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Spring-mass system

Springs make hair bouncy in animations! Hooke's Law explains spring force: it's proportional to stiffness (k) and displacement (change in length). Positive displacement stretches springs (pull force), while negative displacement compresses them (push force). Mass spring systems create realistic hair movement in movies like Brave.

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Video transcript

(bouncy Pixar theme) - In the last exercise you probably noticed that the simulation lacked the natural bounciness of hair. But what makes something look bouncy anyway? To make this model bouncy, we need each strand of hair to expand and contract slightly when forces are applied to them. This required a new physical analogy to base our model on, so we tried springs. Springs are great because they can change length when you pull on them, and to give the hair a little bit of weight, we add a small weight to the end of the spring. This is known as a mass spring system. We can actually draw a mathematical model to explain what happens when a spring expands and contracts. The model we use is based on a law developed by Robert Hooke a 17th century physicist. Hooke noticed that there are two things to consider when a spring expands and contracts. One, if we pull on a spring and it expands, we will increase its length and it will pull back together. Two, if the spring contracts, its length will decrease and it will push apart. How much a spring pushes and pulls is known as the spring force. Robert Hooke was looking for relationship between the spring force in the amount the spring contracts or expands. We call this change in length displacement. Displacement is defined as the current length minus the rest length of the spring. When we stretch a spring, the displacement is positive and the resulting spring force is negative. This is known as the pull force. When we compress a spring, the displacement is negative and the resulting spring force is positive. This is called the push force. So Hooke's observation was quite simple, he noticed that a larger displacement results in a larger force, while a smaller displacement results in a smaller force. That is, he noticed that the displacement is proportional to the force. However, every spring is different, some take a lot of force to displace, and some are really easy to displace. So Robert Hooke introduced the idea of stiffness to account for how hard it is to displace a given spring. It is represented with the letter k. This led Hooke to his final equation which is known as Hooke's Law. The spring force is proportional to a stiffness times the displacement. Notice that there is a negative sign there, that's because we want a positive force when the displacement is negative, and a negative force when it's positive. If we plug this equation into the computer we get this realistic spring behavior. For Brave, we modeled the horse's hair using a mass spring system similar to what we are describing in this tutorial, nearly 10,000 simulated hairs in total. In the next exercise you can explore a simple mass spring system. You'll be able to adjust the following parameters, mass of the particle, spring stiffness, k, and the force of gravity, and we'll ask you some challenge questions to make sure you understand the basics of Hooke's Law.