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 $\mathbf{F}$ along a semicircular path of diameter $O A$. Observing that $r=r_0 \cos \theta$ and using Eq. (12.27), show that the speed of the particle is $v=v_0 / \cos ^2 \theta$.

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 $\theta$.

A series of small packages, each with a mass of $0.5 \mathrm{~kg}$, are discharged from a conveyor belt as shown. Knowing that the coefficient of static friction between each package and the conveyor belt is $0.4$, determine $(a)$ the force exerted by the belt on the package just after it has passed point $A,(b)$ the angle $\theta$ defining the point $B$ where the packages first slip relative to the belt.

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.

Human centrifuges are often used to simulate different acceleration levels for pilots. When aerospace physiologists say that a pilot is pulling $9 \mathrm{~g}$ 's, they mean that the resultant normal force on the pilot from the bottom of the seat is nine times their weight. Knowing that the centrifuge starts from rest and has a constant angular acceleration of $1.5 \mathrm{~rpm}$ per second until the pilot is pulling $9 \mathrm{~g}$ 's and then continues with a constant angular velocity, determine $(a)$ how long it will take for the pilot to reach $9 \mathrm{~g}$ 's $(b)$ the angle $\theta$ of the normal force once the pilot reaches $9 \mathrm{~g}$ 's.

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.

The shareholders of Jolie Company have voted in favor of a buyout offer from Pitt Corporation. Information about each firm is shown below:

Jolie’s shareholders will receive one share of Pitt stock for every three shares they hold in Jolie.

a. What will the EPS of Pitt be after the merger? What will the PE ratio be if the NPV of the acquisition is zero?
b. What must Pitt feel is the value of the synergy between these two firms? Explain how your answer can be reconciled with the decision to go ahead with the takeover.

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.

A $2 \mathrm{~lb}$ sphere is at rest relative to a parabolic dish which rotates at a constant rate about a vertical axis. Neglecting friction and knowing that $r=3 \mathrm{~ft}$, determine $(a)$ the velocity $v$ of the sphere, $(b)$ the magnitude of the normal force exerted by the sphere on the inclined surface of the dish.

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.

A $3 \mathrm{~kg}$ block is at rest relative to a parabolic dish which rotates at a constant rate about a vertical axis. Knowing that the coefficient of static friction is $0.5$ and that $r=2 \mathrm{~m}$, determine the maximum allowable velocity $v$ of the block.