(a) The diameter of neck and bottom of a bottle are 2 cm and 10 cm respectively. The bottle is completely filled with oil. If the cork in the neck is pressed in with a force of `1.2kgf` , what force is exerted on the bottom of the bottle ?
(b) Name the law/ principle you have used to find the force in part (a)

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Determine the radius of the neck and the bottom of the bottle.
- The diameter of the neck is given as 2 cm, so the radius (r₁) of the neck is:
\[ r₁ = \frac{2 \text{ cm}}{2} = 1 \text{ cm} \]
- The diameter of the bottom is given as 10 cm, so the radius (r₂) of the bottom is:
\[ r₂ = \frac{10 \text{ cm}}{2} = 5 \text{ cm} \]

### Step 2: Calculate the area of the cork (neck).
- The area (A₁) of the cork can be calculated using the formula for the area of a circle:
\[ A₁ = \pi r₁^2 = \pi (1 \text{ cm})^2 = \pi \text{ cm}^2 \]

### Step 3: Calculate the pressure applied by the cork.
- The force applied on the cork is given as 1.2 kgf. To find the pressure (P₁) exerted by the cork, we use the formula:
\[ P₁ = \frac{\text{Force}}{\text{Area}} = \frac{1.2 \text{ kgf}}{A₁} = \frac{1.2 \text{ kgf}}{\pi \text{ cm}^2} \]

### Step 4: Calculate the area of the bottom of the bottle.
- The area (A₂) of the bottom can be calculated similarly:
\[ A₂ = \pi r₂^2 = \pi (5 \text{ cm})^2 = 25\pi \text{ cm}^2 \]

### Step 5: Use Pascal's Law to find the force at the bottom of the bottle.
- According to Pascal's Law, the pressure at the cork is equal to the pressure at the bottom of the bottle:
\[ P₁ = P₂ \]
- Therefore, we can set up the equation:
\[ \frac{1.2 \text{ kgf}}{\pi \text{ cm}^2} = \frac{F}{25\pi \text{ cm}^2} \]
- Here, \(F\) is the force exerted at the bottom of the bottle. Rearranging gives:
\[ F = 1.2 \text{ kgf} \times \frac{25\pi \text{ cm}^2}{\pi \text{ cm}^2} = 1.2 \times 25 \text{ kgf} = 30 \text{ kgf} \]

### Final Answer:
(a) The force exerted on the bottom of the bottle is **30 kgf**.

(b) The principle used to find the force in part (a) is **Pascal's Law**.

---