A tunnel of length L=150 m, height H=7.2 m, and width 5.8 m (with a flat roof) is to be constructed at distance d=60 m beneath the ground. (See the Figure.) The tunnel roof is to be supported entirely by square steel columns, each with a cross-sectional area of 960 cm2. The mass of 1.0 cm3 of the ground material is 2.8 g. (a) What is the total weight of the ground material the columns must support? (b) How many columns are needed to keep the compressive stress on each column at one-half its ultimate strength?
Consider the formula for the density as:
Here, is density of the material, m is mass, and v is volume.
Consider the formula for the weight as:
w=mg ……. (ii)
Here, w is weight, m is mass, and g is gravitational acceleration.
Determine the area of the roof as:
Consider the volume of the material is obtained as:
Solve for the density of the material as:
Determine the mass using equation (i) as,
Use the equation (ii) to find the weight.
Therefore, the weight of the ground material the column must support is .
Consider the ultimate strength of the steel is . Multiplying it by area of the column would give us the strength of the columns. All the columns should support the weight of the ground and compressive stress on each column should be less than half of the ultimate strength, so we have:
Substitute the values and solve as:
Therefore, the number of columns that are needed to keep compressive stress on each column should be less than half of the ultimate strength is 75.
The force in Fig. 12-70 keeps the block and the pulleys in equilibrium. The pulleys have negligible mass and friction. Calculate the tension T in the upper cable. (Hint: When a cable wraps halfway around a pulley as here, the magnitude of its net force on the pulley is twice the tension in the cable.)
Figure 12-50 shows a climber hanging by only the crimp hold of one hand on the edge of a shallow horizontal ledge in a rock wall. (The fingers are pressed down to gain purchase.) Her feet touch the rock wall at distance directly below her crimped fingers but do not provide any support. Her center of mass is distance from the wall. Assume that the force from the ledge supporting her fingers is equally shared by the four fingers. What are the values of the(a) horizontal component Fhand (b) vertical component Fv of the force on each fingertip?
Figure 12-19 shows an overhead view of a uniform stick on which four forces act. Suppose we choose a rotation axis through point O, calculate the torques about that axis due to the forces, and find that these torques balance. Will the torques balance if, instead, the rotation axis is chosen to be at
(a) point A (on the stick),
(b) point B (on line with the stick), or
(c) point C (off to one side of the stick)?
(d) Suppose, instead, that we find that the torques about point O does not balance. Is there another point about which the torques will balance?
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