An ice cube of size `a = 10 cm` is floating in a tank (base area `A = 50 cm xx 50 cm`) partially filled with water. The change in gravitational potential energy, when ice melts completely is (density of ice is `900 kg//m^(2)`)

Options

A. `- 0.072 J`
B. `- 0.24 J`
C. `- 0.016 J`
D. `-0.045 J`

Answer

To solve the problem of finding the change in gravitational potential energy when an ice cube melts completely, we can follow these steps:

### Step 1: Calculate the Volume of the Ice Cube
The volume \( V \) of the ice cube can be calculated using the formula for the volume of a cube:

\[ V = a^3 \]

Given that the size of the ice cube \( a = 10 \, \text{cm} = 0.1 \, \text{m} \):

\[ V = (0.1)^3 = 0.001 \, \text{m}^3 \]

### Step 2: Calculate the Mass of the Ice Cube
Using the density of ice, we can find the mass \( m \) of the ice cube:

\[ m = \text{density} \times V \]

Given that the density of ice is \( 900 \, \text{kg/m}^3 \):

\[ m = 900 \, \text{kg/m}^3 \times 0.001 \, \text{m}^3 = 0.9 \, \text{kg} \]

### Step 3: Determine the Volume of Water Produced by Melting Ice
When the ice melts, it turns into water. The volume of water \( V_w \) produced can be calculated using the density of water, which is approximately \( 1000 \, \text{kg/m}^3 \):

\[ V_w = \frac{m}{\text{density of water}} = \frac{0.9 \, \text{kg}}{1000 \, \text{kg/m}^3} = 0.0009 \, \text{m}^3 \]

### Step 4: Calculate the Excess Volume of the Ice Cube Above Water Level
The excess volume \( V_e \) of the ice cube that was above the water level before melting can be calculated as:

\[ V_e = V - V_w = 0.001 \, \text{m}^3 - 0.0009 \, \text{m}^3 = 0.0001 \, \text{m}^3 \]

### Step 5: Calculate the Height of the Excess Volume
The height \( h \) of the excess volume can be calculated using the base area \( A \) of the tank:

\[ h = \frac{V_e}{A} \]

The base area \( A = 50 \, \text{cm} \times 50 \, \text{cm} = 0.25 \, \text{m}^2 \):

\[ h = \frac{0.0001 \, \text{m}^3}{0.25 \, \text{m}^2} = 0.0004 \, \text{m} = 0.04 \, \text{cm} \]

### Step 6: Calculate the Change in Gravitational Potential Energy
The change in gravitational potential energy \( \Delta PE \) when the ice melts can be calculated using the formula:

\[ \Delta PE = \rho \cdot g \cdot h \cdot V_e \]

Where:
- \( \rho \) is the density of water (\( 1000 \, \text{kg/m}^3 \))
- \( g \) is the acceleration due to gravity (\( 9.8 \, \text{m/s}^2 \))
- \( h \) is the height of the excess volume
- \( V_e \) is the excess volume

Substituting the values:

\[ \Delta PE = 1000 \, \text{kg/m}^3 \cdot 9.8 \, \text{m/s}^2 \cdot 0.0001 \, \text{m}^3 \]

\[ \Delta PE = 0.98 \, \text{J} \]

Since the ice cube is floating and the potential energy decreases when it melts, we represent this change as negative:

\[ \Delta PE = -0.98 \, \text{J} \]

### Final Answer
The change in gravitational potential energy when the ice melts completely is \( -0.98 \, \text{J} \).

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