A bathysphere used for deep sea exploration has a radius of 1.50 m and a mass of 1.20 × 104 kg. In order to dive, the sphere takes on mass in the form of sea water. a) Determine the mass the bathysphere must take on so that it can descend at a constant speed of 1.20 m/s when the resistive force on it is 1 100 N upward. The density of sea water is 1.03 × 103 kg/m3.
volume = (4/3) pi r^3 = (4/3) pi (1.5)^3
= 14.1 m^3
buoyant force = rho g V = 1.03*10^3 * 9.8 * 14.1 = 142,700 N
weight down = 1.2*10^4*9.8 = 117,600 N
net positive buoyancy = 142700-117600 = 25,100 N
Force down - 25,100 N = 1,100 N
Force down = 26,200 N
mass = 26,200/9.8 = 2673 kg
Well, water you waiting for? Let's dive right into it!
To determine the mass the bathysphere must take on, we need to consider the forces at play. The gravitational force pulling the bathysphere downward should be equal to the sum of the buoyant force and the resistive force.
First, let's calculate the volume of the bathysphere using its radius. The volume V can be found using the formula V = (4/3)πr^3, where r is the radius. Plugging in the values, we have:
V = (4/3)π(1.50 m)^3
V ≈ 14.1 m^3
Since the density of seawater is given as 1.03 × 10^3 kg/m^3, we can calculate the mass of seawater needed to achieve neutral buoyancy. The mass m can be found using the formula m = ρV, where ρ is the density. Plugging in the values, we have:
m = (1.03 × 10^3 kg/m^3)(14.1 m^3)
m ≈ 1.45 × 10^4 kg
To descend at a constant speed of 1.20 m/s, we need to take into account the upward resistive force of 1,100 N. Since the forces are balanced at constant speed, we have:
Weight of the bathysphere (mg) = Buoyant force (ρVg) + Resistive force
Solving for the mass m, we get:
m = (Resistive force - Buoyant force) / g
Plugging in the values, we have:
m = (1100 N - 14.1 m^3 * 1.03 × 10^3 kg/m^3 * 9.8 m/s^2) / 9.8 m/s^2
m ≈ 128.6 kg
So, the bathysphere must take on approximately 128.6 kg of mass in the form of seawater to descend at a constant speed of 1.20 m/s with an upward resistive force of 1,100 N. Just keep in mind, this is not a "weighty" decision!
To determine the mass the bathysphere must take on, we can use the equation:
Weight of the bathysphere - Buoyant force - Resistive force = Net force
The weight of the bathysphere is given by:
Weight = mass * gravitational acceleration
Weight = (1.20 × 10^4 kg) * (9.8 m/s^2)
Weight = 1.18 × 10^5 N
The buoyant force is given by:
Buoyant force = Volume of displaced water * density of water * gravitational acceleration
Since the bathysphere takes on the mass in the form of sea water, the volume of displaced water is equal to the volume of the bathysphere. The volume of the bathysphere can be calculated using the formula:
Volume = (4/3) * π * radius^3
Volume = (4/3) * π * (1.50 m)^3
Volume = 14.13 m^3
Buoyant force = (14.13 m^3) * (1.03 × 10^3 kg/m^3) * (9.8 m/s^2)
Buoyant force = 1.40 × 10^5 N
The net force is given by:
Net force = Weight - Buoyant force - Resistive force
Using the given values:
Net force = (1.18 × 10^5 N) - (1.40 × 10^5 N) - (1.10 × 10^3 N)
Net force = -2.20 × 10^4 N
Since the bathysphere is descending at a constant speed, the net force must be zero. Therefore, the mass the bathysphere must take on is:
Mass of water = Net force / gravitational acceleration
Mass of water= (-2.20 × 10^4 N) / (9.8 m/s^2)
Mass of water= -2.24 × 10^3 kg
Since mass cannot be negative, we ignore the negative sign. Therefore, the mass the bathysphere must take on is approximately 2.24 × 10^3 kg.
To determine the mass the bathysphere must take on, we need to consider the forces acting on it.
The downward force acting on the bathysphere is the weight of the sphere itself, given by the formula:
Weight = Mass * Acceleration due to gravity
The upward force acting on the bathysphere is the resistive force, which is 1,100 N.
The net force on the bathysphere is the difference between the upward and downward forces.
Net Force = Upward Force - Downward Force
Since the bathysphere is descending at a constant speed, the net force is zero.
Therefore, we can set up the following equation:
Net Force = Upward Force - Downward Force = 0
Rearranging the equation, we get:
Downward Force = Upward Force
Now, let's calculate the downward force:
Downward Force = Weight of the bathysphere
Weight of the bathysphere = Mass of the bathysphere * Acceleration due to gravity
Given:
Radius of the bathysphere = 1.50 m
Mass of the bathysphere = 1.20 × 10^4 kg
Acceleration due to gravity is approximately 9.8 m/s^2.
Weight of the bathysphere = (1.20 × 10^4 kg) * (9.8 m/s^2)
Next, let's determine the volume of sea water the bathysphere needs to take on. The bathysphere is essentially displacing a volume of water equal to its own volume in order to become neutrally buoyant.
The volume of a sphere is given by the formula:
Volume = (4/3) * π * radius^3
Given:
Radius of the bathysphere = 1.50 m
Volume of the bathysphere = (4/3) * 3.14 * (1.50 m)^3
Now, let's calculate the mass of the sea water the bathysphere needs to take on:
Mass = Density * Volume
Given:
Density of sea water = 1.03 × 10^3 kg/m^3
Mass of sea water = (1.03 × 10^3 kg/m^3) * Volume of the bathysphere
Finally, let's substitute the values into the equation:
Downward Force = Upward Force
Weight of the bathysphere = Resistive force
Mass of the bathysphere * Acceleration due to gravity = Resistive force
Mass of the bathysphere = Resistive force / Acceleration due to gravity
Mass of sea water = (1.03 × 10^3 kg/m^3) * Volume of the bathysphere
Therefore, to calculate the mass the bathysphere must take on so it can descend at a constant speed of 1.20 m/s when the resistive force on it is 1,100 N, we need to calculate the mass of the bathysphere and the mass of the sea water separately and add them together.