Find the steady-state temperature distribution u in a semi-infinite solid cylinder (Fig. 2) of radius a if the base (at z-0) is held at 60° and the curved sides at 0'.

2.4 Consider a cell that has a magnetic bead attached to it. We will treat the cell as a Kelvin body. Suppose that the magnetic field has been turned on for a very long time and is producing a constant force on the bead Fo. At time t=0, the force is suddenly switched off. Derive an expression for the resulting displacement of the bead as a function of time, x(t). Note that you do not need to re-derive any formulae given in the text.

J 1. Using Lagrangian mechanics, derive the equations of motion of a cart with two tires under the cart shown in Figure P.4.1.

A metal rod of length L=8.88 has only three forces applied to it in the directions shown in the diagram below. Force #2 is applied at the midpoint of the rod. If the rod is in static equilibrium and the magnitude of force #1 is 7.15, what is the magnitude of force #2?

Please don't provide handwritten solution ....

3. (Problem 2.6) A stenosis is a narrowing of a blood vessel or valve. Stenosis of a blood vessel arises during atherosclerosis and could occlude an artery, depriving the tissue downstream of oxygen. Further, the fluid shear stresses acting on the endothelial cells lining blood vessels may affect the expression of genes that regulate endothelial cell function. Such gene expression can influence whether the stenosis grows or not. Consider a symmetric stenosis as shown in Figure 2.36 (p. 112). Assume the velocity profile within the stenosis of radius Ri(z) has the same shape as the profile outside the stenosis and is represented as Vz(r) = Vmax 1- R2 Outside the stenosis, the radius equals Ro and the maximum velocity is constant. Within the stenosis, the radius of the fluid channel R(z) equals Ro{1–0.5 [1-4(+)³]"} R(z) = Ro The origin of the z axis is the midpoint of the stenosis. (a) Develop an express for Vmax in a stenosis in terms of the volumetric flow rate Q, cylindrical tube of…

2.2 A rectangular plate with a V-notch is shown in Fig. 2.29. Model the plate using triangular elements by taking advan- tage of the symmetry of the system. FIGURE 2.29 A rectangular plate with a notch.

You are watching a new bridge being built near your house. You notice during the construction that two concrete spans of the bridge of total length L; = 270 m are placed end to end so that no room is allowed for expansion (figure (a)). In the opening storyline for the thermodynamics chapter, we talked about buckling sidewalks. The same thing will happen with spans on bridges if allowance is not made for expansion (figure (b)). You want to warn the construction crew about this dangerous situation, so you calculate the height y to which the spans will rise when they buckle in response to a temperature increase of AT = 19.0°C. y b T T + AT 10.26 Enter a number. L Your answer cannot be understood or graded. More Information

A long, thin, wire of uniform cross-section has Young's Modulus Y and mass density D. When laid horizontally on a bench its unstretched length is L. One end is attached to a fixed point and the rest of the wire is allowed to hang freely and vertically in the earth's gravitational field. Find the extension of the lowest end of the wire (i.e., the wire's length additional to L) due to gravitational force. The acceleration due to gravity is g.

Section 1.5 Problem 1. Mark each statement True or False, and justify your answer. (T/F) A homogeneous equation is always consistent. (T/F) If x is a nontrivial solution of Ax=0, then every entry in x is nonzero. (T/F) The equation Ax=0 gives an explicit description of its solution set. (T/F) The equation x = xu+xiv, with xa and xx free (and neither u nor va multiple of the other), describes a plane through the origin. (T/F) The homogeneous equation Ax=0 has the trivial solution if and only if the equation has at least one free variable. (T/F) The equation Ax=b is homogeneous if the zero vector is a solution. (T/F) The equation x = p + tv describes a line through v parallel to p. (T/F) The effect of adding p to a vector is to move the vector in a direction parallel to p.