A wooden cubical block of side 0.1 m has specific gravity (SG) of 0.75. It is held submerged in a pool of oil and water by a massless rigid wire as shown in figure. The density of water is 1000 $kg/m^3$ and acceleration due to gravity is 9.8 $m/s^2$.The tension, in N, in the wire is ________ (round off to 2 decimal places).
Given Parameters
- Side of the cubical block, \( L = 0.1 \, \text{m} \)
- Specific gravity of wood, \( SG_s = 0.75 \)
- Density of water, \( \rho_w = 1000 \, \text{kg/m}^3 \)
- Acceleration due to gravity, \( g = 9.8 \, \text{m/s}^2 \)
- Specific gravity of oil, \( SG_{oil} = 0.7 \) (Note: Since the block is fully submerged in water, the oil layer only affects the absolute pressure but not the net buoyant force).
Step-by-Step Solution
1. Calculate the Volume of the Block
The volume \( V \) of the cubical block is given by the cube of its side length:
$$ V = L^3 = (0.1 \, \text{m})^3 = 0.001 \, \text{m}^3 $$
2. Calculate the Density and Weight of the Block
The density of the wooden block \( \rho_s \) is calculated using its specific gravity:
$$ \rho_s = SG_s \times \rho_w = 0.75 \times 1000 \, \text{kg/m}^3 = 750 \, \text{kg/m}^3 $$
The weight \( W \) of the block acting downwards is:
$$ W = \rho_s \cdot V \cdot g $$ $$ W = 750 \times 0.001 \times 9.8 = 7.35 \, \text{N} $$
3. Calculate the Buoyant Force
The buoyant force \( F_B \) is the weight of the fluid displaced. Since the block is completely submerged in water, we use the density of water:
$$ F_B = \rho_w \cdot V \cdot g $$ $$ F_B = 1000 \times 0.001 \times 9.8 = 9.8 \, \text{N} $$
4. Calculate the Tension in the Wire
For the block to be in equilibrium, the sum of downward forces must equal the sum of upward forces. The upward force is the buoyant force, and the downward forces are the weight and the tension \( T \) in the wire:
$$ \sum F_{up} = \sum F_{down} $$ $$ F_B = W + T $$
Solving for tension \( T \):
$$ T = F_B - W $$ $$ T = 9.8 \, \text{N} - 7.35 \, \text{N} $$ $$ T = 2.45 \, \text{N} $$
Final Answer
The tension in the wire is \( 2.45 \, \text{N} \).
Important Questions from Hydrostatic Force
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In the context of hydrostatics, the resultant hydrostatic force acting on a submerged plane surface passes through which of the following points?
The depth of the center of pressure on a vertical rectangular gate (4 m wide and 3 m high) with water up to top surface is
If a planar surface is immersed in a liquid, the resultant liquid pressure acts at a point called ___________.
The resultant of all normal pressure acts
