Oscillations are a ubiquitous feature of the natural world, from the seasons to

Question

Oscillations are a ubiquitous feature of the natural world, from the seasons to the beating of your heart oscillatory behavior occurs across all time and space scales.

The prototypical example of an oscillation that is frequently explored in introductory physics courses is the mass-spring system:

Where a mass m is attached to the end of a spring with spring constant k (how springy is the spring). x=0 corresponds to the equilibrium or resting point of the spring, x>0 is a stretched spring and x<0 is a compressed spring.

Newton's second law of motion pertains to the behavior of objects for which all existing forces are not balanced. The second law states that the acceleration of an object is dependent upon two variables - the net force acting upon the object and the mass of the object. The acceleration of an object depends directly upon the net force acting upon the object, and inversely upon the mass of the object. This law is frequently stated as F=ma. However what is that net force, F, acting on the object? This is given to us by Hookes law. Hookes law tells us that the restoration force acting on the mass by the spring is proportional to the displacement of the mass from the resting position. That is, F = -kx.

Putting this all together we have ma = -kx. But wait! There is a relationship between position and acceleration. Namely that acceleration is the rate of change of the rate of change of position. That is a long winded way of saying second derivative. This brings us to an example of a second order ordinary differential equation:

This equation states that we wish to find a function x(t) whose second derivative is proportional to minus itself. This week we met functions whose second derivatives are proportional to minus themselves. What are those functions?

Oscillations are a ubiquitous feature of the natural world, from the seasons to the beating of your heart oscillatory behavior occurs across all time and space scales. The prototypical example of an oscillation that is frequently explored in introductory physics courses is the mass-spring system: Oscillations are a ubiquitous feature of the natur This equation states that we wish to find a function x(t) whose second derivative is proportional to minus itself. This week we met functions whose second derivatives are proportional to minus themselves. What are those functions? Oscillations are a ubiquitous feature of the natur Where a mass m is attached to the end of a spring with spring constant k (how springy is the spring). x=0 corresponds to the equilibrium or resting point of the spring, x>0 is a stretched spring and x

Explanation / Answer

The trignometric functions Sin(x) and Cos(x) satisfy the rule.

d(Sin(x))/dx=Cos(x)

d^2(Sin(x))/dx^2=d(Cos(x))/dx=-Sin(x)=-f(x)

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