A tug is capable of pulling a ship with a force of 100 KN. If two such tugs are pulling on one ship, they can produce any force ranging from a minimum of 0 KN to maximum of 200 KN. Give a detailed explanation of how this is possible. Use diagrams to support your result.
A tug is capable of pulling a ship with a force of 100 KN. If two such tugs are pulling on one ship, they can produce any force ranging from a minimum of 0 KN to maximum of 200 KN. Give a detailed explanation of how this is possible. Use diagrams to support your result.
To understand how two tugs can produce a force ranging from 0 KN to 200 KN while pulling a ship, we need to consider the concept of vector addition.
Vector addition involves combining two or more vectors to obtain their resultant vector. In our scenario, each tug exerts a force on the ship, which can be represented as a vector. These force vectors can be added together to determine the total force on the ship.
Let's denote the force exerted by each tug as F1 and F2, and the resultant force as R. Initially, we know that F1 = F2 = 100 KN, as each tug is capable of producing a force of 100 KN.
To understand the range of forces that can be produced, we need to consider the possible combinations of F1 and F2. Since both tugs can exert forces in the same direction or in opposite directions, we have four possible combinations:
1. F1 and F2 act in the same direction:
In this case, the force exerted by each tug adds up to produce the maximum resultant force. Thus, R = F1 + F2 = 100 KN + 100 KN = 200 KN. This is the maximum force that the two tugs can generate together.
2. F1 and F2 act in opposite directions:
Here, the force vectors cancel each other out, leading to a minimum resultant force. R = F1 - F2 = 100 KN - 100 KN = 0 KN. The two forces are equal in magnitude but opposite in direction, resulting in a net force of zero.
3. F1 and F2 act perpendicular to each other:
If the two tugs are pulling the ship perpendicular to each other, the resultant force can be determined using vector addition. In this case, the magnitude of R can be calculated using the Pythagorean theorem, where R^2 = F1^2 + F2^2. Since F1 = F2 = 100 KN, we get R^2 = 100 KN^2 + 100 KN^2 = 20000 KN^2.
By taking the square root of both sides, we find R ≈ 141.4 KN. Therefore, when the tugs are pulling perpendicular to each other, the resultant force is approximately 141.4 KN.
4. F1 and F2 act at an angle between 0 and 90 degrees:
In this case, again, we can determine the resultant force using vector addition. The magnitude of R can be calculated using the formula R^2 = F1^2 + F2^2 + 2F1F2cosӨ, where Ө is the angle between F1 and F2. Since F1 = F2 = 100 KN, we can simplify the equation to R^2 = 200 KN^2 + 2(100 KN)(100 KN)cosӨ = 20000 KN^2 + 20000 KN^2 cosӨ. By taking the square root of both sides, we obtain R = √(20000 KN^2 + 20000 KN^2 cosӨ).
Using a diagram to visualize the scenario, draw two force vectors representing F1 and F2. The angle between them can be adjusted to determine the resultant force magnitude R for any given angle between 0 and 90 degrees.
In summary, by manipulating the direction and angle of the forces exerted by each tug, we can achieve a range of resultant forces from 0 KN to a maximum of 200 KN when two tugs are pulling on a ship. The resultant force can be calculated using vector addition, considering the given magnitudes and angles.