A spherical vessel used for deep-sea exploration has a radius of 1.50 m and a mass of 1.20 × 104 kg. To dive, the vessel takes on mass in the form of seawater. Determine the mass the vessel must take on if it is to descend at a constant speed of 1.20 m/s, when the resistive force on it is 1100 N in the upward direction. The density of seawater is equal to 1.03 × 103 kg/m3.

Σ𝐹𝑦=𝑚𝑎𝑦=0−(1.20×104𝑘𝑔+𝑚)𝑔+𝑝𝑤𝑔𝑉+1100𝑁=0

Where 𝑚 is the mass of the added water and 𝑣 is the sphere’s volume. 1.20×104𝑘𝑔+𝑚=1.03×103[43𝜋(1.50)3]+1100𝑁9.8𝑚/𝑠2=2.67×103𝑘𝑔

To determine the mass the vessel must take on, we can use the concept of buoyancy and gravitational forces.

1. Start by finding the volume of the vessel:
- The volume of a sphere can be calculated using the formula V = (4/3)πr^3, where r is the radius.
- In this case, the radius is 1.50 m, so the volume of the vessel is (4/3)π(1.50)^3.

2. Convert the volume to mass:
- Since the vessel takes on mass in the form of seawater, we need to find the mass of the seawater that occupies the vessel's volume.
- The density of seawater is 1.03 × 10^3 kg/m^3.
- Multiply the volume of the vessel by the density of seawater to find its mass.

3. Calculate the net force acting on the vessel:
- The net force acting on the vessel is the difference between the buoyant force and the resistive force.
- The buoyant force is equal to the weight of the seawater displaced by the vessel, which can be calculated using the formula F_b = ρgV, where ρ is the density of seawater, g is the acceleration due to gravity (approximately 9.8 m/s^2), and V is the volume of the vessel.
- The resistive force is given as 1100 N in the upward direction.

4. Set up an equation to find the mass of the vessel:
- Since the vessel is descending at a constant speed, the net force must be balanced (zero net force).
- The equation is: F_net = F_b - F_resistive = 0
- Rearrange the equation to solve for the mass of the vessel (m_vessel).

5. Solve the equation for the mass of the vessel:
- Substitute the expressions for the buoyant force and resistive force into the equation.
- Solve for m_vessel.

By following these steps, you can determine the mass the vessel must take on to descend at a constant speed of 1.20 m/s.

To solve this problem, we need to use the principles of buoyancy and the concept of net force.

First, let's determine the buoyant force acting on the vessel. The buoyant force is equal to the weight of the seawater displaced by the vessel. The weight of the displaced seawater can be calculated using the volume of the vessel and the density of seawater.

The volume of a sphere can be calculated using the formula V = (4/3) * π * r^3, where r is the radius of the sphere.

For this vessel, the volume is V = (4/3) * π * (1.50 m)^3.

Next, we can calculate the weight of the displaced seawater using the formula weight = volume * density * gravity, where gravity is approximately 9.8 m/s^2.

weight = (4/3) * π * (1.50 m)^3 * (1.03 × 10^3 kg/m^3) * 9.8 m/s^2.

Now, let's determine the net force acting on the vessel. The net force is the difference between the buoyant force and the resistive force.

net force = buoyant force - resistive force.

In this case, the resistive force is given as 1100 N. The buoyant force is equal to the weight of the displaced seawater.

net force = weight - 1100 N.

Since the vessel is descending at a constant speed, the net force must be zero. Therefore,

0 = weight - 1100 N.

Now, we can solve for the weight of the seawater (which is the same as the mass the vessel must take on).

weight = 1100 N.

Substituting the previously calculated weight into the equation:

1100 N = (4/3) * π * (1.50 m)^3 * (1.03 × 10^3 kg/m^3) * 9.8 m/s^2.

Now we can solve this equation to find the weight (and mass) of the seawater that the vessel must take on.

By performing the calculations, the mass of the seawater that the vessel must take on is approximately 1.15 × 10^4 kg.