1) A ball of mass, m = 2.50 kg, floats on a pool of fluid, L-1.00 m wide. It is attached to supports at either end of the pool by springs, as shown in Fig.-1. The left spring has a spring constant, A = 65.0 N/m, and equilibrium length, L1-0.600 m. The spring on the right has a spring constant, k2-75.0N/m, and equilibrium length, L20.800m L = 1.00 m ki, Li Figure 1 -A ball of mass, m = 2.50 kg, floats on a pool of fluid, L = 1.00 m wide. It is attached to supports at either end of the pool by springs. The left spring has a spring constant, A = 65.0 N/m, and equilibrium length, L1 = 0.600 m. The right spring has a spring constant. ka = 75.0 N/m, and equilibrium length, L2 = 0.800 m. (Not to scale.) a) The ball is initially at rest at position, x - d. It's then pulled to the center between the two supports at 0.500 m, and let go. How long after it's let go does it take to return to x- d? (Neglect the drag of the fluid.) Supposing we now consider that the fuld exerts a drag force on the ball of fD b 20.0kg/s. If the ball is again pulled to the center at aand let go, now how long does it take to return to the position, x = d? b) -bu, where c) In Part b, once the ball gets back to the position, x - d, what's the furthest it will get from r-d after that? d) Supposing the drag coefficient is changed to b = 40.0 kg/s. If the ball is again pulled to the center at and let go, now how long does it take to return to the position, x-d? 2 e) Going back to Part b with the drag coefficient given by b -20.0 kg/s, supposing the ball is initially at rest at x - d. But then it's driven with a driving force given by F (50.0N) cos [(8.00s) ]. After a long time has passed, how fast will the ball be traveling whenever it passes through the point where x?

Parcels of air (small volumes of air) in a stable atmosphere (where the temperature increases with height) can oscillate up and down, due to the restoring force provided by the buoyancy of the air parcel. The frequency of the oscillations are a measure of the stability of the atmosphere. Assuming that the acceleration of an air parcel can be modeled as 2zt2=g(z)0zz , prove that z=z0etN2 is a solution, where N is known as the Brunt-Väisälä frequency. Note that in a stable atmosphere, the density decreases with height and parcel oscillates up and down.

Artificial diamonds can be made using high-pressure, high-temperature presses. Suppose an artificial diamond of volume 1.00 106 m3 is formed under a pressure of 5.00 GPa. Find the change in its volume when it is released from the press and brought to atmospheric pressure. Take the diamonds bulk modulus to be B = 194 GPa.

In the short story The Pit and the Pendulum by 19th-century American horror writer Edgar Allen Poe, a man is tied to a table directly below a swinging pendulum that is slowly lowered toward him. The bob of the pendulum is a 1-ft steel scythe connected to a 30-ft brass rod. When the man first sees the pendulum, the pivot is roughly 1 ft above the scythe so that a 29-ft length of the brass rod oscillates above the pivot (Fig. P16.39A). The man escapes when the pivot is near the end of the brass rod (Fig. P16.39B). a. Model the pendulum as a particle of mass ms 5 2 kg attached to a rod of mass mr 5 160 kg. Find the pendulums center of mass and rotational inertia around an axis through its center of mass. (Check your answers by finding the center of mass and rotational inertia of just the brass rod.) b. What is the initial period of the pendulum? c. The man saves himself by smearing food on his ropes so that rats chew through them. He does so when he has no more than 12 cycles before the pendulum will make contact with him. How much time does it take the rats to chew through the ropes? FIGURE P16.39

A spring has a length of 0.200 m when a 0.300kg mass hangs from it, and a length of 0.750 m when a 1.95-kg mass hangs from it. (a) What is the force constant of the spring? (b) What is The unloaded length of the spring?

When an 80.0-kg man stands on a pogo stick, the spring is compressed 0.120 m. (a) What is the force constant of the spring? (b) Will the spring be compressed more when he hops down the road?

In about 1657, Otto von Guericke, inventor of the air pump, evacuated a sphere made of two brass hemispheres (Fig. P15.62). Two teams of eight horses each could pull the hemispheres apart only on some trials and then with greatest difficulty, with the resulting sound likened to a cannon firing. Find the force F required to pull the thin-walled evacuated hemispheres apart in terms of R, the radius of the hemispheres; P, the pressure inside the hemispheres; and atmospheric pressure P0. Figure P15.62

A block of mass M rests on a table. It is fastened to the lower end of a light, vertical spring. The upper end of the spring is fastened to a block of mass m. The upper block is pushed down by an additional force 3mg, so the spring compression is 4mg/k. In this configuration, the upper block is released from rest. The spring lifts the lower block off the table. In terms of m, what is the greatest possible value for m?

Calculate the pressure due to the ocean at the bottom of the Marianas Trench near the Philippines, given its depth is 11.0 km and assuming the density of sea water is constant all the way down. (b) Calculate the percent decrease in volume of sea water due to such a pressure, assuming its bulk modulus is the same as water and is constant. (c) What would be the percent increase in its density? Is the assumption of constant density valid? Will the actual pressure be greater or smaller than that calculated under this assumption?

A spring 1.50 m long with force constant 475 N/m is hung from the ceiling of an elevator, and a block of mass 10.0 kg is attached to the bottom of the spring. (a) By how much is the spring stretched when the block is slowly lowered to its equilibrium point? (b) If the elevator subsequently accelerates upward at 2.00 m/s2, what is the position of the block, taking the equilibrium position found in part (a) as y = 0 and upwards as the positive y-direction. (c) If the elevator cable snaps during the acceleration, describe the subsequent motion of the block relative to the freely falling elevator. What is the amplitude of its motion?