A countershaft carrying two V-belt pulleys is shown in the figure. Pulley A receives power from a motor through a belt with the belt tension. The power is transmitted through the shaft and delivered to the belt on pulley B. Assume the belt tension on the loose side at B is 15 percent of the tension on the tight side.

a.) Determine the tensions in the belt on pulley B, assuming the shaft is running at a constant speed.

b.) Find the magnitudes of the bearing reaction forces, assuming the bearings act as simple supports.

c.) Draw shear-force and bending-moment diagrams for the shaft. If needed, make one set for the horizontal plane and another set for the vertical plane.

d.) At the point of maximum bending moment, determine the bending stress and the torsional shear stress.

e.) At the point of maximum bending moment, determine the principal stresses and the maximum shear stress.

In each case, determine the internal bending moment as a function of x, and state the necessary boundary and/or continuity conditions used to determine the elastic curve for the beam.

La presión de un neumático de automóvil depende de la temperatura del aire que contiene. Cuando la temperatura del aire es de $25^\circ \mathrm{C}$, el manómetro marca $210 \mathrm{~kPa}$. Si el volumen del neumático es de $0.025 \mathrm{~m}^3$, determine el aumento de presión en el neumático cuando la temperatura del aire en el neumático aumenta a $50^\circ \mathrm{C}$. Determine también la cantidad de aire que debe purgarse para restablecer la presión a su valor original a esta temperatura. Supongamos que la presión atmosférica es de $100 \mathrm{~kPa}$.

Using an allowable shear stress of 50 MPa, determine the power that can be transmitted at 2000 rpm through a shaft with a 30-mm diameter.

The state of stress at a point is $\sigma_{x}=-6, \sigma_{y}=18, \sigma_{z}=-12, \tau_{x y}=9, \tau_{y z}=6, \text { and } \tau_{z x}=-15 \mathrm{kpsi}$. Determine the principal stresses, draw a complete Mohr's three-circle diagram, labeling all points of interest, and report the maximum shear stress for this case.

The two V-belt pulleys form an integral unit and rotate about the fixed axis at $O$. At a certain instant, point $A$ on the belt of the smaller pulley has a velocity $v_A=1.5 \mathrm{~m} / \mathrm{s}$, and point $B$ on the belt of the larger pulley has an acceleration $a_B=45 \mathrm{~m} / \mathrm{s}^2$ as shown. For this instant determine the magnitude of the acceleration $\mathbf{a}_C$ of point $C$ and sketch the vector in your solution.

A vertical force P on the foot pedal of the bell crank is required to produce a tension T of 400N in the vertical control rod. Determine the corresponding bearing reactions at A and B.

If block A is moving downward with a speed of 4 ft / s while C is moving up at 2 ft / s, determine the speed of block B.

A gas-turbine power plant operates on a simple Brayton cycle with air as the working fluid. The air enters the turbine at 120 psia and 2000 R and leaves at 15 psia and 1200 R. Heat is rejected to the surroundings at a rate of 6400 Btu/s, and air flows through the cycle at a rate of 40 lbm/s. Assuming the turbine to be isentropic and the compresssor to have an isentropic efficiency of 80 percent, determine the net power output of the plant. Account for the variation of specific heats with temperature.

The cantilevered bar in the figure is made from a ductile material and is statically loaded with $F_{\text{y}} = 250\ \text{lbf}$ and $F_{\text{x}} = F_{\text{z}} = 0$. Analyze the stress situation in the small diameter at the shoulder at A by obtaining the following information.

(a) Determine the precise location of the critical stress element at the cross section at A

(b) Sketch the critical stress element and determine magnitudes and directions for all stresses acting on it. (Transverse shear may be neglected if you can justify this decision

(c) For the critical stress element, determine the principal stresses and the maximum shear stress.

Draw a free-body diagram for the gear and shaft assembly shown in figure. Also sketch free-body diagrams for gear 1, gear 2, and the shaft.

For a solid shaft of diameter d replaced with a hollow shaft of the same material with an outside diameter d, and an inside diameter that is a fraction of the outside diameter, $x \times d$, where x is any value between zero and one. Obtain expressions for percentage reduction in torque transmission and percentage reduction in weight in terms of only x. Notice that the length and diameter of the shaft, and the material, are not needed for this comparison. Plot both results on the same axis for the range $0<x<1$. From the plot, what is the approximate value of x to obtain the greatest difference between the percent decrease in weight and the percent decrease in torque?

Consider a double-pipe counterflow heat exchanger. In order to enhance heat transfer, the length of the heat exchanger is now doubled. Do you think its effectiveness will also double?

Consider a 1-in-square steel thin-walled tube loaded in torsion. The tube has a wall thickness $t=\frac{1}{16} \mathrm{in}$, is 3 6 in long, and has a maximum allowable shear stress of 12 kpsi. Determine the maximum torque that can be applied and the corresponding angle of twist of the tube. a) Assume that the internal radius at the corners $r_{i}=0$. b) Assume that the internal radius at the corners is more realistically $r_{i}=\frac{1}{8} \mathrm{in}$.