For the particle of Prob. $12.74$, determine the tangential component $F_t$ of the central force $\mathbf{F}$ along the tangent to the path of the particle for $(a) \theta=0$, (b) $\theta=45^{\circ}$.

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 $\theta$.

The intensity of the central maximum in the interference pattern of a Young's double-slit experiment is about four times that of either light wave. Is this a violation of the law of conservation of energy?

For the particle of Prob. $12.76$, show $(a)$ that the velocity of the particle and the central force $\mathbf{F}$ are proportional to the distance $r$ from the particle to the center of force $O$, (b) that the radius of curvature of the path is proportional to $r^3$.

Find the length to three significant digits of each arc intercepted by a central angle $\theta$ in a circle of radius r. r=15.1 in., $\theta=210^{\circ}$

A heat engine that propels a ship produces 500 Btu/lbm of work while rejecting 300 Btu/lbm of heat. What is its thermal efficiency?

Slider $C$ has a weight of $0.5 \mathrm{lb}$ and may move in a slot cut in arm $A B$, which rotates at the constant rate $\theta_0=10 \mathrm{rad} / \mathrm{s}$ in a horizontal plane. The slider is attached to a spring of constant $k=2.5 \mathrm{lb} / \mathrm{ft}$, which is unstretched when $r=0$. Knowing that the slider is released from rest with no radial velocity in the position $r=18$ in. and neglecting friction, determine for the position $r=12 \mathrm{in}$. (a) the radial and transverse components of the velocity of the slider, $(b)$ the radial and transverse components of its acceleration, $(c)$ the horizontal force exerted on the slider by arm $A B$.

Show that the radius r of the moon’s orbit can be determined from the radius R of the earth, the acceleration of gravity g at the surface of the earth, and the time $\tau$ required for the moon to complete one full revolution about the earth. Compute r knowing that $\tau$= 27.3 days, giving the answer in both SI and U.S. customary units.

The velocity of block $A$ is $2 \mathrm{~m} / \mathrm{s}$ to the right at the instant when $r=0.8 \mathrm{~m}$ and $u=30^{\circ}$. Neglecting the mass of the pulley and the effect of friction in the pulley and between block $A$ and the horizontal surface, determine, at this instant, $(a)$ the tension in the cable, (b) the acceleration of block $A,(c)$ the acceleration of block $B$.

A $700 \mathrm{~kg}$ horse $A$ lifts a $50 \mathrm{~kg}$ hay bale $B$ as shown. At the instant when $r=8 \mathrm{~m}$ and $\theta=60^{\circ}$, the velocity and acceleration of the horse are $2 \mathrm{~m} / \mathrm{s}$ to the right and $0.5 \mathrm{~m} / \mathrm{s}^2$ to the left, respectively. Neglecting the mass of the pulley, determine at that instant $(a)$ the tension in the cable, $(b)$ the average horizontal force between the ground and the horse's feet.

A tugboat pulls a small barge through a harbor. The propeller thrust minus the drag produces a net thrust that varies linearly with speed. Knowing that the combined weight of the tug and barge is 3600 kN, determine (a) the time required to increase the speed from an initial value $v_1$ = 1.0 m/s to a final value $v_2$ = 2.5 m/s, (b) the distance traveled during this time interval.

The parasailing system shown uses a winch to let rope out at a constant rate so that the $70 \mathrm{~kg}$ rider moves away from the boat, which is traveling with a constant velocity. At the instant shown, the rope has a length of $30 \mathrm{~m}$, it is increasing in length at a constant $1 \mathrm{~m} / \mathrm{s}$, the angle is increasing at a rate of $0.05 \mathrm{rad} / \mathrm{s}$, and $\ddot{\theta}$ is $-0.01 \mathrm{rad} / \mathrm{s}^2$. Knowing that when the rope makes a $30^{\circ}$ angle with respect to the water, the tension in the rope is $10 \mathrm{kN}$, determine the magnitude and direction of the force of the parasail on the parasailor.

Someone has a weighted coin that lands heads up with probability 2/3 and tails up with probability 1/3. If the coin comes up heads, you pay $\$ 1$; if the coin comes up tails, you receive $\$ 1.50$. What is the expected value of this game? Would you play? Why or why not?

The design for a new cementless hip implant is to be studied using an instrumented implant and a fixed simulated femur. Assuming the punch applies an average force of $2\ \mathrm{kN}$ over a time of $2 \mathrm{~ms}$ to the $200 \mathrm{~g}$ implant, determine (a) the velocity of the implant immediately after impact, $(b)$ the average resistance of the implant to penetration if the implant moves $1 \mathrm{~mm}$ before coming to rest.