A solid cube (side=2.54 cm) of aluminum (p=2.70x10^3 kg/m^3) is attached to a vertical cord. (a) Find the tension in the cord if the cube is in air. (b) Find the tension in the cord if the cube is fully immersed in water (p=1.00x10^3 kg/m^3).
To find the tension in the cord, we need to consider the gravitational force acting on the cube for both scenarios: when it is in air and when it is fully immersed in water.
(a) When the cube is in air, the only force acting on it is its weight. The weight of an object can be calculated using the formula:
Weight = mass × gravity
The mass of the cube can be found using its density and volume. The volume of a cube is given by:
Volume = side^3
Substituting the given values, we get:
Volume = (2.54 cm)^3 = 16.387064 cm^3
Converting cm^3 to m^3, we have:
Volume = 16.387064 × 10^(-6) m^3
Now, we can calculate the mass of the cube using its density:
Mass = density × volume
Substituting the given values, we get:
Mass = (2.70 × 10^3 kg/m^3) × (16.387064 × 10^(-6) m^3) = 0.044 kg
Next, we can calculate the weight of the cube using the formula mentioned earlier:
Weight = mass × gravity
Substituting the values, with gravity being approximately 9.8 m/s^2, we have:
Weight = 0.044 kg × 9.8 m/s^2 = 0.4312 N
Therefore, the tension in the cord if the cube is in air is 0.4312 N.
(b) When the cube is fully immersed in water, it experiences an additional buoyant force due to the displaced water. The buoyant force is given by:
Buoyant force = density of fluid × volume of fluid displaced × gravity
The volume of fluid displaced by the cube is equal to its volume since it is fully submerged. We already calculated the volume of the cube earlier, which is:
Volume = 16.387064 × 10^(-6) m^3
Substituting the given values, with the density of water being 1.00 × 10^3 kg/m^3 and gravity being approximately 9.8 m/s^2, we can calculate the buoyant force:
Buoyant force = (1.00 × 10^3 kg/m^3) × (16.387064 × 10^(-6) m^3) × (9.8 m/s^2) = 0.16049 N
Since the buoyant force in water opposes the weight of the cube, the tension in the cord will be reduced by this amount. Therefore, the tension in the cord if the cube is fully immersed in water is:
Tension = Weight - Buoyant force
Tension = 0.4312 N - 0.16049 N = 0.27071 N
Therefore, the tension in the cord if the cube is fully immersed in water is 0.27071 N.