Question

The term oscillator describes any system that has cyclic or near-cyclic behavior. The periodic motion of a planet or a pendulum, the alternating current in an electrical circuit, the vibrations of cesium atoms in an atomic clock are examples of oscillators. It turns out that many oscillators are governed by differential equations- equations that involve a function and its derivatives. In this project we explore solutions of a common oscillator equation. Imagine a block hanging vertically from a spring that is attached to a solid support. If the block is pulled down or pushed up from its rest position and then released, it bounces back and forth, becoming an oscillator. The goal is to describe the vertical displacement of the block for all times after it is released. The displacement at time to is y(t). WO)>0 where y is positive when the block is above its rest y=0) - position (Figure 1). (1)<0 Newton's Second Law of motion says that mass x acceleration = sum of all external forces. The force that makes the block bounce back and Figure 1 forth is the restoring force of the spring. The restoring force obeys Hooke's Law: The force acts in the direction opposite to that of the displacement (if y> 0, the restoring force acts in the negative direction and vice versa). Furthermore, the restoring force is proportional to the displacement (the farther the spring is stretched, the stronger the force). Therefore, the restoring force has the form F, = -ky, where k> 0 is a spring constant that measures the strength of the spring. The moving block is also subject to some kind of frictional force, perhaps air resistance or energy loss in the spring. The frictional force opposes the motions and is commonly assumed to be proportional to the velocity of the block, which is y'(t). Therefore the frictional force is F;= -cy', where c> is a constant. Therefore, the equation of motion for the block is my"(t) F,+F, =-ky(t)-cy'). massacceleration restoring force friction force Dividing through this equation by m and rearranging, we have y"(t)+ay'(t)+wy(t) = 0, where co?= k/m and a=c/m are positive constants. The goal is to find displacement functions y that satisfy this equation. 1. First consider the ideal (unrealistic) case in which there are no frictional forces, which means we seek functions that satisfy the equation y"(t)+oʻy(t) = 0 or y"(t)=-oʻy(t). The solutions are functions whose second derivative is a multiple of itself. Verify that the functions y=sin ot and y = cos ot satisfy this equation.

Answer 1

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