While vacationing at the Outer Banks of North Carolina, you find an old coin that looks like it is made of gold. You know there were many shipwrecks here, so you take the coin home to check the possibility of it being gold. You suspend the coin from a spring scale and find that it has a weight in air of 49.7 g. You then let the coin hang submerged in a glass of water and find that the scales reads 47.1 g. Should you get excited about the possibility that this coin might really be gold?
Answers
m1 = 49.7
g = 0.0497 KG is the spring scale reading when the coin is suspended in the air.
m2 = 47.1
g = 0.0471 kg is the spring scale reading when the coin is submerged in the water.
ρwater = 1,000 kg/m3 is the density of water.
ρgold = 19,300 kg/m3 is the density of the gold
g = 9.81 m/S2 is the acceleration of gravity
R is the scale reading(spiring force)
To determine whether a coin is made of gold, we need to determine the density of the coin. The true weight of the coin when suspended in the air is:
W air = m1g
= 0.0497
kg X ×9.81
m/s2 = 0.487557
=0.487557 N
When a coin is submerged in water, the spring scale reading is equal to the true weight of the coin minus the buoyant force applied by the water. Using Newton's second law:
∑F vertical
= 0
Scale reading - True Weight + Buoyant Force = 0
0m2g − W air + FB = 0
FB = W air - M2g
=0.487557 N - 0.0471 KG X 9.81 m/S2
=0.025506 N
This is the buoyant force acting on the coin.
The volume of the coin is calculated as follows:
FB = V ρWater g
V = FB/ρWater g
= 0.025506 N/1,000 KG/M3 X 9.81 m/S2
=0.0000026 M3
The density of the coin is calculated as follows:
Density= mass in the air/volume
ρ= 0.0497 KG/0.0000026 m3
Therefore,the density of the material used fro the coin is 19,100 KG/m3