If the elastic limit of steel is 5.0\times10^8 Pa,determine the minimum diameter a steel wire can have if it is tosupport a 60 kg circus performerwithout its elastic limit being exceeded.

Write in words how to read each of the following out loud.
a. ${\displaystyle \{x\in R^{\prime }\mid 0<x<1\}}$
b. ${\displaystyle \{x\in R\mid x\le 0\text{or}x\Rightarrow 1\}}$
c. ${\displaystyle \{n\in Z\mid nisafactorof6\}}$
d. ${\displaystyle \{n\in Z\cdot \mid nisafactorof6\}}$

A stunt man whose mass is 70 kg swings from the end ofa 4.0 m long rope along thearc of a vertical circle. Assuming that he starts from rest whenthe rope is horizontal, find the tensions in the rope that are required to make him follow his circular path at each of the following points.
(a) at the beginning of his motion N
(b) at a height of 1.5 m above the bottom of the circular arc N
(c) at the bottom of the arc N

A 2.0- kg piece of wood slides on the surface. The curved sides are perfectly smooth, but the rough horixontal bottom is 30 m long and has a kinetic friction coefficient of 0.20 with the wood. The iece of wood starts fromrest 4.0 m above the rough bottom (a) Where will this wood eventually come to rest? (b) For the motion from the initial release until the piece of wood comes to rest, what is the total amount of work done by friction?

A stream of water strikes a stationary turbine bladehorizontally, as the drawing illustrates. The incident water streamhas a velocity of +18.0 m/s, while the exiting water stream has avelocity of -18.0 m/s. The mass of water per second that strikesthe blade is 17.0 kg/s. Find themagnitude of the average force exerted on the water by theblade.

Suppose you have 1.0 mol of ${\displaystyle O_2}$ gas. How many coulombs of positive charge are contained in the atomic nuclei of this gas?

A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. Four hundred feet of fencing is used. Find the dimensions of the playground that maximize the total enclosed area. What is the maximum area?

Casting of molten metal is important in many industrialprocesses. Centrifugal casting is used for manufacturing pipes,bearings and many other structures. A veriety of sophisticatedtechniques have been invented. A cylindrical enclosure is rotated rapidly and steadily about a horizontal axis. Molten metal is poured into therotating cylinder and then cooled, forming the finished product.Turning the cylinder at a high rotation rate forces the solidifying metal strongly to the outside. Any bubbles are displace toward the axis, so unwanted voids will not be present in the casting. Sometimes it is desirable to form a composite casting, such as fora bearing. Here a stron steel outer surface is poured, followed byan inner lining of special low- friction metal. In some applications a very strong metal is given a coating of corrosion-resistan metal. Centrifugal casting results in stronbonding between the layers.
Suppose that a copper sleeve of iner radius 2.10 cm and outer radius 2.20 cm is to be cast. To eleminate bubbles and give high structural intgrity, the centripetal acceleration of each bit of metal should be 100g what rate of rotation is required? State theanswer in revolutions per minute.