A metal cube of edge 5 cm and density `9.0cm^(-3)` is suspended by a thread so as to be completely immersed in a liquid of density `1.2gcm^(-3)`. Find the tension in thread. (Take `g=10ms^(-2)`)

Answer

To find the tension in the thread suspending the metal cube, we will follow these steps:

### Step 1: Calculate the Volume of the Cube
The volume \( V \) of a cube can be calculated using the formula:
\[ V = \text{edge}^3 \]
Given that the edge of the cube is 5 cm:
\[ V = 5 \, \text{cm} \times 5 \, \text{cm} \times 5 \, \text{cm} = 125 \, \text{cm}^3 \]

### Step 2: Calculate the Mass of the Cube
The mass \( m \) of the cube can be calculated using the formula:
\[ m = V \times \text{density} \]
Given that the density of the cube is \( 9 \, \text{g/cm}^3 \):
\[ m = 125 \, \text{cm}^3 \times 9 \, \text{g/cm}^3 = 1125 \, \text{g} \]

### Step 3: Calculate the Weight of the Cube
The weight \( W \) of the cube can be calculated using the formula:
\[ W = m \times g \]
Given that \( g = 10 \, \text{m/s}^2 \) and converting grams to kilograms (1 g = 0.001 kg):
\[ W = 1125 \, \text{g} \times 10^{-3} \, \text{kg/g} \times 10 \, \text{m/s}^2 = 1125 \times 0.01 = 112.5 \, \text{N} \]

### Step 4: Calculate the Buoyant Force
The buoyant force \( F_b \) can be calculated using Archimedes' principle:
\[ F_b = \rho \times V \times g \]
Where \( \rho \) is the density of the liquid, given as \( 1.2 \, \text{g/cm}^3 \):
\[ F_b = 1.2 \, \text{g/cm}^3 \times 125 \, \text{cm}^3 \times 10^{-3} \, \text{kg/g} \times 10 \, \text{m/s}^2 \]
Calculating this:
\[ F_b = 1.2 \times 125 \times 0.001 \times 10 = 1.5 \, \text{N} \]

### Step 5: Calculate the Tension in the Thread
Using the equilibrium of forces, we can express the tension \( T \) in the thread:
\[ T = W - F_b \]
Substituting the values we found:
\[ T = 112.5 \, \text{N} - 1.5 \, \text{N} = 111 \, \text{N} \]

### Final Answer
The tension in the thread is \( 111 \, \text{N} \).

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