The value of $g$ at any latitude $\phi$ may be obtained from the formula

$$g=32.09\left(1+0.0053 \sin ^2 \phi\right) \mathrm{ft} / \mathrm{s}^2$$

which takes into account the effect of the rotation of the earth, as well as the fact that the earth is not truly spherical. Determine to four significant figures $(a)$ the weight in pounds, $(b)$ the mass in pounds, $(c)$ the mass in $\mathrm{lb} \cdot \mathrm{s}^2 / \mathrm{ft}$, at the latitudes of $0^{\circ}, 45^{\circ}, 60^{\circ}$, of a silver bar, the mass of which has been officially designated as $5\ \mathrm{lb}$.

The roller-coaster track shown is contained in a vertical plane. The portion of track between $A$ and $B$ is straight and horizontal, while the portions to the left of $A$ and to the right of $B$ have radii of curvature as indicated. A car is traveling at a speed of $72 \mathrm{~km} / \mathrm{h}$ when the brakes are suddenly applied, causing the wheels of the car to slide on the track $\left(\mu_k=0.25\right)$. Determine the initial deceleration of the car if the brakes are applied as the car $(a)$ has almost reached $A,(b)$ is traveling between $A$ and $B,(c)$ has just passed $B$.

A 5000-lb truck is being used to lift a 1000-lb boulder B that is on a 200-lb pallet A. Knowing the acceleration of the truck is 1 $\mathrm{ft} / \mathrm{s}^{2}$, determine (a) the horizontal force between the tires and the ground, (b) the force between the boulder and the pallet.

It was observed that as the Galileo spacecraft reached the point on its trajectory closest to Io, a moon of the planet Jupiter, it was at a distance of $1750 \mathrm{mi}$ from the center of Io and had a velocity of $49.4 \times 10^3 \mathrm{ft} / \mathrm{s}$. Knowing that the mass of Io is $0.01496$ times the mass of the earth, determine the eccentricity of the trajectory of the spacecraft as it approached $Io$.

A $12-\mathrm{lb}$ block $B$ rests as shown on the upper surface of a $30-\mathrm{lb}$ wedge $A$. Neglecting friction, determine immediately after the system is released from rest $(a)$ the acceleration of $A,(b)$ the acceleration of $B$ relative to $A$.

Disk $A$ rotates in a horizontal plane about a vertical axis at the constant rate $\dot{\theta}_0=10 \mathrm{rad} / \mathrm{s}$. Slider $B$ has mass $1 \mathrm{~kg}$ and moves in a frictionless slot cut in the disk. The slider is attached to a spring of constant $k$, which is undeformed when $r=0$. Knowing that the slider is released with no radial velocity in the position $r=500 \mathrm{~mm}$, determine the position of the slider and the horizontal force exerted on it by the disk at $t=0.1 \mathrm{~s}$ for $(a) k=100$ $\mathrm{N} / \mathrm{m},(b) k=200 \mathrm{~N} / \mathrm{m}$

Show that the angular momentum per unit mass $h$ of a satellite describing an elliptic orbit of semimajor axis $a$ and eccentricity $\varepsilon$ about a planet of mass $M$ can be expressed as

$$h=\sqrt{G M a\left(1-\varepsilon^2\right)}$$

Derive Kepler's third law of planetary motion from $Eqs.$ $\frac{1}{r}=\frac{G M}{h^2}(1+\varepsilon \cos \theta)$ and $\tau=\frac{2(\pi a b)}{h}$.

(a) Express the eccentricity $\varepsilon$ of the elliptic orbit described by a satellite about a planet in terms of the distances $r_0$ and $r_1$ corresponding, respectively, to the perigee and apogee of the orbit. (b) Use the result obtained in part $a$ and the data given in Prob. 12.111, where $R_E=$ $149.6 \times 10^6 \mathrm{~km}$, to determine the approximate maximum distance from the sun reached by comet Hyakutake.

Telemetry technology is used to quantify kinematic values of a $200 \mathrm{~km}$ roller-coaster cart as it passes overhead. According to the system, $r=25 \mathrm{~m}$, $\dot{r}$ $=-10 \mathrm{~m} / \mathrm{s}$, $\ddot{r}=-2 \mathrm{~m} / \mathrm{s}^2$, $\theta=90^{\circ}$, $\dot{\theta}=-0.4 \mathrm{rad} / \mathrm{s}$, and $\ddot{\theta}=-0.32 \mathrm{rad} / \mathrm{s}^2$. At this instant, determine $(a)$ the normal force between the cart and the track, $(b)$ the radius of curvature of the track.

A satellite describes an elliptic orbit about a planet. Denoting by $r_0$ and $r_1$ the distances corresponding, respectively, to the perigee and apogee of the orbit, show that the curvature of the orbit at each of these two points can be expressed as

$$\frac{1}{\rho}=\frac{1}{2}\left(\frac{1}{r_0}+\frac{1}{r_1}\right)$$

A space probe is describing a circular orbit of radius $n R$ with a velocity $v_0$ about a planet of radius $R$ and center $O$. As the probe passes through point $A$, its velocity is reduced from $v_0$ to $\beta v_0$, where $\beta<1$, to place the probe on a crash trajectory. Express in terms of $n$ and $\beta$ the angle $A O B$, where $B$ denotes the point of impact of the probe on the planet.

As a space probe approaching the planet Venus on a parabolic trajectory reaches point $A$ closest to the planet, its velocity is decreased to insert it into a circular orbit. Knowing that the mass and the radius of Venus are $4.87 \times 10^{24} \mathrm{~kg}$ and $6052 \mathrm{~km}$, respectively, determine the time needed for the space probe to travel from $B$ to $C$.

The Clementine spacecraft described an elliptic orbit of minimum altitude $h_A=400 \mathrm{~km}$ and maximum altitude $h_B=2940 \mathrm{~km}$ above the surface of the moon. Knowing that the radius of the moon is $1737 \mathrm{~km}$ and that mass of the moon is $0.01230$ times the mass of the earth, determine the periodic time of the spacecraft.