Suppose an airplane is acted on by three forces: gravity, wind and engine thrust. Assume that the force vector for gravity is $m \mathbf{g}=m\langle 0,0,-32\rangle$, the force vector for wind is $\mathbf{w}=\langle 0,1,0\rangle$ for $0 \leq t \leq 1$ and $\mathbf{w}=\langle 0,2,0\rangle$ for t>1, and the force vector for engine thrust is $\mathbf{e}=\langle 2 t, 0,24\rangle$. Newton's second law of motion gives us ma=mg+w+e. Assume that m=1 and the initial velocity vector is $\mathbf{v}(0)=\langle 100,0,10\rangle$. Show that the velocity vector for $0 \leq t \leq 1 \text { is } \mathbf{v}(t)=\left\langle t^{2}+100, t, 10-8 t\right\rangle$. For t>1, integrate the equation a=g+w+e, to get $\mathbf{v}(t)=\left\langle t^{2}+a, 2 t+b,-8 t+c\right\rangle$, for constants a, b and c. Explain (on physical grounds) why the function v(t) should be continuous and find the values of the constants that make it so. Show that v(t) is not differentiable. Given the nature of the force function, why does this make sense?

Solve for x, assuming a, b, and c are negative constants. $a x+b<c$

Using an allowable stress of $155\ \text{MPa}$, determine the largest bending moment $M_x$ that can be applied to the wide-flange beam shown. Neglect the effect of the fillets.

Identify the constants and variables in the given expression.

65qrs + 12x

(a) Determine the maximum bending moment in the beam and the value of $xx$ where it occurs. (b) Show that the equations for $VV$ and $MM$ as functions of $xx$ satisfy the equation $V=dM/dxV=d M / d x$.

A tension member in a machine is filleted as shown in figure. The member has a manufacturing defect that causes the fluctuating tension load to be applied eccentrically resulting in a fluctuating bending load as well. Measurements indicate that the maximum bending stress is $16.4\ \mathrm{MPa}$ and the minimum is $4.1\ \mathrm{MPa}$. The tensile load fluctuates from a high of $3.6\ \mathrm{kN}$ to a low of $0.90\ \mathrm{kN}$ and is in phase with the bending stress. The member is machined and its dimensions are $D=33 \mathrm{~mm}$, $d=25 \mathrm{~mm}, h=3 \mathrm{~mm}$ and $r=3 \mathrm{~mm}$. The material is SAE 1020 cold-rolled steel. Determine the infinite-life fatigue safety factor for the member for $90 \%$ reliability.

The shaft is rotating at $1750\ \mathrm{rpm}$, and it receives $7.5\ \mathrm{hp}$ through a $5.00-\text{in}$ V-belt drive from a mating belt sheave directly below. The power is delivered by a worm with a pitch diameter of $2.00\ \text{in}$. The forces on the worm were used earlier and are shown in the figure. Draw the complete free-body diagrams for the shaft in both the vertical plane and the horizontal plane. Then draw the complete shearing force and bending moment diagrams for the shaft in both planes.

Find the derivatives. Assume that $a$ and $b$ are constants.

$$y = \pi^{(x+2)}$$

What are two advantages of using named constants?

Estimate the endurance strength of a 1.5-in-diameter rod of AISI 1040 steel having a machined finish and heat-treated to a tensile strength of 110 kpsi, loaded in rotating bending.

A part is loaded with a combination of bending, axial, and torsion such that the following stresses are created at a particular location: Bending: Completely reversed, with a maximum stress of 60 MPa Axial: Constant stress of 20 MPa Torsion: Repeated load, varying from 0 MPa to 50 MPa Assume the varying stresses are in phase with each other. The part contains a notch such that Kf , bending= 1.4, Kf, axial =1.1 and Kf, torsion =2.0. The material properties are Sy = 300 MPa and Su = 400 MPa. The completely adjusted endurance limit is found to be Se = 200 MPa. Find the factor of safety for fatigue based on infinite life. If the life is not infinite, estimate the number of cycles. Be sure to check for yielding.

The antenna on a car is 10 mm in diameter and 1.8 m long. Estimate the bending moment that tends to snap it off if the car is driven at 120 km/hr on a standard day.

Prove the statement by mathematical induction.

$(a b)^n=a^n b^n$ (Assume $a$ and $b$ are constants.)

An S8 $\times 23$ steel beam (see Appendix A) is loaded and supported as shown in given figure. (a) Using the coordinate axes shown, write equations for the shear force $V$ and the bending moment $M$ for any section of the beam in the interval $6\ \mathrm{ft}<x<$ $8\ \mathrm{ft}$. (b) Find the flexural stress at a point $1 \mathrm{in}$. below the top of the beam on a section at $x=3\ \mathrm{ft}$. (c) Find the maximum flexural stress on a section at $x=3\ \mathrm{ft}$.