A spiral spring loaded with a piece of metal, extends by 10.5cm in air, when the metal is fully submerged
in water, the springs extends by 6.8cm. Calculate the relative density of the metal (assume Hook’s law is
Obeyed). (4mks)
Well, it sounds like the metal is a real dive enthusiast! It's just diving down and making the spring go boing! Let's see if we can calculate its relative density.
First, let's define the displacement in air as x1 (10.5 cm) and the displacement in water as x2 (6.8 cm).
According to Hooke's law, the force exerted by the spring is directly proportional to the displacement. So, we can say that:
F1 = k1 * x1 (Equation 1)
F2 = k2 * x2 (Equation 2)
Where F1 is the force in air, F2 is the force in water, k1 and k2 are the spring constants in air and water, respectively.
Now, since the spring constant is a property of the spring itself, it remains constant regardless of where it is located (air or water). So, we can say that:
k1 = k2 (Equation 3)
Using equations 1, 2, and 3, we can rewrite the equations as:
F1 = k * x1 (Equation 4)
F2 = k * x2 (Equation 5)
Now, let's consider the forces acting on the metal in water. We have the force due to gravity (mg) and the buoyant force (ρVg), where m is the mass of the metal, g is the acceleration due to gravity, ρ is the density of water, and V is the volume of the metal.
Since the net force on the metal is zero, we have:
F2 = mg - ρVg (Equation 6)
Rearranging equation 5 and substituting equation 6 into it, we get:
k * x2 = m(g - ρVg) (Equation 7)
Now, let's consider the forces acting on the metal in air. We have only the force due to gravity (mg). So, we have:
F1 = mg (Equation 8)
From equations 4 and 8, we get:
k * x1 = mg (Equation 9)
Dividing equation 7 by equation 9, we can eliminate the mass term (mg):
(x2 / x1) = (g - ρVg) / g
Now, let's solve this equation for ρ, the relative density of the metal:
ρ = 1 - (x2 / x1)
Plugging in the given values for x1 (10.5 cm) and x2 (6.8 cm):
ρ = 1 - (6.8 / 10.5)
Doing the math, we find:
ρ ≈ 0.3524
Therefore, the relative density of the metal is approximately 0.3524.
To calculate the relative density of the metal, we need to use Hook's Law:
F = -kx
Where:
F is the force applied on the spring
k is the spring constant
x is the displacement of the spring
In this case, we know that the spring extends by 10.5 cm in air, so we can use this information to calculate the spring constant.
Let's assume the spring constant for the metal in air is k1.
Using Hook's Law for the metal in air:
F1 = -k1 * x1
Next, we have the spring extended by 6.8 cm when the metal is fully submerged in water. We can use this information to calculate the spring constant when submerged in water.
Let's assume the spring constant for the metal submerged in water is k2.
Using Hook's Law for the metal submerged in water:
F2 = -k2 * x2
Now, we can use the relationship between the two forces and the two displacements to determine the relative density of the metal.
Since the force is directly proportional to the displacement, we can write:
F1 / F2 = x1 / x2
Now substituting the equations for force and displacement:
(-k1 * x1) / (-k2 * x2) = x1 / x2
We can cross multiply to get:
k1 / k2 = x1 / x2
Now, substituting the given values:
10.5 cm / 6.8 cm = k1 / k2
Simplifying the equation gives us:
k1 / k2 = 210 / 17
So, the relative density of the metal can be calculated as:
Relative Density = k1 / k2 = 210 / 17