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?

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.

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 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.

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 $3.80\text{-kg}$ 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?

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.

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 $2 \mathrm{~kg}$ 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 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.

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.

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?

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 $t=3 \mathrm{~s}$.