A ruler of mass m = 75g and L=3.0meters is placed so that it is exactly balanced

Question

A ruler of mass m = 75g and L=3.0meters is placed so that it is exactly balanced across two spinning cylinders a distance d = 1.0meters apart. The coefficient of kinetic friction between the ruler and each cylinder is ?k = 0.2. You move the ruler 12 cm to the right and discover that it pulls back to the left. It then overshoots the center point, goes a maximum of 11 cm to the left and then continues to move back and forth with smaller and smaller amplitude. a) What is the frequency of oscillation in the ruler? b) What is the maximum distance it goes to the right just after it has gone a maximum of 11 cm to the left? c) Create an equation of motion that describes the motion of the ruler.

Explanation / Answer

a) If your FBD boils down to an equation of the form: d2x/dt2=-w2x then you know that you have "simple harmonic motion" and the equation of motion that describes the motion of the object is given as: x(t)=A*cos(wt+?). b) In the case of objects that have small oscillations (i.e. T < 10 degrees) then you can use the "small angle approximation" where sin(T) = tan(T) (T has to be expressed in radians). c) In the case of a damped oscillator, the equation of motion is going to have the form x(t) = (Ae^(-t/t))cost(wt + ?) where t is the time constant.

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