A mass m is gently placed on the end of a freely hanging spring. The mass then falls 33 cm before it stops and begins to rise. What is the frequency of the oscillation?

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.

An elevator (mass $4850 \mathrm{~kg}$ ) is to be designed so that the maximum acceleration is $0.0680 \mathrm{~g}$. What are the maximum and minimum forces the motor should exert on the supporting cable?

A 220-kg wooden raft floats on a lake. When a 75-kg man stands on the raft, it sinks 4.0 cm deeper into the water. When he steps off, the raft vibrates for a while. What is the frequency of vibration?

We usually neglect the mass of a spring if it is small compared to the mass attached to it. But in some applications, the mass of the spring must be taken into account. Consider a spring of unstretched length $\ell$ and mass $M_S$ uniformly distributed along the length of the spring. A mass $m$ is attached to the end of the spring. One end of the spring is fixed and the mass $m$ is allowed to vibrate horizontally without friction (Fig. 14-47). Each point on the spring moves with a velocity proportional to the distance from that point to the fixed end. For example, if the mass on the end moves with speed $v_0$, the midpoint of the spring moves with speed $v_0 / 2$. Show that the kinetic energy of the mass plus spring when the mass $m$ is moving with velocity $v$ is

$$K=\frac{1}{2} M v^2$$

where $M=m+\frac{1}{3} M_{\mathrm{S}}$ is the "effective mass" of the system. [Hint: Let $D$ be the total length of the stretched spring. Then the velocity of an infinitesimal length $d x$ of spring, of mass $d M$, located at $x$ is $v(x)=v_0(x / D)$. Note also that $d M=d x\left(M_{\mathrm{S}} / D\right)$.]

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.

A thin, straight, uniform rod of length $\ell=1.00 \mathrm{~m}$ and mass $m=185 \mathrm{~g}$ hangs from a pivot at one end. What is the length of a simple pendulum that will have the same period?

A thin, straight, uniform rod of length $\ell=1.00 \mathrm{~m}$ and mass $m=185 \mathrm{~g}$ hangs from a pivot at one end. What is its period for small-amplitude oscillations?

Jupiter is about $320$ times as massive as the Earth. Thus, it has been claimed that a person would be crushed by the force of gravity on a planet the size of Jupiter because people cannot survive more than a few $g$ 's. Calculate the number of $g$ 's a person would experience at Jupiter's equator, using the following data for Jupiter: mass = $1.9 \times 10^{27} \mathrm{~kg}$, equatorial radius $=7.1 \times 10^4 \mathrm{~km}$, rotation period $=9 \mathrm{~hr~} 55 \mathrm{~min}$. Take the centripetal acceleration into account.

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 mass attached to the end of a spring is stretched a distance $x_0$ from equilibrium and released. At what distance from equilibrium will it have velocity equal to half its maximum velocity?

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.

A mass attached to the end of a spring is stretched a distance $x_0$ from equilibrium and released. At what distance from equilibrium will it have acceleration equal to half its maximum acceleration?

One possibility for a low-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel "spinup." What is the angular velocity of the flywheel when it has a full "energy charge"?