A cylinder shaped buoy is stationary out on a lake. The height of the buoy is 1 m and the diameter is 30 cm and it‘s mass is 50 kg. Now a whale swims under the buoy and hit it. The buoy raises up and begins to oscillate. a) Find the total force on the buoy if the displacement from the start is x and make a derivative equation that describes the oscillation. b) If the initial displacement of the buoy was 10 cm, find the amplitude, angular frequency ω and sketch x(t).
To solve this problem, we will need to consider the forces acting on the buoy, namely the buoyant force and the force due to the whale's impact.
a) Finding the total force on the buoy:
The buoyant force is equal to the weight of the displaced water, which can be calculated using Archimedes' principle. The buoy's weight can be calculated using its mass and the acceleration due to gravity (9.8 m/s^2).
1. Find the volume of the cylinder-shaped buoy:
The buoy has a height of 1 m and a diameter of 30 cm. Using the formula for the volume of a cylinder (V = π * r^2 * h), where r is the radius and h is the height, we can calculate the volume:
V = π * (15 cm)^2 * 100 cm = π * (0.15 m)^2 * 1 m = 0.0707 m^3.
2. Find the weight of the buoy:
The weight of an object is given by the formula W = m * g, where m is the mass and g is the acceleration due to gravity. In this case, the buoy's mass is given as 50 kg:
W = 50 kg * 9.8 m/s^2 = 490 N.
3. Find the buoyant force:
The buoyant force is equal to the weight of the displaced water. The density of water is approximately 1000 kg/m^3. The volume of water displaced is the same as the volume of the buoy (0.0707 m^3):
Buoyant Force = Density of Water * Volume of Water * g
Buoyant Force = 1000 kg/m^3 * 0.0707 m^3 * 9.8 m/s^2 = 689.74 N.
4. Find the total force on the buoy:
The total force on the buoy is the sum of the buoyant force and the force due to the whale's impact:
Total Force = Buoyant Force + Force due to whale's impact.
b) Making a derivative equation that describes the oscillation:
To describe the oscillation, we can consider the forces acting on the buoy and use Newton's second law of motion, F = ma, where F is the net force, m is the mass of the object, and a is the acceleration:
Total Force = ma
Since the total force on the buoy is in the opposite direction to the displacement x, we can substitute F with -kx, where k is the spring constant that relates the displacement and the force:
-kx = ma
To describe the motion of the buoy, we can obtain the second derivative of x with respect to time, which will provide an equation that describes the oscillation:
-kx = m(d^2x/dt^2)
b) Finding the amplitude, angular frequency, and sketching x(t):
First, we need to determine the spring constant k. We can use Hooke's law, which states that the force exerted by a spring is directly proportional to the displacement from its rest position:
F = -kx
Using the given information, we have the total force on the buoy as -kx. Comparing this with Hooke's law, we find that k is equal to the negative of the total force divided by x.
Once we have the spring constant, we can determine the angular frequency ω, which is related to k and the mass m using the formula:
ω = √(k/m)
With the amplitude, angular frequency, and equation for oscillation, we can sketch x(t) by using the equation x(t) = Acos(ωt), where A is the amplitude and t is time.
a) To find the total force on the buoy, we need to consider two forces: the buoyant force and the gravitational force.
1. Buoyant force (F_b):
The buoyant force is equal to the weight of the water displaced by the buoy. We can calculate it using Archimedes' principle:
F_b = ρ * V * g
where:
ρ is the density of the water (assumed to be constant),
V is the volume of the cylinder-shaped buoy,
g is the acceleration due to gravity.
The volume of a cylinder is given by:
V = π * r^2 * h
where:
r is the radius of the buoy (half of the diameter),
h is the height of the buoy.
Given:
diameter = 30 cm = 0.3 m
radius = 0.3 m / 2 = 0.15 m
height = 1 m
mass = 50 kg
The volume can be calculated as:
V = π * (0.15 m)^2 * 1 m = 0.0707 m^3
Now we can calculate the buoyant force:
F_b = ρ * V * g
2. Gravitational force (F_g):
The gravitational force acting on the buoy is equal to its weight:
F_g = m * g
Given:
mass = 50 kg
g ≈ 9.8 m/s^2
Now we can calculate the total force on the buoy:
F_total = F_b - F_g
b) To find the amplitude, angular frequency (ω), and sketch x(t), we need to derive the equation of motion for the buoy. We'll consider simple harmonic motion, assuming no damping or external forces other than the gravitational and buoyant forces.
The equation of motion for simple harmonic motion is:
m * d^2x/dt^2 = -k * x
where:
m is the mass of the buoy,
x is the displacement of the buoy from its equilibrium position,
k is the spring constant related to the buoyant and gravitational forces (k = F_total / x).
To find the amplitude (A) and angular frequency (ω), we can rewrite the equation of motion as:
d^2x/dt^2 + (k/m) * x = 0
Comparing this equation to the general equation for simple harmonic motion:
d^2x/dt^2 + ω^2 * x = 0
We can find:
ω^2 = k/m
A = x_max
where:
ω is the angular frequency,
A is the amplitude,
x_max is the maximum displacement (initial displacement in this case).
To sketch x(t), we need the equation:
x(t) = A * cos(ω * t)
Let's plug in the provided values to calculate the total force and find the amplitude, angular frequency, and sketch x(t).