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 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 testing laboratory wants to determine if a new widget can withstand large accelerations and decelerations. To find out, they glue a $5.0 \mathrm{~kg}$ widget to a test stand that will drive it vertically up and down. The graph shows its acceleration during the first second, starting from rest. a. Identify the forces acting on the widget and draw a free-body diagram. b. Determine the value of $n_y$, the $y$-component of the normal force acting on the widget, during the first second of motion. Give your answer as a graph of $n_y$ versus $t$. c. Your answer to part b should show an interval of time during which $n_y$ is negative. How can this be? Explain what it means physically for $n_y$ to be negative. d. At what time is the weight of the widget a maximum? What is the acceleration at this time? e. Is the weight of the widget ever zero? If so, at what time does this happen? What is the acceleration at that time? f. Suppose the technician forgets to glue the widget to the test stand. Will the widget remain on the test stand throughout the first second, or will it fly off the stand at some instant of time? If so, at what time will this occur?

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 $1360-\mathrm{kg}$ 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.)

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 train consists of a 50-Mg engine and three cars, each having a mass of 30 Mg. If it takes 80 s for the train to increase its speed uniformly to 40 km/h, starting from rest, determine the force T developed at the coupling between the engine E and the first car A. The wheels of the engine provide a resultant frictional tractive force F which gives the train forward motion, whereas the car wheels roll freely. Also, determine F acting on the engine wheels.

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 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.

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 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 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 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?