A crate, initially traveling horizontally with a speed of 18 ft/s, is made to slide down a 14 ft chute inclined at $35^\circ.$ The surface of the chute has a coefficient of kinetic friction $\mu_k,$ and at its lower end, it smoothly lets the crate onto a horizontal trajectory. The horizontal surface at the end of the chute has a coefficient of kinetic friction $\mu_k^2.$ Model the crate as a particle, and assume that gravity and the contact forces between the crate and the sliding surface are the only relevant forces. Let $\mu_k = 0.5$ and suppose that once the crate reaches the bottom of the chute and after sliding horizontally for 5 ft, the crate runs into a bumper. If the weight of the crate is W = 110 lb, $\mu_k^2 = 0.33,$ and you model the bumper as a linear spring with constant k and neglect the mass of the bumper, determine the value of k so that the crate comes to a stop 2 ft after impacting with the bumper.

A crate, initially traveling horizontally with a speed of 18 ft/s, is made to slide down a 14 ft chute inclined at $35^\circ.$ The surface of the chute has a coefficient of kinetic friction $\mu_k,$ and at its lower end, it smoothly lets the crate onto a horizontal trajectory. The horizontal surface at the end of the chute has a coefficient of kinetic friction $\mu_k^2.$ Model the crate as a particle, and assume that gravity and the contact forces between the crate and the sliding surface are the only relevant forces. Find $\mu_k$ such that the crate’s speed at the bottom of the chute (immediately before the crate’s trajectory becomes horizontal) is 15 ft/s.

A 5.00-kg crate is suspended from the end of a short vertical rope of negligible mass. An upward force F(t) is applied to the end of the rope, and the height of the crate above its initial position is given by $y(t)=(2.80 m / s) t+\left(0.610 m / s^{3}\right) t^{3}$. What is the magnitude of F when t = 4.00 s?

The crate moves up the incline a distance d = 4.5 ft due to the action of the constant force P = 100 lb. The weight of the crate is W = 65 lb, and the spring with constant k = 10 lb/ft is unstretched before the crate starts moving. Friction between the crate and the incline is negligible, and the angle between the force P and the surface on which the crate slides is $\beta = 30^\circ.$ Letting $\theta = 30^\circ,$ determine the total work done by all forces after the crate has moved the distance d.

Packages for transporting delicate items (a laptop or glass) are designed to “absorb” some of the energy of the impact in order to protect their contents. These energy absorbers can get very complicated (the mechanics of Styrofoam peanuts can be complex), but we can begin to understand how they work by modeling them as a linear elastic spring of constant k that is placed between the contents (an expensive vase) of mass m and the package P. Assume that the vase’s mass is 3 kg and that the box is dropped from rest from a height of 1.5 m. Treating the vase as a particle and neglecting all forces except for gravity and the spring force, determine the value of the spring constant k so that the maximum displacement of the vase relative to the box is 0.15 m. Assume that the spring relaxes after the box is dropped and that it does not oscillate.

The linkage is made using two A992 steel rods, each having a circular cross section. Determine the diameter of each rod to the nearest $\frac{1}{8} \mathrm{in}$. that will support a load of $P=6$ kip. Assume that the rods are pin connected at their ends. Use a factor of safety with respect to buckling of 1.8.

Packages for transporting delicate items (a laptop or glass) are designed to “absorb” some of the energy of the impact in order to protect their contents. These energy absorbers can get pretty complicated (the mechanics of Styrofoam peanuts is not easy), but we can begin to understand how they work by modeling them as a linear elastic spring of constant k that is placed between the contents (an expensive vase) of mass m and the package P. Assume that the vase weighs 6 lb and that the box is dropped from a height of 5 ft. Treat the vase as a particle, and neglect all forces except for gravity and the spring force. Determine the maximum displacement of the vase relative to the box and the maximum force on the vase due to the spring if k = 264 lb/ft.

A vehicle A is stuck on the railroad tracks as a train B approaches with a speed of 120 km/h. As soon as the problem is detected, the train’s emergency brakes are activated, locking the wheels and causing the train’s wheels to slide relative to the tracks. If the coefficient of kinetic friction between the wheels and the track is 0.2, use the workenergy principle to determine the minimum distance dmin at which the brakes must be applied to avoid a collision under the following circumstances: (a) The train consists of just a 195,000 kg locomotive. (b) The train consists of a 195,000 kg locomotive and a string of cars whose mass is $10 \times 106 kg$, all of which can apply brakes and lock their wheels. (c) The train consists of a 195,000 kg locomotive and a string of cars whose mass is $10 \times 106 kg$, but only the locomotive can apply its brakes and lock its wheels. Treat the train as a particle, and assume that the railroad tracks are rectilinear and horizontal.

Indicate whether the statement is true or false. If $z=p(A, B)=(1 / 2)\left(A^{2}-B\right)$ then $\frac{\partial z}{\partial A}=A$.

Show that every polynomial function of degree 3 $f(x)=a x^{3}+b x^{2}+c x+d$ has exactly one inflection point.

Find an equation of the line that passes through the points. Then sketch the line. $(-5,5),(10,-1)$

A 350 kg crate is sliding down a rough incline with a constant speed v = 7 m/s. Assuming that the angle of the incline is $\theta=33^{\circ}$ and that the only forces acting on the crate are gravity, friction, and the normal force between the crate and the incline, determine the work done by friction over every meter slid by the crate.

A rigid tank whose volume is unknown is divided into two parts by a partition. One side of the tank contains an ideal gas at 927$^\circ{}$C. The other side is evacuated and has a volume twice the size of the part containing the gas. The partition is now removed and the gas expands to fill the entire tank . Heat is now transferred to the gas until the pressure equals the initial pressure. Determine the final temperature of the gas. Answer: 3327$^\circ{}$C

A tank whose volume is unknown is divided into two parts by a partition. One side of the tank contains 0.03 m^3 of refrigerant-134a that is a saturated liquid at 0.9 MPa, while the other side is evacuated. The partition is now removed, and the refrigerant fills the entire tank. If the final state of the refrigerant is 20$^\circ{}$C and 280 kPa, determine the volume of the tank.