1 Math 171 Group Worksheet over Chapter 3 Please submit work on a separate sheet of paper. Problem 1. A peg on the end of a rotating crank slides freely in the vertical guide shown in the diagram. The guide is rigidly connected to a piston which moves horizontally. The crank rotates at a constant angular speed of Rad/s in a circle of radius r. Placing the origin at the point O indicated on the diagram, then the location of the vertical guide at time t is given by x(t) = r cos(t). (assuming it starts at = 0). The location of the right face of the piston at time t is given by P (t) = L + x(t). Find the velocity, acceleration, and jerk for the piston. Problem 2. Visual angles are important to predator avoidance. Here we describe part of a calculation done by Larry Dill , a biologist at Simon Fraser University in his study of the Zebra danio, a small tropical sh, which has many predators (larger sh) eager to have it for dinner. The gure below shows the relation between the angle subtended at the Danio's eye, , and the prole size S of an approaching predator, located at a distance x away. Let = /2. d S = dx 2(x2 + S 2 /4) b. From the gure, it seems clear that = 2 increases as the predator approaches the prey. d Why, then, is negative? dx a. Show that 2 Problem 3. Using a simple energy balance of input energy from the sun and energy lost form the earth due to radiation into space one nd that the temperature of the earth, T is given by: 1/4 (1 a)S T = where is the emissivity of the Earth's atmosphere, which which represents the earth's tendency to emit radiation energy. This constant depends on cloud cover, water vapor as well as on greenhouse gas concentration in the atmosphere. is a physical constant (the Stephan-Boltzmann constant) which is xed for the purpose of our discussion. S is the incoming radiation energy per unit area (also called the solar constant) and 0 a 1 is the fraction of that energy reected. a is also called the albedo and depends on cloud cover, and other aspects of the planet such as percent forest, snow, desert and ocean as well as the level of greenhouse gases. a. Suppose that a depends on G the level of green house gases, and one has an estimate of da/dG. Use the change rule to compute dT /dG. b. If da/dG is negative, is dT /dG positive or negative? d is c. Suppose that a is constant but depends on G. Assume that DG given. Determine the rate change of temperature with respect to the level of greenhouse gasses in this case. dT d. Suppose that both a and depend ion G. Find in this more general dG case. (Hint: the quotient rule as well as the chain rule will be needed in this case.)