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?

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.

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.

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 $2 \mathrm{~kg}$ 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?

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

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

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 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 $4.00 \hat{\mathrm{i}} \mathrm{V} / \mathrm{m}$ ?

The mass of the helicopter is $9300 \mathrm{~kg}$. It takes off vertically at time $t=0$. The pilot advances the throttle so that the upward thrust of its engine (in $\mathrm{kN}$ ) is given as a function of time in seconds by $T=100+2 t^2$. (a) Determine the magnitude of the linear impulse due to the forces acting on the helicopter from $t=0$ to $t=3 \mathrm{~s}$. (b) Use the principle of impulse and momentum to determine how fast the helicopter is moving at $t=3 \mathrm{~s}$.

A uniform sphere of mass 2.50 kg and radius 15.0 cm is released from rest at the top of an incline that is 5.25 m long and makes an angle of $35^{\circ}$ with the horizontal. Assuming it rolls without slipping, (a) determine its total kinetic energy at the bottom of the incline. (b) Determine its rotational kinetic energy at the bottom of the incline. (c) What type of friction, static or kinetic, is acting on the surface of the sphere? Explain. (d) Determine the force of friction in part (d).

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?

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 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 composite beam is made of two brass $[E=110 \mathrm{GPa}]$ bars to two aluminum $[E=70 \mathrm{GPa}]$ bars, as shown in Figure P8.44. The beam is subjected to a bending moment of $380 \mathrm{~N}$-m acting about the $z$ axis. Using $a=5 \mathrm{~mm}, b=40 \mathrm{~mm}, c=10 \mathrm{~mm}$, and $d=$ $25 \mathrm{~mm}$, calculate (a) the maximum bending stresses in the aluminum bars. (b) the maximum bending stress in the brass bars.