To determine if the crown is made of real gold, we can use the principle of buoyancy. When an object is submerged in a fluid, it experiences an upward buoyant force equal to the weight of the fluid displaced by the object.
Let's consider the forces acting on the crown when it is submerged in water:
- The force of tension on the scale (Ft)
- The buoyant force due to the water (Fb)
- The force of gravity acting on the crown (Fg)
Based on the given information, the scale reads a weight of 13.4 kg, which is less than the actual weight of the crown (14.7 kg). This indicates that the buoyant force (Fb) is greater than the actual weight of the crown.
We can calculate the buoyant force using the formula: Fb = (density of fluid) * g * (volume of fluid)
The volume of fluid displaced by the crown is equal to the volume of the crown itself. So, we need to find the volume of the crown.
To find the volume of the crown, we can use the density of an object formula: density = (mass of object) / (volume of object)
Rearranging this formula, we get: volume of object = (mass of object) / (density of object)
Since we know the mass of the crown (14.7 kg) and the density of gold (19,300 kg/m^3), we can calculate the volume of the crown.
Now, substitute the values into the formula for the buoyant force.
Fb = (density of fluid) * g * (volume of crown)
The density of water is approximately 1000 kg/m^3, and g is the acceleration due to gravity, which is 9.8 m/s^2.
With these values, you can solve for Fb.
Next, we can find the force of tension on the scale (Ft). Since the scale measures the apparent weight of the crown, Ft is equal to the difference between the actual weight of the crown (Fg) and the buoyant force (Fb).
Ft = Fg - Fb
If Ft is positive, it means the scale is measuring a greater weight than the actual weight of the crown. If Ft is negative, it means the scale is measuring a lower weight than the actual weight of the crown.
If Ft is negative, then the crown is less dense than water (density of object < density of water), which means it is not made of real gold.