Saturday, May 15, 2010
A nearly frictionless near-ramp is manufactured to have an angle of 25 ± 0.01º with the horizontal in one direction, and 0 ± 0.023º in a horizontal perpendicular to that direction. Its profile is a flat plane within a tolerance of 0.1%. It has a linear thermal expansion coefficient of 0.12 ± 0.04 mm/K. A block which is a rectangular prism whose corners are 90º to within 0.134% and whose edges are flat to within 0.072% with dimensions specified to within a tolerance 0.234% is placed on the ramp at a distance along the ramp of 1.43 ± 0.06 m from a fixed point that has been defined as "the ground." The block has a linear thermal expansion coefficient of 0.23 ± 0.03 mm/K. The initial average temperature of the ramp is 301.4 ± 0.2 K fluctuating with a standard deviation of 2.3 ± 0.26 K over the volume of the ramp. The initial average temperature of the block is 301.2 ± 0.3 K fluctuating with a standard deviation of 3.4 ± 0.15 K over the volume of the block. The air within a sphere with a radius of 5.0 ± 0.1 m centered at the initial point of contact between the block and the ramp has a temperature of 301.5 ± 0.1 K fluctuating with a standard deviation of 0.4 ± 0.1 K over the volume of the air. The air in the immediate vicinity of the block has a viscosity of 14.2 ± 0.1 × 10^-6 m²/s, however this will change with the temperature of the air. The block has a cross-sectional area of 223 ± 0.32 cm². The block and the ramp have a coefficient of kinetic friction of 0.0056 ± 0.0004 at their initial temperatures, however this will decrease with increasing temperature of either. The block has a heat capacity of 1,561 ± 1.2 J/K, and the ramp has a heat capacity of 12,341 ± 3.6 J/K. These will also increase with increasing temperature. The block/ramp system is located at an altitude of 1,223 ± 2.4 m above a sphere defined as "sea level." The sun is located 18.1 ± 0.6º from the vertical axis on a line 126.1 ± 0.2º from the down-angle of the ramp. The moon is located 163.6 ± 0.5º from the vertical on a line 32.4 ± 0.3º from the down-angle of the ramp. Taking all of these factors into account, as well heat generated by friction between the ramp and the block, between the block and the air, and between the ramp and the air, calculate the time it will take the block to reach the plane defined as "the ground" as well as its final velocity when it does so. List appropriate tolerances.
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