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.

A boat is pulled into a dock by means of a winch $12$ feet above the deck of the boat .

Suppose the boat is moving at a constant rate of $4$ feet per second. Determine the speed at which the winch pulls in rope when there is a total of $13$ feet of rope out. What happens to the speed at which the winch pulls in rope as the boat gets closer to the dock?

A boat is being pulled into a dock by means of a winch 12 feet above the dock of the boat. a) The winch pulls the rope in at a rate of 4 feet per second. Determine the speed of the boat when there is 15 feet of rope out. What happens to the speed of the boat as it gets closer to the boat? b) Suppose the boat is moving at a constant rate of 4 feet per second. Determine the speed at which the winch pulls in the rope when there is a total of 15 feet of rope out. What happens to the speed at which the winch pulls in rope as the boat gets closer to the dock?

The initial angular velocity and the angular acceleration of four rotating objects at the same instant in time are listed in the table that follows. For each of the objects (a), (b), (c), and (d), determine the final angular speed after an elapsed time of $2.0 \mathrm{~s}$.

The initial angular velocity and the angular acceleration of four rotating objects at the same instant in time are listed in the table that follows. For each of the objects (a), (b), (c), and (d), determine the final angular speed after an elapsed time of 5 s.

a) initial angular velocity 18 rad/s, angular acceleration 3 rad/s^2

b) initial angular velocity 18 rad/s, angular acceleration -3 rad/s^2

c) initial angular velocity -18 rad/s, angular acceleration 3 rad/s^2

d) initial angular velocity -18 rad/s, angular acceleration -3 rad/s^2

Some parasailing systems use a winch to pull the rider back to the boat. During the interval when $\theta$ is between $20^{\circ}$ and $40^{\circ}$ (where t = 0 at $\theta$ = $20^{\circ}$), the angle increases at the constant rate of 2°/s. During this time, the length of the rope is defined by the relationship where r and t are expressed in feet and seconds, respectively. Knowing that the boat is travelling at a constant rate of 15 knots (where 1 knot = 1.15 mi/h), (a) plot the magnitude of the velocity of the parasailer as a function of time, (b) determine the magnitude of the acceleration of the parasailer when t = 5 s.

A car (mass = 1100 kg) is traveling at 32 m/s when it collides head-on with a sport utility vehicle (mass = 2500 kg) traveling in the opposite direction. In the collision, the two vehicles come to a halt. At what speed was the sport utility vehicle traveling?

A small block $B$ fits inside a slot cut in arm $OA$ that rotates in a vertical plane at a constant rate. The block remains in contact with the end of the slot closest to $A$ and its speed is $1.4 \mathrm{~m} / \mathrm{s}$ for $0 \leq \theta \leq 150^{\circ}$. Knowing that the block begins to slide when $\theta=150^{\circ}$, determine the coefficient of static friction between the block and the slot.

One object is at rest, and another is moving. Furthermore, one object is more massive than the other. The two collide in a one -dimensional completely inelastic collision. In other words, they stick together after the collision and move off with a common velocity. Momentum is conserved. (a) Consider the final momentum of the two-object system after the collision. Is it greater when the large-mass object is moving initially or when the small-mass object is moving initially? (b) Is the final speed after the collision greater when the large-mass or the small-mass object is moving initially? The speed of the object that is moving initially is 25 m/s. The masses of the two object s are 3.0 and 8.0 kg. Determine the final speed of the two-object system after the collision for the case when the large-mass object is moving initially and the case when the small-mass object is moving initially

After sliding down a snow-covered hill on an inner tube, Ashley is coasting across a level snowfield at a constant velocity of +2 .7 m/s. Miranda runs after her at a velocity of +4.5 m/s and hops on the inner tube. How fast do the two slide across the snow together on the inner tube? Ashley's mass is 71 kg and Miranda's is 58 kg. Ignore the mass of the inner tube and any friction between the inner tube and the snow.

A wagon is rolling forward on level ground. Friction is negligible. The person sitting in the wagon is holding a rock. The total mass of the wagon, ride, and rock is 95.0 kg. The mass of the rock is 0.300 kg. Initially the wagon is rolling forward at a speed of 0.500 m/s. Then the person throws the rock with a speed of 16.0 m/s. Both speeds are relative to the ground. Find the speed of the wagon after the rock is thrown directly forward in one case and directly backward in another.

The internal loadings at a critical section along the steel drive shaft of a ship are calculated to be a torque of 2300 lb · ft, a bending moment of 1500 lb · ft, and an axial thrust of 2500 lb. If the yield points for tension and shear are

$$σ_γ = 100 ksi and τ_γ = 50 ksi,$$

respectively, determine the required diameter of the shaft using the maximum shear stress theory.

A tractor-trailer rig with a $2000~\mathrm{kg}$ tractor, a $4500~\mathrm{kg}$ trailer, and a $3600~\mathrm{kg}$ trailer is traveling on a level road at $90 \mathrm{~km} / \mathrm{h}$. The brakes on the rear trailer fail, and the antiskid system of the tractor and front trailer provide the largest possible force that will not cause the wheels to slide. Knowing that the coefficient of static friction is $0.75$, determine $(a)$ the shortest time for the rig to a come to a stop, $(b)$ the force in the coupling between the two trailers during that time. Assume that the force exerted by the coupling on each of the two trailers is horizontal.

As the mentioned drawing illustrates, two disks with masses $m_1$ and $m_2$ are moving horizontally to the right at a speed of $v_0$. They are on an air-hockey table, which supports them with an essentially frictionless cushion of air. They move as a unit, with a compressed spring between them, which has a negligible mass. Then the spring is released and allowed to push the disks outward. Consider the situation where disk 1 comes to a momentary halt shortly after the spring is released. Assuming that $m_1=1.2 \mathrm{~kg}, m_2=2.4 \mathrm{~kg}$, and $v_0=+5.0 \mathrm{~m} / \mathrm{s}$, find the velocity of disk 2 at that moment.