Consider the differential equation dy dy md2++ky Fo cos(wot) where y = y(t) is a function...

 

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Consider the differential equation dy dy md2++ky Fo cos(wot) where y = y(t) is a function of the independent variable t. This equation models the motion of an object at the end of a spring that is being affected by several forces. y represents the displacement of the object; m is a constant representing the mass of the object; j and k are positive real-valued constants measuring the affect of damping and restoring forces, respectively; and Fo cos(wot) is a function representing the affect of the external force, where Fo and wo are also positive real-valued constants. (a) Suppose that y is measured in meters (m), t is measured in seconds (s), and the external force (Fo cos(wot)) is measured in newtons (kg-m/s). What must the units of m be so that the model makes sense? What about k? Give a brief explanation. (b) Suppose that we rescale the variables y and t according to y=yY and t = tct as was done in class. Use the Chain Rule to show that: dy ye dY dt2 = t dr (c) Find a rescaling of y and t that recasts/nondimensionalizes the original equation into the form: dy dy dr+dr ++aY = cos(T) Where Y is a function of the new independent variable t, and a and are positive, real-valued constants. Clearly state what ye, te, a, and are in relation to the variables and constants of the original equation. Activate Wine Consider the differential equation dy dy md2++ky Fo cos(wot) where y = y(t) is a function of the independent variable t. This equation models the motion of an object at the end of a spring that is being affected by several forces. y represents the displacement of the object; m is a constant representing the mass of the object; j and k are positive real-valued constants measuring the affect of damping and restoring forces, respectively; and Fo cos(wot) is a function representing the affect of the external force, where Fo and wo are also positive real-valued constants. (a) Suppose that y is measured in meters (m), t is measured in seconds (s), and the external force (Fo cos(wot)) is measured in newtons (kg-m/s). What must the units of m be so that the model makes sense? What about k? Give a brief explanation. (b) Suppose that we rescale the variables y and t according to y=yY and t = tct as was done in class. Use the Chain Rule to show that: dy ye dY dt2 = t dr (c) Find a rescaling of y and t that recasts/nondimensionalizes the original equation into the form: dy dy dr+dr ++aY = cos(T) Where Y is a function of the new independent variable t, and a and are positive, real-valued constants. Clearly state what ye, te, a, and are in relation to the variables and constants of the original equation. Activate Wine