A crystalline solid consists of atoms stacked up in a repeating lattice structure. Consider a crystal as shown inFigure P1.1a. The atoms reside at the corners of cubes of side L = 0.200nm. One piece of evidence for the regular arrangement of atoms comes from the flat surfaces along which a crystal separates, or cleaves, when it is broken.Suppose this crystal cleaves along a face diagonal, as shown in Figure P1.1b. Calculate the spacing d between two adjacent atomic planes that separate when the crystal cleaves.

You are making a sculpture that is a pyramid with a square base . You want the height of the pyramid to be 4 inches less than the length of a side of the base . You want the volume of the sculpture to be 200 cubic inches . a . Let x represent the length ( in inches ) of a side of the sculpture's base . Draw a diagram of the sculpture , and label the dimensions in terms of x . b . Write a function that gives the volume V of the sculpture in terms of x .

One of the most well-known equations in fluid mechanics is the Bernoulli equation. The standard form of the Bernoulli equation for incompressible irrotational fluid flow is (see pic below)(P is pressure, u is flow velocity of the fluid, z is the height of fluid relative to a reference line)a. Verify that each additive term in the Bernoulli equation has the samedimensions. b. What are the dimensions of the constant C?

A bacteria colony which is unrestricted in its growth will grow out in a circular manner. The population ?(?) of the colony at time t is directly proportional to the area of the circle. Suppose that a particular bacteria grows in such a way that the population is 1.3 times larger each day, and that initially (t=0) the colony has a diameter of 4 centimeter and a population of 7 million. a) Find the Population and Diameter of the colony on the fifth day. b) Find the Rate of Change of the Population and Diameter of the colony on the fifth day. c) Unrestricted growth models are often impractical as values will quickly become unreasonable. On what day does our model predict that the area of the bacteria colony will be 1 square mile?

Schwarzschild radius RS of a black hole is the maximum distance from the black hole’s center at which light cannot escape its gravitational field. The quantity RS (with dimensions of length) is dependent on the mass of the black hole M, the speed of light c, and the gravitational constant G. Based on the dimensions of these four parameters, predict an equation for the Schwarzschild radius. Hint: G has dimensions of [L3/MT2]

A hopskip is a (fictional) unit of length equal to 1.5 meters (1 hs = 1.5 m). The volume of the Earth’s oceans is estimated to be ~ 1.3X 10^24cm  . Use dimensional analysis to express the volume of the Earth’s oceans in hs^3.