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).

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).

Name: Solve and show solution for answer given -Show that for a "broken in"
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Description: Solve and show solution for answer given -Show that for a "broken in" clutch, the maximum torque transfer through the clutch occurswhen the inner radius of the clutch "r" is equal to the outer radius of the clutch "ro" dividedby the "√3" (i.e. r₁ = 1

Principles of Physics: A Calculus-Based Text

5th Edition

ISBN:9781133104261

Author:Raymond A. Serway, John W. Jewett

Publisher:Raymond A. Serway, John W. Jewett

Chapter10: Rotational Motion

Section: Chapter Questions

Problem 78P: Review. A string is wound around a uniform disk of radius R and mass M. The disk is released from...

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Review. A string is wound around a uniform disk of radius R and mass M. The disk is released from rest with the string vertical and its top end tied to a fixed bar (Fig. P10.78). Show that (a) the tension in the string is one third of the weight of the disk, (b) the magnitude of the acceleration of the center of mass is 2g/3, and (c) the speed of the center of mass is (4gh/3)1/2 after the disk has descended through distance h. (d) Verify your answer to part (c) using the energy approach. Figure P10.78
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A nylon siring has mass 5.50 g and length L = 86.0 cm. The lower end is tied to the floor, and the upper end is tied to a small set of wheels through a slot in a track on which the wheels move (Fig. P18.76). The wheels have a mass that is negligible compared with that of the siring, and they roll without friction on the track so that the upper end of the string is essentially free. Figure P18.76 At equilibrium, the string is vertical and motionless. When it is carrying a small-amplilude wave, you may assume the string is always under uniform tension 1.30 N. (a) Find the speed of transverse waves on the siring, (b) The string's vibration possibilities are a set of standing-wave states, each with a node at the fixed bottom end and an antinode at the free top end. Find the node-antinode distances for each of the three simplest states, (c) Find the frequency of each of these states.
In Figure P10.40, the hanging object has a mass of m1 = 0.420 kg; the sliding block has a mass of m2 = 0.850 kg; and the pulley is a hollow cylinder with a mass of M = 0.350 kg, an inner radius of R1 = 0.020 0 m, and an outer radius of R2 = 0.030 0 m. Assume the mass of the spokes is negligible. The coefficient of kinetic friction between the block and the horizontal surface is k = 0.250. The pulley turns without friction on its axle. The light cord does not stretch and does not slip on the pulley. The block has a velocity of vi = 0.820 m/s toward the pulley when it passes a reference point on the table. (a) Use energy methods to predict its speed after it has moved to a second point, 0.700 m away. (b) Find the angular speed of the pulley at the same moment. Figure P10.40
an irregularly shaped plastic plate withuniform thickness and density (mass per unit volume) is to berotated around an axle that is perpendicular to the plate faceand through point O. The rotational inertia of the plate about that axle is measured with the followingmethod. A circular disk of mass 0.500 kg andradius 2.00 cm is glued to the plate, with itscenter aligned with point O . Astring is wrapped around the edge of the diskthe way a string is wrapped around a top.Then the string is pulled for 5.00 s. As a result,the disk and plate are rotated by a constantforce of 0.400 N that is applied by thestring tangentially to the edge of the disk.The resulting angular speed is 114 rad/s.What is the rotational inertia of the plateabout the axle?