A particle of mass $m$ is projected from point $A$ with an initial velocity $\mathbf{v}_0$ perpendicular to line $O A$ and moves under a central force $\mathbf{F}$ directed away from the center of force $O$. Knowing that the particle follows a path defined by the equation $r=r_0 \sqrt{\cos 2 \theta}$ and using Eq. (12.27), express the radial and transverse components of the velocity $\mathbf{v}$ of the particle as functions of $\theta$.

To transport a series of bundles of shingles $A$ to a roof, a contractor uses a motor-driven lift consisting of a horizontal platform $B C$ which rides on rails attached to the sides of a ladder. The lift starts from rest and initially moves with a constant acceleration $\mathbf{a}_1$ as shown. The lift then decelerates at a constant rate $\mathbf{a}_2$ and comes to rest at $D$, near the top of the ladder. Knowing that the coefficient of static friction between a bundle of shingles and the horizontal platform is $0.30$, determine the largest allowable acceleration $\mathbf{a}_1$ and the largest allowable deceleration $\mathbf{a}_2$ if the bundle is not to slide on the platform.

A nozzle discharges a stream of water in the direction shown with an initial velocity of $24 \mathrm{ft} / \mathrm{s}$. Determine the radius of curvature of the stream $(a)$ as it leaves the nozzle, $(b)$ at the maximum height of the stream.

A race car enters the circular portion of a track that has a radius of $70 \mathrm{~m}$. When the car enters the curve at point $P$, it is traveling with a speed of 120 $\mathrm{km} / \mathrm{h}$ that is increasing at $5 \mathrm{~m} / \mathrm{s}^2$. Three seconds later, determine the $x$ and $y$ components of velocity and acceleration of the car.

By itself, the 2500-kg pickup truck executes a 0–100 km/h acceleration run in 10 s along a level road. What would be the corresponding time when pulling the 500-kg trailer? Assume constant acceleration and neglect all retarding forces.

The system is released from rest with the cable taut. For the friction coefficients $\mu_s$ = 0.25 and $\mu_k$ = 0.20, calculate the acceleration of each body and the tension T in the cable. Neglect the small mass and friction of the pulleys.

An object is projected upward with initial velocity v0 meters per second from a point s0 meters above the ground. Show that

$$[v(t)]^2=v0^2-19.6[s(t)-so]$$

A roller coaster starts from rest at $A$, rolls down the track to $B$, describes a circular loop of $40 \mathrm{~ft}$ diameter, and moves up and down past point $E$. Knowing that $h=60 \mathrm{ft}$ and assuming no energy loss due to friction, determine $(a)$ the force exerted by his seat on a $160 \mathrm{lb}$ rider at $B$ and $D$, $(b)$ the minimum value of the radius of curvature at $E$ if the roller coaster is not to leave the track at that point.

A satellite is placed into a circular orbit about the planet Saturn at an altitude of 2100 mi. The satellite describes its orbit with a velocity of 54.7 $\times$ 103 mi/h. Knowing that the radius of the orbit about Saturn and the periodic time of Atlas, one of Saturn’s moons, are 85.54 $\times$ 103 mi and 0.6017 days, respectively, determine (a) the radius of Saturn, (b) the mass of Saturn. (The periodic time of a satellite is the time it requires to complete one full revolution about the planet.)

An advanced spatial disorientation trainer is programmed to only rotate and translate in the horizontal plane. The pilot’s location is defined by the relationships $r=8\left(1-e^{-t}\right)$ and $\theta=2 / \pi\left(\sin \frac{\pi}{2} t\right)$ where r, $\theta$, and t are expressed in feet, radians, and seconds, respectively. Determine the radial and transverse components of the force exerted on the 175-lb pilot at t = 3 s.

Two identical cars A and B are at rest on a loading dock with brakes released. Car C, of a slightly different style but of the same weight, has been pushed by dockworkers and hits car B with a velocity of 1.5 m/s. Knowing that the coefficient of restitution is 0.8 between B and C, and it is 0.5 between A and B, determine the velocity of each car after all collisions have taken place.

Two identical cars $A$ and $B$ are at rest on a loading dock with brakes released. Car $C$, of a slightly different style but of the same weight, has been pushed by dockworkers and hits car $B$ with a velocity of $1.5 \mathrm{~m} / \mathrm{s}$. Knowing that the coefficient of restitution is $0.8$ between $B$ and $C$ and $0.5$ between $A$ and $B$, determine the velocity of each car after all collisions have taken place.

The 8-kg mass $A$ and the $12-\mathrm{kg}$ mass $B$ slide on the smooth horizontal bar with the velocities shown. The coefficient of restitution is $e=0.2$. Determihe the velocities of the masses after they collide.

The system shown is at rest when a constant $150-\mathrm{N}$ force is applied to collar $B$. Neglecting the effect of friction, determine $(a)$ the time at which the velocity of collar $B$ will be $2.5 \mathrm{~m} / \mathrm{s}$ to the left, (b) the corresponding tension in the cable.