A marine bacterium is isolated and shown to contain an inducible operon whose genetic products metabolize oil when it is encountered in the environment. Investigation demonstrates that the operon is under positive control and that there is a reg gene whose product interacts with an operator region $(o)$ to regulate the structural genes, designated $s g$.
In an attempt to understand how the operon functions, a constitutive mutant strain and several partial diploid strains were isolated and tested with the results shown here:

$$\begin{array}{lll} \text { Host Chromosome } & \text { F' }^{\prime} \text { Factor } & \text { Phenotype } \\ \text { wild type } & \text { none } & \text { inducible } \\ \text { wild type } & \text { reg gene from mutant strain } & \text { inducible } \\ \text { wild type } & \text { operon from mutant strain } & \text { constitutive } \\ \text { mutant strain } & \text { reg gene from wild type } & \text { constitutive } \end{array}$$

Draw all possible conclusions about the mutation as well as the nature of regulation of the operon. Is the constitutive mutation in the trans-acting reg element or in the cis-acting $o$ operator element?

A proton travels through uniform magnetic and electric fields. The magnetic field is $\vec{B}=-3.25 \hat{\mathrm{i}} \mathrm{m} \mathrm{T}$. At one instant the velocity of the proton is $\vec{v}=2000 \hat{\mathrm{j}} \mathrm{m} / \mathrm{s}$. At that instant and in unit-vector notation, what is the net force acting on the proton if the electric field is (a) $4.00 \hat{\mathrm{k}} \mathrm{V} / \mathrm{m}$, (b) $-4.00 \hat{\mathrm{k}} \mathrm{V} / \mathrm{m}$, and $4.00 \hat{\mathrm{i}} \mathrm{V} / \mathrm{m}$ ?

A person of mass $M$ lies on the floor doing leg raises. His leg is $95 \mathrm{~cm}$ long and pivots at the hip. Treat his legs (including feet) as uniform cylinders, with both legs comprising $34.5 \%$ of body mass, and the rest of his body as a uniform cylinder comprising the rest of his mass. He raises both legs $50.0^{\circ}$ above the horizontal. (c) Since the center of mass of his body rises, there must be an external force acting. Identify this force.

One kind of baseball pitching machine works by rotating a light and stiff rigid rod about a horizontal axis until the ball is moving toward the target. Suppose a $144 \mathrm{~g}$ baseball is held $85$ $\mathrm{cm}$ from the axis of rotation and released at the major league pitching speed of $85 \mathrm{mph}$.
a. What is the ball's centripetal acceleration just before it is released?
b. What is the magnitude of the net force that is acting on the ball just before it is released?

Consider the branched pathway in problem 15. The first common step $(\mathrm{A} \rightarrow \mathrm{B})$ is partly inhibited by both of the final products, each acting independently of the other. Suppose that a high level of $\mathrm{Y}$ alone decreased the rate of the $\mathrm{A} \rightarrow \mathrm{B}$ step from 100 to 60 $\mathrm{s}^{-1}$ and that a high level of $\mathrm{Z}$ alone decreased the rate from 100 to $40 \mathrm{~s}^{-1}$. What would the rate be in the presence of high levels of both $\mathrm{Y}$ and $\mathrm{Z}$ ?

A system consists of a force $\mathrm{F}$ acting at the origin $O$ and a couple $\mathbf{M}$, where

$$\mathbf{F}=\mathbf{i}+2 \mathbf{j}+5 \mathbf{k}\ (\mathrm{N}), \quad \mathbf{M}=10 \mathbf{i}+8 \mathbf{j}-4 \mathbf{k}\ (\mathrm{N}-\mathrm{m})$$

If you represent it by a wrench consisting of the force $\mathbf{F}$ and a parallel couple $\mathbf{M}_p$, (a) determine $\mathbf{M}_p$, and determine where the line of action of $\mathbf{F}$ intersects (b) the $x-z$ plane, (c) the $y-z$ plane.

A $2 \mathrm{~kg}$ book sits at rest on a horizontal table. The coefficient of static friction between the book and the surface is $0.40$, and the coefficient of kinetic friction is $0.20$.
(a) What is the normal force acting on the book?
(b) Is there a friction force on the book?
(c) What minimum horizontal force would be required to cause the book to slide on the table?
(d) If you give the book a strong horizontal push so that it begins sliding, what kind of force will cause it to come to rest?
(e) What is the magnitude of this force?

