeter.

Density is a property inherent to pure substances. That is, the density of gold is always going to be 19.3 grams per cubic centimeter. The densest naturally occurring element is osmium, which has a density of 22.6 grams per cubic centimeter.

To measure the density of a substance, first measure the mass of the object. To measure volume, there are two methods that can be used. The first is used with regularly shaped objects where we can measure a length, width, and height. Use a ruler or calipers to measure each dimension and multiply them together to find the volume.

This acetate cube has a mass of 7.5 grams. I used calipers to measure the length, width and height of the cube. These dimensions are 1.87 cm, 1.85 cm, and 1.86 cm. The volume is length times width times height, which is 6.43 cubic centimeters. Using the equation density equals mass divided by volume, we can calculate the density to be 1.17 grams per cubic centimeter.

What if we have an irregularly shaped object whose volume cannot be measured with a ruler? We need another way to measure the volume. We can use the liquid displacement method. Fully submerge the object in a volume of water and see how much the water level rises. Every milliliter of water rise equals a volume of one cubic centimeter.

This piece of granite has a mass of 44.6 grams. When I placed it into a graduated cylinder, the water level increased by 20 milliliters. Therefore the density of this piece of granite is 2.23 grams per cubic centimeter.

Elasticity

Elasticity is the property of a solid object to return to its original shape after becoming deformed by a force. If you’ve played with a Slinkey before, you’ve probably seen this property. If a slinkey is deformed by stretching it out lightly, once that force is removed, it will return to its original shape and configuration.

The application of force causes an object to deflect. When the force is removed, the object will go back to its original shape. However, after a certain amount of force is applied, that object can no longer go back to its original configuration. This happens when we exceed the yield strength of an object. If you’ve ever seen a Slinkey that won’t quite go back to its coiled up position, then you’ve seen this before.

Adding even MORE force will eventually cause an object to break. This happens when we exceed the ultimate strength of an object. Enough force will cause any elastic object to break: a Slinkey, a rubber band, or a spring.

Elasticity is defined mathematically for springs using Hooke’s law. Hooke’s law states that F equals k times delta x. F is the force applied to the spring. Delta x is the distance that the spring deflects as a result of that force. The variable lowercase k is called the spring constant. The units of the spring constant are Newtons per meter. You can think of the spring constant as being like the stiffness of the spring: the larger the spring constant, the more force that needs to be applied to make it stretch.

Let’s calculate the spring constant of a spring.

In this demo, I pull on a spring with a force probe. It takes about 0.8 newtons of force to stretch it 10 centimeters, and 1.6 newtons to stretch 20 centimeters, and so on. The spring constant k is equal to force divided by deflection distance. 0.8 newtons divided by 10 centimeters is 0.08 newtons per centimeter.

What if we connect springs together? Springs can be connected in series such that the springs join together like one long line, each spring connected together end to end. When a mass is placed on series springs, each spring will stretch out based on its spring constant, as both of the springs feel the full force of that mass.

The effective spring constant of springs in series is smaller than the individual spring constants: the stiffness is lower because the spring combination stretches more than any one individual spring would stretch.

Springs can also be connected in parallel such that the springs join together side-by-side. When a mass is placed on parallel springs, the force from that mass is distributed between both springs. Therefore they will stretch out less than they would do if they were individual or connected in series.

The effective spring constant of springs in parallel is larger than the individual spring constants: the stiffness is higher because the spring combination stretches less than any one individual spring would stretch.

Forces in Solids

Solids are present in all aspects of our lives. Possibly, as you watch this video, you’re in a house or a building of some sort. That structure is built out of solid objects that are engineered to create a durable, robust building that does not move, even when people walk around in it, work in it, or when weather interacts with the outside of the building. How does that work?

All solid objects deflect to some degree when a force is placed on them. Take this meter stick. As I place masses on it, the meter stick deflects. When a lot of mass is placed on top, the meter stick bends noticeably.

When I stand on the floor, or when I sit in a chair, or when forces act on any solid objects, those solids will deflect. The materials used in houses, chairs, tables, and other things that don’t visibly bend when we use them have been chosen because they don’t deflect much under the influence of the types of forces they will be subjected to. But just because we can’t see it, doesn’t mean it isn’t happening.

In this demo, a laser has been set up to reflect off of a mirror on the door frame and return to a solar panel. That solar panel is then connected to a speaker. The speaker emits noises any time the distance between the laser and the door frame changes, using a principle called interference that we will discuss in future lectures. This interference principle is capable of detecting really small changes in distance.

When Dr. Fazzini pushes on the door frame, the speaker emits noises, indicating that the distance has changed. When he pushes on the door frame, the metal deflects. Dr. Fazzini has literally bent the wall by pushing on it! Thankfully, the deflection is small and the building stays upright.

The two major forces that act on solid objects are tension and compression. Tension is the force that exists when something is pulled or stretched apart. Compression is the opposite: it is the force that exists when something is pushed or squeezed together.

Going back to our meter stick example, when forces are placed on the meter stick and it deflects, the force causes the top of the meter stick to stretch out: that is a tension force. But the bottom of the meter stick is squeezed together: that is a compression force.

