What is the acceleration of an 0.4m aluminum cube with a density of 2.72 in a beaker of methanol, which has a density of .791?
To determine the acceleration of an aluminum cube in methanol, we need to calculate the net force acting on the cube and then apply Newton's second law of motion (F = ma), where a is the acceleration.
First, let's calculate the weight of the aluminum cube using its density and volume. The formula for calculating weight is weight = mass x acceleration due to gravity. The acceleration due to gravity is approximately 9.8 m/s².
The density of aluminum is 2.72 g/cm³ or 2,720 kg/m³. Since the aluminum cube has a dimension of 0.4m, the volume of the cube can be calculated as 0.4m x 0.4m x 0.4m = 0.064m³.
The mass of the aluminum cube can be calculated using the formula mass = density x volume. Therefore, the mass of the aluminum cube is 2,720 kg/m³ x 0.064m³ = 174.08 kg.
Now, let's calculate the buoyant force acting on the aluminum cube submerged in methanol. The formula for calculating buoyant force is buoyant force = weight of the fluid displaced.
The density of methanol is 0.791 g/cm³ or 791 kg/m³. The weight of the fluid displaced can be calculated using the formula weight = density x volume x acceleration due to gravity. Therefore, the weight of the fluid displaced is 791 kg/m³ x 0.064m³ x 9.8 m/s² = 494.12 N.
The net force acting on the aluminum cube is the difference between the weight of the cube and the weight of the fluid displaced. Therefore, the net force is 174.08 N - 494.12 N = -320.04 N (negative sign indicates an upward force).
Finally, we can use Newton's second law of motion to calculate the acceleration of the cube. Rearranging the formula F = ma, we have a = F/m. Plugging in the values, we find a = -320.04 N / 174.08 kg ≈ -1.839 m/s².
Hence, the acceleration of the 0.4m aluminum cube in methanol is approximately -1.839 m/s². The negative sign indicates that the cube is being accelerated in the upward direction due to the buoyant force being greater than its weight.