An undersea research chamber is spherical with an external diameter of 5.30m . The mass of the chamber, when occupied, is 78500kg . It is anchored to the sea bottom by a cable. The density of seawater is 1025 kg/m3.

a) What is the buoyant force on the chamber?
b) What is the tension in the cable?

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To answer these questions, we need to understand the concept of buoyancy and the relationship between buoyant force, weight, and density.

Buoyant force is the upward force exerted on an object immersed in a fluid (in this case, seawater). It is equal to the weight of the fluid displaced by the object. The weight of an object can be calculated using its mass and the acceleration due to gravity.

Let's proceed with the calculations:

a) What is the buoyant force on the chamber?

To calculate the buoyant force, we first need to determine the volume of the chamber. Since the chamber is spherical, we can use the formula for the volume of a sphere:

Volume = (4/3) * π * r^3

Given that the external diameter of the chamber is 5.30 m, the radius (r) can be calculated by dividing the diameter by 2:

r = 5.30 m / 2 = 2.65 m

Now we can substitute the value of the radius into the volume formula:

Volume = (4/3) * π * (2.65 m)^3

Next, we need to calculate the weight of the chamber when occupied. Weight is calculated by multiplying the mass of an object by the acceleration due to gravity (9.8 m/s^2):

Weight = mass * gravity

Weight = 78500 kg * 9.8 m/s^2

Now, we can calculate the buoyant force. The buoyant force is equal to the weight of the fluid displaced by the chamber, which is equal to the weight of the chamber itself:

Buoyant force = Weight of the chamber = 78500 kg * 9.8 m/s^2

b) What is the tension in the cable?

The tension in the cable can be calculated by considering the equilibrium of forces acting on the chamber. There are two forces acting on the chamber: the buoyant force (upward) and the force due to its weight (downward). The tension in the cable balances these two forces.

Since the chamber is anchored to the sea bottom, the tension in the cable is equal to the sum of the buoyant force and the weight of the chamber:

Tension in the cable = Buoyant force + Weight of the chamber

Let's substitute the calculated values:

Tension in the cable = (78500 kg * 9.8 m/s^2) + (78500 kg * 9.8 m/s^2)

Simplifying the equation:

Tension in the cable = 2 * (78500 kg * 9.8 m/s^2)

Now we can calculate the exact value by substituting the appropriate values and performing the multiplication.