Three forces of 5N, 8N, and 10N act from the same corner of a rectangular solid along its three edges.
a) Calculate the magnitude of the equilibrant of these three forces.
b) Determine the angle that the equilibrant makes with each of the three forces.
a) The magnitude of the equilibrant of these three forces is 23N.
b) The angle that the equilibrant makes with each of the three forces is 120°.
To solve this problem, we need to first understand what an equilibrant is. An equilibrant is a single force that can bring a system of forces into equilibrium or balance. In other words, it is a force that can counteract the combined effect of the given forces and make the net force on the object zero.
a) To find the magnitude of the equilibrant, we can use the concept of vector addition. The equilibrant is equal in magnitude but opposite in direction to the resultant of the three forces.
Let's label the forces as follows:
- Force 1: 5N
- Force 2: 8N
- Force 3: 10N
To find the resultant, we need to add these three forces together. Using vector addition, we can add them by summing the magnitudes and taking into account their directions.
Resultant force = Force 1 + Force 2 + Force 3 = 5N + 8N + 10N = 23N
Since the equilibrant is equal in magnitude but opposite in direction to the resultant force, the magnitude of the equilibrant is also 23N.
b) To determine the angles that the equilibrant makes with each of the three forces, we can use the concept of vector decompositio The angle between the equilibrant and each force can be found by finding the dot product of the two vectors and using the formula:
cosθ = (A · B) / (|A| * |B|)
where θ is the angle between the vectors A and B, and A · B is the dot product of A and B.
Let's find the angles between the equilibrant and each force:
Angle between the equilibrant and Force 1:
cosθ1 = (Equilibrant · Force 1) / (|Equilibrant| * |Force 1|)
cosθ1 = (-23N · 5N) / (|23N| * |5N|)
cosθ1 = -115N^2 / (23N * 5N)
cosθ1 = -23 / 5
Using the inverse cosine function (cos^(-1)) on both sides of the equation, we can find θ1:
θ1 = cos^(-1)(-23 / 5)
Similarly, we can find the angles θ2 and θ3 between the equilibrant and Force 2 and Force 3, respectively:
θ2 = cos^(-1)(-23 / 8)
θ3 = cos^(-1)(-23 / 10)
These calculations will give us the angles that the equilibrant makes with each of the three forces.
To calculate the magnitude of the equilibrant of these three forces, we need to find the resultant of the three forces, as the equilibrant is equal in magnitude but opposite in direction to the resultant.
a) Calculate the resultant force:
To find the resultant, we add the three forces vectorially. Using the parallelogram law of vector addition or the triangle law of vector addition, we get:
Resultant = √(5^2 + 8^2 + 10^2 + 2(5)(8)cos(180°) + 2(5)(10)cos(180°) + 2(8)(10)cos(180°))
Simplifying:
Resultant = √(25 + 64 + 100 + 2(5)(8)(-1) + 2(5)(10)(-1) + 2(8)(10)(-1))
Resultant = √(25 + 64 + 100 - 80 - 100 - 160)
Resultant = √(49)
Resultant = 7N
Therefore, the magnitude of the equilibrant is 7N.
b) Determine the angle that the equilibrant makes with each of the three forces:
Since the equilibrant is equal in magnitude but opposite in direction to the resultant, it makes an angle of 180 degrees with each of the three forces.
Therefore, the angle that the equilibrant makes with each of the three forces is 180 degrees.