A 2-m diameter log having a specific gravity of 0.9 is 10-m long. It is acted upon by water pressure as shown, compute the net vertical reaction at point A?
90.01 KN
Well, well, well, looks like we have a log in a watery situation here! I hope it's not feeling too "logged down".
To find the net vertical reaction at point A, we need to consider the water pressure acting on the log.
First, let's find the volume of the log using its diameter and length. The volume of a cylinder is given by the formula V = πr^2h, where r is the radius (half of the diameter) and h is the height.
In this case, the radius (r) is 1 meter (half of the diameter) and the height (h) is 10 meters.
So, V = π × 1^2 × 10 = 10π cubic meters.
Now, let's calculate the weight of the log. The weight of an object is equal to its volume multiplied by the specific gravity (SG) and the density of water (ρ).
The density of water is approximately 1000 kg/m^3. Since the specific gravity is given as 0.9, this means it is 90% of the density of water.
So, the weight of the log = V × SG × ρ = 10π × 0.9 × 1000 = 9000π N.
Finally, we can find the net vertical reaction at point A, which is equal to the weight of the log minus the buoyant force.
Now, the buoyant force is equal to the weight of the displaced water. The weight of the displaced water is equal to the volume of the log (V) multiplied by the density of water (ρ) times the gravitational acceleration (g).
So, the buoyant force = V × ρ × g = 10π × 1000 × 9.8 = 98,000π N.
The net vertical reaction at point A is then equal to the weight of the log minus the buoyant force.
Net vertical reaction at point A = 9000π - 98000π = -89000π N.
Therefore, the net vertical reaction at point A is approximately -89,000π N.
And there you have it! The log is feeling a negative reaction at point A. I hope this water pressure situation doesn't dampen its day too much!
To determine the net vertical reaction at point A, we need to consider the forces acting on the log.
1. Weight of the log: The weight of the log can be calculated using the formula:
Weight = Volume × Density × Gravity
The volume of the log can be calculated using the formula for the volume of a cylinder:
Volume = π × Radius^2 × Height
Substituting the given values, we have:
Volume = π × (1m)^2 × 10m = 10π m^3
The density is given as a specific gravity of 0.9, which means it is 0.9 times the density of water.
Density = 0.9 × Density of Water
The density of water is approximately 1000 kg/m^3.
Substituting the values, we have:
Density = 0.9 × 1000 kg/m^3 = 900 kg/m^3
Now, we can calculate the weight of the log:
Weight = 10π m^3 × 900 kg/m^3 × 9.8 m/s^2
2. Buoyant force: The buoyant force acting on the log is equal to the weight of the water displaced by the submerged portion of the log.
Buoyant Force = Volume of Submerged Portion × Density of Water × Gravity
The volume of the submerged portion can be calculated using the formula for the volume of a cylinder:
Volume of Submerged Portion = π × Radius^2 × Height of Submerged Portion
The height of the submerged portion can be calculated using the given information:
Height of Submerged Portion = Diameter of Log - Height above the water level
= 2m - 1m
= 1m
Substituting the values, we have:
Volume of Submerged Portion = π × (1m)^2 × 1m = π m^3
Now, we can calculate the buoyant force:
Buoyant Force = π m^3 × 1000 kg/m^3 × 9.8 m/s^2
Now, to determine the net vertical reaction at point A, we can subtract the buoyant force from the weight of the log:
Net Vertical Reaction at Point A = Weight - Buoyant Force
To compute the net vertical reaction at point A, we need to consider the forces acting on the log.
First, let's determine the weight of the log:
The weight of the log can be calculated using the formula:
Weight = Volume x Density x g,
where g is the acceleration due to gravity, and the density is the specific gravity multiplied by the density of water.
The volume of the log is given by the formula:
Volume = πr^2h,
where r is the radius of the log (half of the diameter), and h is the length of the log.
In this case, the diameter of the log is 2 m, so the radius is 1 m. The length of the log is 10 m.
So, the volume of the log is:
Volume = π x (1 m)^2 x 10 m = 10π m^3.
Since the specific gravity is 0.9, the density of the log is:
Density = 0.9 x density of water.
The density of water is approximately 1000 kg/m^3.
Therefore, the density of the log is:
Density = 0.9 x 1000 kg/m^3 = 900 kg/m^3.
Now, we can calculate the weight of the log:
Weight = Volume x Density x g = 10π m^3 x 900 kg/m^3 x 9.8 m/s^2.
The value of π (pi) is approximately 3.14159, so substituting the values:
Weight = 10π m^3 x 900 kg/m^3 x 9.8 m/s^2.
After calculating this expression, the weight of the log is given in Newtons.
Next, let's determine the water pressure acting on the log:
The water pressure is equal to the weight of the water displaced by the log.
The weight of the water displaced is equal to the weight of an equivalent volume of water. The volume of water displaced is equal to the volume of the log.
So, the weight of the water displaced is:
Weight of water displaced = Volume x Density x g,
where the density is the density of water (1000 kg/m^3).
Therefore, the weight of the water displaced is:
Weight of water displaced = 10π m^3 x 1000 kg/m^3 x 9.8 m/s^2.
After calculating this expression, the weight of the water displaced is given in Newtons.
Finally, the net vertical reaction at point A can be computed by subtracting the weight of the water displaced from the weight of the log. The net vertical reaction is the force exerted by the log on point A.
Net Vertical Reaction at A = Weight of the log - Weight of water displaced.
After performing the subtraction, the net vertical reaction at point A will be calculated, and the value will be in Newtons.