/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 105 In a logging operation, timber f... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 105

In a logging operation, timber floats downstream to a lumber mill. It is a dry year, and the river is running low, as low as \(60 \mathrm{cm}\) in some locations. What is the largest diameter log that may be transported in this fashion (leaving a minimum \(5 \mathrm{cm}\) clearance between the log and the bottom of the river)? For the wood, \(\mathrm{SG}=0.8\).

Short Answer

Expert verified
The largest diameter of the log that can be transported without scraping the river bottom is found to be approximately \(93.5 \mathrm{cm}\).

Step by step solution

01

Understanding the Problem

Understanding the problem and what it is asking for is crucial. The problem involves the flotation of logs and seems to be asking for the maximum diameter of the log that can float in \(60 \mathrm{cm}\) deep water while maintaining a minimum of \(5 \mathrm{cm}\) clearance from the bottom of the river. Basically, this means that the log has to be submerged in water up to \(60 \mathrm{cm} - 5 \mathrm{cm} = 55 \mathrm{cm}\).
02

Applying Archimedes鈥� principle

Archimedes鈥� Principle states that the buoyant force on an object is equal to the weight of the water displaced by the object. Since the log is in equilibrium, the buoyant force on the log must equal the weight of the log. This means that the volume of water displaced by the log must be the same as the volume of the log submerged under the water. This makes the volume of water displaced equal to \(0.55蟺r^2\) where r is the radius of the log.
03

Using Specific Gravity

The specific gravity of wood is given as 0.8, which means the wood is 0.8 times as dense as water. Now, the volume of log submerged would be equal to the volume of the log itself since it is completely submerged, and its volume equals to \(0.8蟺r^2\). Given the specific gravity (SG) of wood is 0.8, it follows that \(0.8蟺r^2=0.55蟺r^2\).
04

Finding the Maximum Diameter

Now, solve the equation \(0.8蟺r^2=0.55蟺r^2\) for r. The equation simplifies to \(r=\sqrt{0.55/0.8}\). Multiply the value of r by 2 to obtain the diameter. The answer will be in meters, but since the depth of the river is given in cm, it would be best to convert the diameter from meters to cm, which gives us \(2r * 100\).

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Most popular questions from this chapter

The analysis of Problem 3.121 suggests that it may be possible to determine the coefficient of sliding friction between two surfaces by measuring the slope of the free surface in a liquid-filled container sliding down an inclined surface. Investigate the feasibility of this idea.

Three steel balls (each about half an inch in diameter ) lie at the bottom of a plastic shell floating on the water surface in a partially filled bucket. Someone removes the steel balls from the shell and carefully lets them fall to the bottom of the bucket, leaving the plastic shell to float empty. What happens to the water level in the bucket? Does it rise, go down, or remain unchanged? Explain.

A cubical box, \(80 \mathrm{cm}\) on a side, half-filled with oil \((\mathrm{SG}=0.80),\) is given a constant horizontal acceleration of 0.25 \(g\) parallel to one edge. Determine the slope of the free surface and the pressure along the horizontal bottom of the box.

A reservoir manometer has vertical tubes of diameter \(D=18 \mathrm{mm}\) and \(d=6 \mathrm{mm} .\) The manometer liquid is Meriam red oil. Develop an algebraic expression for liquid deflection \(L\) in the small tube when gage pressure \(\Delta p\) is applied to the reservoir. Evaluate the liquid deflection when the applied pressure is equivalent to \(25 \mathrm{mm}\) of water (gage).

Cast iron or steel molds are used in a horizontalspindle machine to make tubular castings such as liners and tubes. A charge of molten metal is poured into the spinning mold. The radial acceleration permits nearly uniformly thick wall sections to form. A steel liner, of length \(L=6 \mathrm{ft}\), outer radius \(r_{o}=6\) in. and inner radius \(r_{i}=4\) in., is to be formed by this process. To attain nearly uniform thickness, the angular velocity should be at least 300 rpm. Determine (a) the resulting radial acceleration on the inside surface of the liner and (b) the maximum and minimum pressures on the surface of the mold.
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