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 $64~\text{kg}$ sprinter, starting from rest, runs $44~\text{m}$ in $7.0~\text{s}$ at constant acceleration.

• Part A What is the magnitude of the horizontal force acting on the sprinter? Express your answer to two significant figures and include the appropriate units.
• Part B What is the sprinter's power output at $2.0~\text{s}$, $4.0~\text{s}$, and $6.0~\text{s}$? Express your answers using two significant figures separated by commas.

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?

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?

If a single constant force acts on an object that moves on a straight line, the object’s velocity is a linear function of time. The equation v = vi + at gives its velocity v as a function of time, where a is its constant acceleration. What if velocity is instead a linear function of position? Assume that as a particular object moves through a resistive medium, its speed decreases as described by the equation v = vi − kx, where k is a constant coefficient and x is the position of the object. Find the law describing the total force acting on this object.

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.

The shaft rotating at $240\ \mathrm{rpm}$ carries a spur gear $D$ having 48 teeth with a diametral pitch of 6. The teeth are of the $20^{\circ}$, full-depth, involute form. The gear receives $15\ \mathrm{hp}$ from pinion $Q$ located as shown. Compute the torque delivered to the shaft and the tangential and radial forces exerted on the shaft by the gear. Resolve the tangential and radial forces into their horizontal and vertical components, and determine the net forces acting on the shaft at $D$ in the horizontal and vertical directions.

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?

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 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?

The $1360-\mathrm{kg}$ car travels along a straight road of increasing grade whose vertical profile is given by the equation shown. The magnitude of the car's velocity is a constant $100 \mathrm{~km} / \mathrm{h}$. When $x=200 \mathrm{~m}$, what are the $x$ and $y$ components of the total force acting on the car (including its weight)?

Strategy: You know that the tangential component of the car's acceleration is zero. You can use this condition together with the equation for the profile of the road to determine the $x$ and $y$ components of the car's acceleration.

Gravity If an imaginary line segment is drawn between the centers of the earth and the moon, then the net gravitational force F acting on an object situated on this line segment is $F=\frac{-K}{x^{2}}+\frac{0.012 K}{(239-x)^{2}}$ where K>0 is a constant and x is the distance of the object from the center of the earth, measured in thousands of miles. How far from the center of the earth is the "dead spot' where no net gravitational force acts upon the object? (Express your answer to the nearest thousand miles.)

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?

An object with mass $m=1.0 \mathrm{~g}$ and charge $q$ is placed at point $A$, which is $0.05 \mathrm{~m}$ above an infinitely large, uniformly charged, nonconducting sheet $\left(\sigma=-3.5 \cdot 10^{-5} \mathrm{C} / \mathrm{m}^2\right)$. Gravity is acting downward $\left(g=9.81 \mathrm{~m} / \mathrm{s}^2\right)$. Determine the number, $N$, of electrons that must be added to or removed from the object for the object to remain motionless above the charged plane.