PROBLEMS, SECTION 41. Given a circle of radius a, find by integration using polar coordinates(a) its area;(b) the centroid of one quadrant;(c) the moment of inertia of a circular lamina about a diameter;(d) its circumference;(e) the centroid of a quarter circle arc.2. Using polar coordinates :(a) Show that the equation of the circle sketched is1= 2a cos 0. Hint : Use the right triangle OPQ.(b) Find its area by integration.(c) Find the centroid of the first quadrant semicircular area.(d) Find the moments of inertia of a circular lamina in thisposition about each of the three coordinate axes, as-suming constant density.(e) Find the length and the centroid of the semicircular arc in the first quadrant.(f) Find the center of mass and the moments of inertia of the circular area if the density is r.(g) Find the area common to the circle sketched and the circle r s a.P(r, a)2a

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Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section11.6: Polar Equations Of Conics

Problem 40E

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Gear Trains Figure 8 shows a single-stage gear train. Gear trains are used in many products, such as clocks and automotive transmissions, to reduce or increase the angular velocity of a component. The size of each gear is measured by the number of teeth rather than the radius. Suppose the first gear has n1 and the second gear has n2 teeth. Because the spacing of the teeth is the same for both gears, the ratio of their radii will be equivalent to the corresponding ratio of the number of teeth. When two gears are meshed together, they share the same linear velocity. If 1 and 2 are the angular velocities of the first and second gears, respectively, then v2=v1r22=r112=r1r212=n1n21 The first gear in a single-stage gear train has 42 teeth and an angular velocity of 2 revolutions per second. The second gear has 7 teeth. Find the angular velocity of the second gear.
Gear Trains Figure 8 shows a single-stage gear train. Gear trains are used in many products, such as clocks and automotive transmissions, to reduce or increase the angular velocity of a component. The size of each gear is measured by the number of teeth rather than the radius. Suppose the first gear has n1 and the second gear has n2 teeth. Because the spacing of the teeth is the same for both gears, the ratio of their radii will be equivalent to the corresponding ratio of the number of teeth. When two gears are meshed together, they share the same linear velocity. If 1 and 2 are the angular velocities of the first and second gears, respectively, then v2=v1r22=r112=r1r212=n1n21 The second gear in a single-stage gear train has 6 teeth and an angular velocity of 90 revolutions per minute. The first gear has 54 teeth. Find the angular velocity of the first gear.

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1. Given a circle of radius a, find by integration using polar coordinates (a) its area; (b) the centroid of one quadrant; (c) the moment of inertia of a circular lamina about a diameter;