A spiral spring work with a piece of metal extent by 10.5 in arise when the metal is fully submerged in water,the spring extended by 6.8cm cal. the R.O of the metal (assumed hook's law is obeyed).

To find the unknown quantity (R.O of the metal) in this problem, we can use Hooke's Law. Hooke's Law states that the force required to extend or compress a spring is directly proportional to the displacement of the spring from its equilibrium position.

Let's go step by step to solve the problem:

1. First, convert all the given measurements to a consistent system of units. In this case, we will convert the inches to centimeters since the spring extension is also given in centimeters.

Given:
Spring extension when the metal is fully submerged in water = 6.8 cm
Metal extension = 10.5 inches

Converting inches to centimeters:
1 inch = 2.54 centimeters
Metal extension = 10.5 inches × 2.54 cm/inch ≈ 26.7 cm

Now we have the spring extension (6.8 cm) and the metal extension (26.7 cm).

2. Apply Hooke's Law to find the spring constant (k):
Hooke's Law can be written as F = k * x, where F is the force applied to the spring, k is the spring constant, and x is the displacement or extension of the spring.

Since the metal extension is causing the spring extension, we can equate the forces exerted by the spring and the metal:
F_spring = F_metal

Now, we can rewrite Hooke's Law using the given values:
k * x_spring = k * x_metal

3. Rearrange the equation to solve for the spring constant (k):
k = k * x_metal / x_spring

4. Find the R.O of the metal using the spring constant (k):
Since the spring constant (k) is equal to the R.O of the metal, we can substitute it into the equation:
R.O of the metal = k * x_metal / x_spring

Plugging in the values:
R.O of the metal = (k * 26.7 cm) / (6.8 cm)

Now, to find the R.O of the metal, we need the spring constant (k). If the spring constant is given or can be calculated, we can substitute it into the equation to get the answer.

To find the radius of the metal, we can use Hooke's Law, which states that the extension of a spring is directly proportional to the force acting on it.

Let's break down the given information:
- The spring extends by 10.5 inches (in) when the metal is fully submerged in water.
- The spring extends by 6.8 centimeters (cm) when a certain force is applied.

Since we need to find the radius of the metal, we can set up the equation using Hooke's Law:

F = k * x

Where:
F is the force applied to the spring
k is the spring constant
x is the extension of the spring

Now, let's convert the given measurements to the same unit. We'll use centimeters since they are mentioned in the equation already.

1 inch = 2.54 cm

10.5 inches = 10.5 * 2.54 = 26.67 cm

So, the extension of the spring when the metal is fully submerged in water is 26.67 cm.

Now, we can set up the first equation:

F₁ = k * x₁
Where:
F₁ is the force when the metal is fully submerged in water.
x₁ is the extension when the metal is fully submerged in water.

Similarly, we can set up the second equation:

F₂ = k * x₂
Where:
F₂ is the force when a certain force is applied.
x₂ is the extension when a certain force is applied.

Since we assume Hooke's Law is obeyed, the spring constant (k) will be the same for both cases. So, we can equate the equations:

F₁ = * x₁ = k * x₂

Now, let's substitute the values we have:

k * 26.67 = k * 6.8

Since k is common in both sides of the equation, we can cancel it out:

26.67 = 6.8

This is not possible, so there seems to be an error or discrepancy in the given information. Please make sure the values provided are accurate or provide additional information if available.