Search: The barrels of the 16 in guns (bore diameter = 16 in. = 41 cm) on the World War II battleship U.S.S. Massachusetts were each 15 m long. The shells each had a mass of 1050 kg and were fired with sufficient explosive force to provide them with a muzzle velocity of 730 m/s. Use the work-energy principle to determine the explosive force (assumed to be a constant) that was applied to the shell within the barrel of the gun. Express your answer in both (a) newtons and (b) in pounds
(avg. force)*(barrel length) = (kinetic energy at muzzle velocity)
F = (1/2)M V^2/L
The barrel diameter does not enter into the calculation. To convert Newtons to pounds of force, use
1 N = 0.225 lb
37500 N
166666.6667 LBS
To determine the explosive force applied to the shell within the gun barrel, we can use the work-energy principle. The work done on an object is equal to the change in its kinetic energy. In this case, the work done on the shell by the explosive force is equal to the change in its kinetic energy as it accelerates from rest to a muzzle velocity of 730 m/s.
The work done on the shell can be calculated using the formula:
Work = (1/2) * mass * velocity^2
Given:
Bore diameter = 16 in. = 41 cm
Length of each barrel = 15 m
Mass of the shell = 1050 kg
Muzzle velocity = 730 m/s
First, let's convert the bore diameter from centimeters to meters:
Bore diameter = 41 cm = 41/100 = 0.41 m
Next, calculate the cross-sectional area of the barrel using the formula for the area of a circle:
Area = π * (radius)^2
radius = bore diameter / 2 = 0.41 / 2 = 0.205 m
Area = π * (0.205)^2 = 0.1324 m^2
Now, calculate the force exerted on the shell within the gun barrel using the formula:
Force = Pressure * Area
To find the pressure, we can apply the work-energy principle with the assumption that no energy is lost due to friction or other factors:
Work = Force * Distance
Where the work done is equal to the change in kinetic energy:
(1/2) * mass * velocity^2 = Force * Distance
Rearranging the equation, we can solve for the Force:
Force = (1/2) * mass * velocity^2 / Distance
Substituting the given values:
Force = (1/2) * 1050 kg * (730 m/s)^2 / 15 m
Now, we can calculate the force:
Force = (1/2) * 1050 * (730^2) / 15
= 134,679 Newtons (to three significant figures)
To convert Newtons to pounds, we use the conversion factor:
1 Newton = 0.224808943 pounds
So, we multiply the force in Newtons by the conversion factor to get the force in pounds:
Force in pounds = 134,679 Newtons * 0.224808943
≈ 30,259 pounds (to three significant figures)
Therefore, the explosive force applied to the shell within the gun barrel was approximately 134,679 Newtons or 30,259 pounds.
To determine the explosive force applied to the shell within the barrel of the gun, we can use the work-energy principle. The work-energy principle states that the work done on an object is equal to the change in its kinetic energy.
The work done on the shell can be calculated as the product of the force and the distance over which the force is applied. In this case, the distance is equal to the length of the barrel, which is 15 m.
First, let's calculate the initial kinetic energy of the shell using its mass and muzzle velocity. The kinetic energy (K) is given by the equation:
K = (1/2) * m * v^2
where m is the mass of the shell and v is its muzzle velocity.
Given that the mass of the shell (m) is 1050 kg and the muzzle velocity (v) is 730 m/s, we can calculate the initial kinetic energy:
K = (1/2) * 1050 kg * (730 m/s)^2
Now we have the initial kinetic energy of the shell.
Next, we need to find the work done on the shell. According to the work-energy principle, this work is equal to the change in kinetic energy.
Since the shell starts from rest inside the barrel, its initial kinetic energy is zero. Therefore, the work done equals the final kinetic energy of the shell.
To find the work done, we need to know the final velocity of the shell. However, this information is not provided in the question. Hence, we need to make an assumption in order to proceed with the calculation.
For simplicity, let's assume that the final velocity of the shell is also 730 m/s (as stated in the question). This assumption means that the shell is not affected by air resistance, which is a reasonable approximation for this calculation.
Now, the final kinetic energy of the shell is calculated using the same equation as before:
K = (1/2) * 1050 kg * (730 m/s)^2
The work done on the shell is then equal to this final kinetic energy.
Now, let's calculate the work done:
Work = (1/2) * 1050 kg * (730 m/s)^2
This work represents the force applied to the shell multiplied by the distance traveled by the shell through the barrel.
Finally, to find the explosive force, we divide the work done by the distance traveled:
Explosive force = Work / Distance
Explosive force = [(1/2) * 1050 kg * (730 m/s)^2] / 15 m
Now, to express the answer in newtons, we can simply evaluate the expression. To express the answer in pounds, we will need to convert newtons to pounds.
1 pound (lb) is equal to 4.448 newtons (N). Hence, to convert the answer from newtons to pounds, we divide it by 4.448.
So, to summarize:
(a) To find the explosive force in newtons, evaluate [(1/2) * 1050 kg * (730 m/s)^2] / 15 m.
(b) To find the explosive force in pounds, evaluate [(1/2) * 1050 kg * (730 m/s)^2] / 15 m / 4.448.