Derive Lagrange’s equation of motion for the rolling cylinder.

Derive Lagrange’s equation of motion for the rolling cylinder. A homogeneous circular cylindrical segment of radius R, length L, height h, and mass m perform s rocking oscillations without slipping on a rough horizontal surface. The center of mass is at r from the center O. The segment is released from rest at the placement 9(0) = 90

• (a) Derive the differential equation for the finite angular motion 9

(1) by

(i) application of Lagrange’s equations, and

(ii) by use of the Newton- Euler equations.

(b) Determine the first integral of the equation of motion.

(c) Derive an equation for the period of the large amplitude oscillations.

(d) Find the circular frequency for small oscillations.

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The agave bush and the desert tortoise are both found in the Sonora Desert in the Southwest of North America. Known as the century plant, the agave blooms in June to August but only once after about 25 years and then dies. In contrast, tortoises may live in excess of 100 years and when they reach the age of around 15 years they lay eggs between May and July at 2-3 year intervals. What type of reproductive strategy is demonstrated by each of these species? View keyboard shortcuts As previous stated, tortoises may lay multiple clutches of eggs with varying numbers per clutch depending on the amount of rainfall during the summer monsoon season in the desert. In general more rain results in 1 or two more clutches per season and more eggs per clutch. Would this be a density dependent or density independent factor with respect to population growth of tortoises and what specifically in the information above made you come to this conclusion? Desert tortoises may live up to 100 years and don't reach sexual maturity typically until 15 years of age. Based on all the information presented on tortoises in this quiz, would consider them to be more or a r or K selected species and what are two specific facts from the information presented that support your conclusion? Assume a population of desert tortoises has a per capita growth rate (r) or 0.039. If this rate could be sustained, how long would it take for the population that is currently 186 individuals to reach 450 individuals? (just give the simplest expression with the data provided)

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