A thin spherical shell of mass 0.400 kg and diameter 0.190 m is filled with alcohol (ρ = 806 kg/m3). It is then released from rest on the bottom of a pool of water. Find the acceleration of the alcohol filled shell as it rises toward the surface of the water.
To find the acceleration of the alcohol-filled shell as it rises toward the surface of the water, we can use the concept of buoyancy and the force equations involved.
First, let's determine the buoyant force acting on the shell. The buoyant force is the upward force exerted on an object submerged in a fluid and can be calculated using the formula:
Buoyant Force = Weight of the fluid displaced
In this case, the fluid displaced is alcohol, so we need to calculate the weight of the alcohol displaced by the shell. The volume of a sphere can be calculated by the formula:
Volume of a Sphere = (4/3) * π * r^3
Given that the diameter of the shell is 0.190 m (which means the radius is 0.095 m), the volume of the shell is:
Volume = (4/3) * π * (0.095)^3 = 0.000420 m^3
Next, we need to calculate the weight of the alcohol displaced. The weight is the mass of the alcohol times the acceleration due to gravity. We can find the mass of the alcohol using its density:
Mass = Density * Volume = 806 kg/m^3 * 0.000420 m^3 = 0.338 kg
Weight of the alcohol displaced = Mass * Acceleration due to gravity = 0.338 kg * 9.8 m/s^2 = 3.3164 N
Since the shell is released from rest on the bottom of the pool, the only force acting on it initially is its weight, which can be calculated by:
Weight of the shell = Mass * Acceleration due to gravity = 0.400 kg * 9.8 m/s^2 = 3.92 N
Now, let's calculate the net force acting on the shell as it rises toward the surface of the water:
Net force = Buoyant force - Weight of the shell
Net force = 3.3164 N - 3.92 N = -0.6036 N
The negative sign indicates that the net force is acting downward. According to Newton's second law of motion, the net force is equal to the mass of the object multiplied by its acceleration. So we can rearrange the equation to solve for acceleration:
Net force = Mass * Acceleration
-0.6036 N = 0.400 kg * Acceleration
Acceleration = -0.6036 N / 0.400 kg = -1.509 m/s^2
Therefore, the acceleration of the alcohol-filled shell as it rises toward the surface of the water is approximately -1.509 m/s^2.