A thin-walled cylindrical pressure vessel of a radius $r$ is subjected simultaneously to internal gas pressure $p$ and a compressive force $F$ acting at the ends. (a) What should be the magnitude of the force $F$ in order to produce pure shear in the wall of the cylinder? (b) If force $F=190\ \mathrm{kN}$, internal pressure $p=12\ \mathrm{MPa}$, inner diameter $=200 \mathrm{~mm}$, and allowable normal and shear stresses are $110\ \mathrm{MPa}$ and $60\ \mathrm{MPa}$, respectively, what is the required thickness of the vessel?

A string passing over a pulley has a $3.80\text{-kg}$ mass hanging from one end and a $3.05\text{-kg}$ mass hanging from the other end. The pulley is a uniform solid cylinder of radius $4.0 \text{~cm}$ and mass $0.60 \text{~kg}$. $(b)$ In fact, it is found that if the heavier mass is given a downward speed of $0.20 \text{~m/s}$, it comes to rest in $6.2 \text{~s}$. What is the average frictional torque acting on the pulley?

Newton's law of motion for a particle of mass $m$ moving in a straight line says that $F=m a$, where $F$ is the force acting on the particle and $a$ is the acceeleration of the particle. In relativistic mechanics, this law is replaced by $F=\frac{m_0 \frac{d}{d t} v}{\sqrt{1-\left(\frac{v}{c}\right)^2}}$, where $m_0$ is the mass of the particle measured at rest, $v$ is the velocity of the particle and $c$ is the speed of light. Show that

$$F=\frac{m_0 a}{\left(1-\left(\frac{v}{c}\right)^2\right)^{\frac{3}{2}}} .$$

A string passing over a pulley has a $3.80-\mathrm{~kg}$ mass hanging from one end and a $3.15-\mathrm{~kg}$ mass hanging from the other end. The pulley is a uniform solid cylinder of radius $4.0 \mathrm{~cm}$ and mass $0.80 \mathrm{~kg}$.In fact, it is found that if the heavier mass is given a downward speed of $0.20 \mathrm{~m} / \mathrm{~s}$, it comes to rest in $6.2 \mathrm{~s}$. What is the average frictional torque acting on the pulley?

An object with weight $W$ is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle $\theta$ with the plane, then the magnitude of the force is

$$F=\frac{\mu W}{\mu \sin \theta+\cos \theta}$$

where $\mu$ is a constant called the coefficient of friction. If $W=50 ~\mathrm{lb}$ and $\mu=0.6$, draw the graph of $F$ as a function of $\theta$ and use it to locate the value of $\theta$ for which $d F / d \theta=0$. Is the value consistent with your answer to previous part?

A body of mass $m$ moves in an elliptical path with a constant angular speed $\omega$ (see the figure). It can be shown that the force acting on the body is always directed toward the origin and has magnitude given by

$$F=m \omega^2 \sqrt{a^2 \cos ^2 \omega t+b^2 \sin ^2 \omega t} \qquad t \geq 0$$

where $a$ and $b$ are constants with $a>b$. Find the points on the path where the force is greatest and where it is smallest. Does your result agree with your intuition?

In an oscillating incompressible flow field the force per unit mass acting on a fluid particle is obtained from Newton's second law in intensive form,

$$\frac{\vec{F}}{m}=\frac{\partial \vec{V}}{\partial t}+(\vec{V} \cdot \vec{V}) \vec{V}$$

Suppose the characteristic speed and characteristic length for a given flow field are $V_{\infty}$ and $L$, respectively. Also suppose that $\omega$ is a characteristic angular frequency $(\mathrm{rad} / \mathrm{s})$ of the oscillation. Define the following nondimensionalized variables,

$$t^*=\omega t, \quad \vec{x}^*=\frac{\vec{x}}{L}, \quad \vec{\nabla}^*=L \vec{\nabla}, \quad \text { and } \quad \vec{V}^*=\frac{\vec{V}}{V_{\infty}}$$

Since there is no given characteristic scale for the force per unit mass acting on a fluid particle, we assign one, noting that $\{\vec{F} / m\}=\left\{\mathrm{L} / t^2\right\}$. Namely, we let

$$(\vec{F} / m)^*=\frac{1}{\omega^2 L} \vec{F} / m$$

Nondimensionalize the equation of motion and identify any established (named) dimensionless parameters that may appear.