The equation of the deflection curve for a simply supported beam is

$$v(x)=\frac{q_0}{24 E I}\left(2 L x^3-x^4-L^3 x\right)$$

(a) Derive the slope equation of the beam. (b) Derive the bending-moment equation of the beam. (c) Derive the shear-force equation of the beam. (d) Describe the loading acting on the beam.

An element in biaxial stress is subjected to stresses $\sigma_{x}=6250$ psi and $\sigma_{y}=-1750$ psi. Using Mohr’s circle, determine: (a) The stresses acting on an element oriented at a counterclockwise angle $\theta=55^{\circ}$ from the x axis. (b) The maximum shear stresses and associated normal stresses. Show all results on sketches of properly oriented elements.

A rocket is at rest on the launch pad. When the rocket is launched, its kinetic energy increases. Consider all of the forces acting on the rocket during the launch, and decide whether the following statement is true or false: The amount by which the kinetic energy of the rocket increases during the launch is equal to the work done by the force generated by the rocket's engine.

Resistance of air acting on a falling body can be taken into account by the approximate relation for the acceleration:

$$a=\frac{d v}{d t}=g-k v,$$

where $k$ is a constant. Derive a formula for the velocity of the body as a function of time assuming it starts from rest $(v=0$ at $t=0)$. [Hint: Change variables by setting $u=g-k v$.]

The deflection curve for a cantilever beam AB is given by the following equation: $v=-\frac{q_{0} x^{2}}{360 L^{2} E I}\left(45 L^{4}-40 L^{3} x+15 L^{2} x^{2}-x^{4}\right)$ (a) Describe the load acting on the beam. (b) Determine the reactions $R_{A}$ and $M_{A}$ at the support.

Suppose the force acting on a column that helps to support a building is a normally distributed random variable X with mean value 15.0 kips and standard deviation 1.25 kips. Compute the following probabilities by standardizing. a. $P(X \leq 15)$ b. $P(X \leq 17.5)$ c. $P(X \geq 10)$ d. $P(14 \leq X \leq 18)$ e. $P(|X-15| \leq 3)$

A projectile is fired with initial speed $v_{0}$ feet per second from a height of h feet at an angle of $\theta$ above the horizontal. Assuming that the only force acting on the object is gravity, find the maximum altitude, horizontal range and speed at impact. $v_{0}=80, h=6, \theta=\frac{\pi}{4}$

Decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem. If false, provide an example, illustration, or brief explanation of why the statement is false. The impulse of a force F(t) acting on an object is obtained by integrating F(t) over all values of t.

A long piece of wire with a mass of 0.100 kg and a total length of 4.00 m is used to make a square coil with a side of 0.100 m. The coil is hinged along a horizontal side, carries a 3.40-A current, and is placed in a vertical magnetic field with a magnitude of 0.010 0 T. Find the torque acting on the coil due to the magnetic force at equilibrium.

A half-full recycling bin has mass 3.0 kg and is pushed up a 40.0° incline with constant speed under the action of a 26-N force acting up and parallel to the incline. The incline has friction. What magnitude force must act up and parallel to the incline for the bin to move down the incline at constant velocity?

Like Harry Potter and Lord Voldemort. Match each term with its description.
(a) Transcription factor _______
(b) Enhancer _______
(c) Mediator _______
(d) Combinatorial control _______
(e) Zinc-finger domain _______
(f) Coactivator _______
(g) Selective estrogenreceptor modulator _______
(h) Histone acetyltransferase _______
(i) Acetyllysine-binding domain _______
(j) Chromatin-remodeling engine _______

1. Depends on ATPcitrate lyase for its substrate
2. Acts as a bridge between transcription factors and RNA polymerase II
3. Based on the fact that the regulatory properties of a protein depend on what otherproteins are present
4. Facilitates estrogenreceptor action
5. Binds to cis-acting elements
6. Bromodomain
7. Shifts the location of nucleosomes
8. Tamoxifen
9. DNA-binding domain in some proteins
10. Cis-acting element that may be far from the gene regulated

A long, straight wire lies on a horizontal table in the xy-plane and carries a current of 1.20 $\mu \mathrm { A }$ in the positive x-direction along the x-axis. A proton is traveling in the negative x-direction at speed $2.30 \times 10 ^ { 4 }$ m/s a distance d above the wire (i.e., z = d). What is the direction of the magnetic force acting on the proton?

A paper tube is formed by rolling a paper strip in a spiral and then gluing the edges together as shown. Determine the shear stress acting along the seam, which is at $30^{\circ}$ from the vertical, when the tube is subjected to an axial force of $10 \mathrm{~N}$. The paper is $1 \mathrm{~mm}$ thick and the tube has an outer diameter of $30 \mathrm{~mm}$.

A simple log bridge in a remote area consists of two parallel logs with planks across them. The logs are Douglas fir with average diameter 300 mm. A truck moves slowly across the bridge, which spans 2.5 m. Assume that the weight of the truck is equally distributed between the two logs. Because the wheelbase of the truck is greater than 2.5 m, only one set of wheels is on the bridge at a time. Thus, the wheel load on one log is equivalent to a concentrated load W acting at any position along the span. In addition, the weight of one log and the planks it supports is equivalent to a uniform load of $850\ \mathrm{N} / \mathrm{m}$ acting on the log. Determine the maximum permissible wheel load W based upon (a) an allowable bending stress of 7.0 MPa, and (b) an allowable shear stress of 0.75 MPa.