A bathysphere used for deep sea exploration has a radius of 1.60 m and a mass of 1.20 104 kg. In order to dive, the sphere takes on mass in the form of sea water. Determine the mass the bathysphere must take on so that it can descend at a constant speed of 1.00 m/s when the resistive force on it is 1179 N upward. The density of sea water is 1.03 103 kg/m3.
To determine the mass the bathysphere must take on, we can use the following steps:
Step 1: Calculate the gravitational force acting on the bathysphere:
The gravitational force can be calculated using the formula: F_gravity = m * g,
where F_gravity is the gravitational force, m is the mass, and g is the acceleration due to gravity.
Given:
Mass of the bathysphere, m = 1.20 x 10^4 kg
Acceleration due to gravity, g = 9.8 m/s^2
F_gravity = 1.20 x 10^4 kg * 9.8 m/s^2
F_gravity = 1.176 x 10^5 N
Step 2: Calculate the buoyant force acting on the bathysphere:
The buoyant force can be calculated using the formula: F_buoyant = V * ρ * g,
where F_buoyant is the buoyant force, V is the volume displaced by the bathysphere, ρ is the density of the fluid (sea water), and g is the acceleration due to gravity.
Given:
Radius of the bathysphere, r = 1.60 m
Density of sea water, ρ_water = 1.03 x 10^3 kg/m^3
Acceleration due to gravity, g = 9.8 m/s^2
The volume displaced by the bathysphere is calculated using the formula: V = (4/3) * π * r^3.
V = (4/3) * 3.14 * (1.60 m)^3
V ≈ 17.093 m^3
F_buoyant = V * ρ_water * g
F_buoyant = 17.093 m^3 * 1.03 x 10^3 kg/m^3 * 9.8 m/s^2
F_buoyant ≈ 167,604 N
Step 3: Determine the net force acting on the bathysphere:
The net force acting on the bathysphere can be calculated by subtracting the resistive force from the gravitational force:
Net Force = F_gravity - F_buoyant - Resistive Force
Given:
Resistive force, F_resistive = 1179 N
Net Force = 1.176 x 10^5 N - 1.676 x 10^5 N - 1179 N
Net Force = -128,794 N
Step 4: Determine the mass of the additional sea water needed:
The net force acting on the bathysphere is the product of the additional mass (m_water) and the acceleration due to gravity (g):
Net Force = m_water * g
-128,794 N = m_water * 9.8 m/s^2
m_water = -128,794 N / 9.8 m/s^2
m_water ≈ -13,157 kg (negative sign indicates it acts opposite to the downward direction)
The negative value indicates that the bathysphere already has excess mass and does not require additional sea water to descend at a constant speed of 1 m/s.
To find the mass the bathysphere must take on in order to descend at a constant speed, we can use the concept of buoyancy. Buoyancy is the upward force exerted on an object immersed in a fluid that is equal to the weight of the fluid displaced by the object.
In this case, the bathysphere is filled with sea water, so the buoyant force acting on it can be calculated using the formula:
Buoyant force = Volume of displaced fluid * Density of fluid * Acceleration due to gravity
The volume of displaced fluid is equal to the volume of the bathysphere, which can be calculated using the formula:
Volume = (4/3) * π * r^3
Let's start by finding the volume of the bathysphere using its given radius:
Volume = (4/3) * π * (1.60 m)^3
Volume ≈ 17.06 m^3
Next, we can calculate the buoyant force:
Buoyant force = Volume * Density of sea water * Acceleration due to gravity
Buoyant force = 17.06 m^3 * (1.03 * 10^3 kg/m^3) * (9.81 m/s^2)
Buoyant force ≈ 170,535.25 N
Since the resistive force acting on the bathysphere is upward, we need to subtract it from the buoyant force to determine the net force acting on the bathysphere:
Net force = Buoyant force - Resistive force
Net force = 170,535.25 N - 1179 N
Net force ≈ 169,356.25 N
Now, we can use the formula for weight to find the mass the bathysphere must take on:
Weight = mass * Acceleration due to gravity
Rearranging the formula, we can solve for mass:
mass = Weight / Acceleration due to gravity
Let's plug in the values:
mass = 169,356.25 N / 9.81 m/s^2
mass ≈ 17,263.33 kg
Therefore, the bathysphere must take on a mass of approximately 17,263.33 kg in order to descend at a constant speed of 1.00 m/s when the resistive force on it is 1179 N upward.