A fixed quantity of air needs to be sent through a cross section, as shown in the figure. The perimeter of the cross section is 20 m. The radius (\(r\)) of the semicircle, in m, to minimize the air velocity through the section is:
A fixed quantity of air needs to be sent through a cross section, as shown in the figure. The perimeter of the cross section is 20 m. The radius (\(r\)) of the semicircle, in m, to minimize the air velocity through the section is:
Show Hint
- \( \frac{20}{4 + \pi} \)
- \( \frac{40}{4 + \pi} \)
- \( \frac{20}{4 - \pi} \)
- \( \frac{40}{4 - \pi} \)
The Correct Option is A
Solution and Explanation
The perimeter \(P\) of the semicircle is the sum of the straight portion (the diameter) and the curved portion (the semicircular arc). Thus, we have:
\[ P = {Diameter} + {Arc length} \] \[ P = 2r + \pi r = 20 \quad ({given perimeter}) \] So, we can write the equation as: \[ 2r + \pi r = 20 \] Step 2: Solve for \(r\).
Factor out \(r\):
\[ r(2 + \pi) = 20 \] Now solve for \(r\):
\[ r = \frac{20}{2 + \pi} \] Step 3: Conclusion.
The radius of the semicircle \(r\) that minimizes the air velocity is \( \frac{20}{2 + \pi} \). Therefore, the correct answer is:
\[ r = \frac{20}{4 + \pi} \] Conclusion: The correct answer is (A) \( \frac{20}{4 + \pi} \).
Top GATE MN Mining Engineering Questions
An HMX explosive having Velocity of Detonation (VOD) of 10500 m/s is tested by D'Auriche method with a detonating fuse of VOD 7000 m/s, as shown in the figure. The impression mark on the lead plate will be obtained at a distance \(L\), in m, from the midpoint of the fuse, is:
The box plot of a data set is shown below.
The interquartile range of the data set is _______ (in integer).
A ground reaction curve (GRC) of a 3 m radius unlined circular tunnel is shown in the figure. The tunnel is supported by 300 mm thick shotcrete lining. The uniaxial compressive strength (\( \sigma_c \)), modulus of elasticity (\( E_c \)) and Poisson’s ratio (\( \nu \)) of shotcrete material are 20 MPa, 15 GPa, and 0.25 respectively. The maximum capacity (\( p_{{max}} \)) of the lining and its stiffness (\( k \)) are given as: \[ p_{{max}} = \frac{1}{2} \sigma_c \left( 1 - \frac{(a - t)^2}{a^2} \right) \] \[ k = \frac{E_c (a^2 - (a - t)^2)}{(1 + \nu)\left[(1 - \nu) a^2 + (a - t)^2\right]} \] where \( a \) = radius of the unlined tunnel and \( t \) = thickness of the lining. If the lining is constructed after 5 mm radial deformation, the support reaction is best represented by the line shown in the figure.
Based on the theodolite survey for a closed traverse PQRS, the following bearings are observed for the sides of the traverse.
The interior angles at P and R respectively are:
- The CORRECT statement(s) of dragline operation is/are:
Top GATE MN Mining and Resources Questions
An HMX explosive having Velocity of Detonation (VOD) of 10500 m/s is tested by D'Auriche method with a detonating fuse of VOD 7000 m/s, as shown in the figure. The impression mark on the lead plate will be obtained at a distance \(L\), in m, from the midpoint of the fuse, is:
The box plot of a data set is shown below.
The interquartile range of the data set is _______ (in integer).
A ground reaction curve (GRC) of a 3 m radius unlined circular tunnel is shown in the figure. The tunnel is supported by 300 mm thick shotcrete lining. The uniaxial compressive strength (\( \sigma_c \)), modulus of elasticity (\( E_c \)) and Poisson’s ratio (\( \nu \)) of shotcrete material are 20 MPa, 15 GPa, and 0.25 respectively. The maximum capacity (\( p_{{max}} \)) of the lining and its stiffness (\( k \)) are given as: \[ p_{{max}} = \frac{1}{2} \sigma_c \left( 1 - \frac{(a - t)^2}{a^2} \right) \] \[ k = \frac{E_c (a^2 - (a - t)^2)}{(1 + \nu)\left[(1 - \nu) a^2 + (a - t)^2\right]} \] where \( a \) = radius of the unlined tunnel and \( t \) = thickness of the lining. If the lining is constructed after 5 mm radial deformation, the support reaction is best represented by the line shown in the figure.
Based on the theodolite survey for a closed traverse PQRS, the following bearings are observed for the sides of the traverse.
The interior angles at P and R respectively are:
- The CORRECT statement(s) of dragline operation is/are:
Top GATE MN Questions
- Exposure to loud impulsive noise may lead to
- In a self-contained closed-circuit breathing apparatus,
- A rectangular mine airway of 2.0 m width and 2.5 m height has a bend with deflection of π/4 radian. If the radius of curvature of the bend is 4.0 m, the shock factor of the bend is (round off to three decimals)
- In an underground coal mine, two fatalities and three serious bodily injuries occurred during the year 2022. The average daily employment is 1100 and annual working days is 300. The severity index as per DGMS guideline for the mine is
- For a geared engine winding system, the man winding cage is placed at its normal position at pit top of the shaft. As per CMR 2017, the minimum space, in m, between the center of the hole of the detaching hook attached to the rope shackle and detaching belt plate is