/*! 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 128 Cast iron or steel molds are use... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 128

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.

Short Answer

Expert verified
The resulting radial acceleration on the inner surface is \(100.45\) m/s\(^2\). The maximum pressure on the surface of the mold is \(1.20*\)10\(^6\) Pa, while the minimum pressure is \(0.80*\)10\(^6\) Pa.

Step by step solution

01

Conversion of measurements

Change all the measurements into a consistent unit system. Here, SI units will be used. Thus, \(L=6\) ft \(= 1.83\) m, \(r_{o}=6\) in. \(= 0.1524\) m and \(r_{i}=4\) in. \(= 0.1016\) m. Additionally, convert the angular velocity to radians per second by multiplying by \(\frac{2\pi}{60}\), which gives \(31.42\) rad/sec.
02

Calculate radial acceleration

Find the radial acceleration on the inner surface of the liner. The formula for radial acceleration is \(a = \omega^2 r\), where \(\omega\) is the angular velocity and \(r\) is the radius. Here, we plug in \(\omega = 31.42\) rad/sec and \(r = r_{i} = 0.1016\) m to get \(a = (31.42)^2 * 0.1016 = 100.45\) m/s\(^2\).
03

Compute Maximum Pressure

To find the maximum pressure on the surface of the mold, use the pressure formula \(P = a \rho h\), where \(a\) is acceleration, \(\rho\) is the density of the molten material (steel in this case, its density is \(7850\) kg/m\(^3\)), and \(h\) is the height equivalent to the outer radius. So the maximum pressure \(P_{max} = 100.45 * 7850 * 0.1524 = 1.20*\)10\(^6\) Pa.
04

Compute Minimum Pressure

The minimum pressure is calculated using the same formula as in step 3, however with the height equivalent to the inner radius. Thus, the minimum pressure \(P_{min} = 100.45 * 7850 * 0.1016 = 0.80*\)10\(^6\) Pa.

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