What happens in between the top and bottom of the meter stick? The space in between is known as the neutral layer. There are no forces acting in this part of the meter stick. This interesting fact explains the design of i-beams. An i-beam is used in the construction of buildings. If they are bolted together at both ends, the forces of tension and compression will exist on the top and bottom if the beam is deflected. Because the tension and compression forces don’t exist in at the center of the beam, engineers can cut down on the construction materials used in this area. Notice that the top and bottom of the i-beam, which will be subjected to the most forces, has the most material present. In the center, where there are no forces, less material is needed, so the center is carved out. This reduces the cost and weight of the building materials.

Another place you may notice compression and tension forces is if you ever see an empty flatbed truck driving down the highway. Notice that the bed of the flatbed truck is curved upward. When a heavy load of objects is placed on the flatbed, it deflects and becomes flat. What would happen if the flatbed truck started out flat, and then deflected downward when a heavy load was placed on top? Think about it!

Scaling

Let’s talk about scaling. This topic has a lot of important ramifications in chemistry, biology, and physics. Scaling has to do with the properties of a solid object and how they change when we make an object larger or smaller. The properties we are interested in are: the surface area of the object, the volume of the object, and the weight of the object. We’ll measure these properties for a few cubes to come up with scaling parameters.

Surface area gives us an indication of how strong an object is. If you’ve ever played sports or done weight training, you may have noticed your muscles increase in surface area as they become stronger. We can expect that an object with a larger surface area will be stronger than an object with a smaller surface area – if all other things are kept equal.

Volume and density tell us how massive an object is, or how much it weighs when it is subject to the force of gravity.

Surface area divided by volume tells us how strong something is compared to its weight. This is an important concept from a structural standpoint. An object that is very strong but weighs a lot will be less structurally sound than something that has the same strength and weighs less. An ant is very strong: the surface area of an ant’s legs is small, but they have very little volume, so the surface area to volume ratio is rather high. When we look at larger insects and animals, we see that the larger an animal is, the larger the surface area of its supporting limbs to support their weight. An elephant has very thick legs to keep the surface area to volume ratio high enough for the elephant to be able to walk around.

Let’s see how surface area divided by volume changes as we scale something larger and larger.

Let’s start with a cube that measures one centimeter by one centimeter by one centimeter. This cube has six faces each with a face area of one square centimeter. The total surface area is six square centimeters. The volume of the cube is length times width times height: 1 cubic centimeter. We can calculate the surface area to volume ratio as 6 inverse centimeters.

Let’s say we put the cube in a machine that can make an object larger. The new cube is two centimeters by two centimeters by two centimeters. Each face has an area of 4 square centimeters, so this new cube has a surface area of 24 square centimeters. That’s much bigger than before! But now the volume has increased to 8 cubic centimeters, making the surface area to volume ratio a mere 3 inverse centimeters.

Let’s make the cube even larger. The new cube is three centimeters by three centimeters by three centimeters. The surface area is now 54 square centimeters. Now the volume is 27 cubic centimeters, so the surface area to volume ratio is now only two inverse centimeters.

As we continue to make the cube larger and larger, we see that surface area increases… but it does not increase as fast as the volume! A cube with an edge length of x centimeters (where x can be any number we choose) will have a surface area of 6 times x-squared, but a volume of x-cubed. This means the surface area to volume ratio is 6 divided by x. The larger x becomes, the smaller the surface area to volume ratio.

From a structural standpoint, this means that we cannot simply make things larger and expect them to be as structurally sound. An ant, blown up to the size of an elephant, would not be able to support its own weight. Sci-fi movies featuring giant spiders destroying cities are getting physics wrong: those giant insects would have a very small surface area to volume ratio, making those giant insects completely unable to move!

Surface area and volume don’t only matter to structural and civil engineers. It also matters in chemistry, biology, and other related fields. Think of a cell. As a cell becomes larger in volume, it requires more nutrients. But nutrients can only be absorbed through the cell wall, which is only as large as the surface area. At a certain point, if a cell becomes too big, it cannot obtain enough nutrients or other chemicals to carry out its functionality. Instead of becoming larger and larger, cells divide. This keeps the surface area to volume ratio sufficient for cells to continue to survive.

Surface area also plays a large role in heat transfer. Many computer processors have a heat sink placed over the processor. As you browse the internet, play games on your computer, or write up an essay for your English class, the computations that your computer performs causes the processor to heat up. That heat is then dissipated through the heat sink, which turns on a fan to blow cooling air over the system and keep things from melting. A heat sink that uses airflow to dissipate heat usually has a large surface area. The more surface area there is, the easier it is for heat to get transported away. You also see this in air-cooled cylinders in airplane engines, among other things.

As a final example, I will show the effect that surface area has on chemical reactions. In this demo, I start with a tin plate covered with lycopodium powder. This is a dry powder that is also very flammable. When I try to ignite the lycopodium powder, I have a hard time doing so. Only a little bit of it burns before the fire goes out. This is because there is very little surface area available to combust with the surrounding fire and oxygen.

When I instead blow individual lycopodium powder into an open flame, the result is that the lycopodium powder explodes. By exposing more of the surface area, the chemical reaction is more readily enabled, and we can see that very dramatically in this example.

Chemical reactions will be more vigorous when the reactants have more surface area than when they have less. This is true for experiments in test tubes as well as in fires. Grain elevators are at risk of explosion after dust has been stirred up when grains have been newly introduced, stirring up lots of particles and exposing a lot of surface area. If there are any sparks that occur during these times, grain elevators can explode.

Thanks for taking the time to learn about solids. Until next time, stay well.