Show that for a "broken in" clutch, the maximum torque transfer through the clutch occurs when the inner radius of the clutch "r" is equal to the outer radius of the clutch "ro" divided by the "√3" (i.e. r = 12). Start with equation 18.6 (i.e. T = (л)(Pmax)(µ)(N)(ro²ri — r₁³)). Treat "r" as the independent variable, "T" as the dependent variable, and the remaining parameters as "constants" (Note: Essentially, the torque "T" is a cubic function (3rd degree polynomial) of the inner radius "r"). "Maximize" this "cubic function" over a realistic domain using differential calculus. Be sure to show the “critical point" r₁ = 1/3 corresponds to a "relative maximum" (Hint: This can be done by using either the 1st derivative test or the 2nd derivative test).
Simple harmonic motion
Simple harmonic motion is a type of periodic motion in which an object undergoes oscillatory motion. The restoring force exerted by the object exhibiting SHM is proportional to the displacement from the equilibrium position. The force is directed towards the mean position. We see many examples of SHM around us, common ones are the motion of a pendulum, spring and vibration of strings in musical instruments, and so on.
Simple Pendulum
A simple pendulum comprises a heavy mass (called bob) attached to one end of the weightless and flexible string.
Oscillation
In Physics, oscillation means a repetitive motion that happens in a variation with respect to time. There is usually a central value, where the object would be at rest. Additionally, there are two or more positions between which the repetitive motion takes place. In mathematics, oscillations can also be described as vibrations. The most common examples of oscillation that is seen in daily lives include the alternating current (AC) or the motion of a moving pendulum.
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