Answer :
Answer:
Explanation:
To calculate the righting arm (GZ) at a given angle of heel (in this case, 45 degrees), you need the following information:
The center of gravity (G) and center of buoyancy (B) of the vessel.
Metacentric height (GM).
Length of the ship's beam (B).
The righting arm (GZ) can be found using the formula:
=
⋅
sin
(
)
GZ=GM⋅sin(θ)
where:
GM is the metacentric height.
θ is the angle of heel in radians.
However, if the information about the metacentric height is not available, you may need to use more detailed hydrostatic calculations or ship stability data, often found in the ship's stability book or calculated using software designed for naval architecture.
Example Calculation
Assume the following data:
Metacentric height (GM): 2 meters
Angle of heel (
θ): 45 degrees (which is
/
4
π/4 radians)
First, convert the angle from degrees to radians:
4
5
∘
×
180
=
4
radians
45
∘
×
180
π
=
4
π
radians
Then, calculate the sine of the angle:
sin
(
4
5
∘
)
=
sin
(
4
)
=
2
2
≈
0.707
sin(45
∘
)=sin(
4
π
)=
2
2
≈0.707
Now, apply the formula:
=
⋅
sin
(
)
GZ=GM⋅sin(θ)
=
2
meters
⋅
0.707
GZ=2 meters⋅0.707
≈
1.414
meters
GZ≈1.414 meters
So, the righting arm (GZ) at 45 degrees of heel is approximately 1.414 meters.
Important Notes
The actual GZ calculation can be more complex for real vessels due to the shape of the hull and the distribution of weight.
In practical applications, naval architects use detailed stability curves and hydrostatic tables to determine the GZ at various angles of heel.
If you have access to specific hydrostatic data for the vessel, you should use that for more accurate calculations.
