Three forces of magnitude 6N,6N,√72N act at a corner of a cube along three sides .resultant of the force is a)12N along on b)18N along oa c)18N along of d)12N along oe
To find the resultant of multiple forces, you need to determine the vector sum of these forces. The vector sum can be computed by adding the forces together using vector addition.
In this case, let's consider the three forces acting on the corner of the cube. We have three forces with magnitudes of 6N, 6N, and √72N.
Step 1: Decompose the forces into their x, y, and z-components.
Given that the forces are acting along three sides of the cube, we can assume that each force is acting along one of the coordinate axes (x, y, z). Therefore, we can first decompose each force into its components along these axes.
Force 1 (6N): This force can be decomposed into (6N, 0N, 0N), as it is acting only along the x-axis.
Force 2 (6N): This force can also be decomposed into (0N, 6N, 0N), as it is acting only along the y-axis.
Force 3 (√72N): To decompose this force, we need to determine its components along the x, y, and z-axes. Given that √72 = √(36 * 2) = 6√2, we can decompose this force into (6√2N, 6√2N, 0N).
Step 2: Add the corresponding components of the forces.
To find the resultant of the three forces, we can add the corresponding components together.
x-component: 6N + 6√2N = 6N + 6√2N = 6N + 6√2N.
y-component: 6N + 6N = 12N.
z-component: 0N + 0N = 0N.
Therefore, the resultant force is given by (6N + 6√2N, 12N, 0N).
Step 3: Find the magnitude and direction of the resultant force.
To find the magnitude of the resultant force, we can calculate the magnitude of the vector sum using the Pythagorean theorem:
Magnitude = √((6N + 6√2N)^2 + (12N)^2 + (0N)^2)
= √(36N^2 + 72N + 72N + 144N^2)
= √(180N^2 + 72N√2)
To simplify further, we can factor out 36N:
Magnitude = √(36N(5N + 2√2))
To determine the direction of the resultant force, we can look at the x, y, and z-components we calculated earlier. Since all the forces in the x and z directions cancel out (resulting in 0N), the resultant force is solely in the y-direction.
Therefore, the resultant force is (6N + 6√2N, 12N, 0N), which simplifies to (6 + 6√2)N in the y-direction.
Comparing the options provided:
a) 12N along on
b) 18N along oa
c) 18N along of
d) 12N along oe
None of the given options match the resultant force calculated, which is (6 + 6√2)N along the y-direction.
To find the resultant of the three forces, we can use the Pythagorean theorem for the two perpendicular components of the forces. Let's name the sides of the cube as A, B, and C.
The forces acting along sides AB and AC are both 6N, so their resultant can be found using the Pythagorean theorem:
Resultant_AB_AC = √(6^2 + 6^2)
= √(36 + 36)
= √(72)
= 6√2 N
Now, the remaining force acting along side BC has a magnitude of √72N.
To find the total resultant force, we can again use the Pythagorean theorem with the resultant_AB_AC and the force along side BC:
Resultant_Total = √(Resultant_AB_AC^2 + (√72)^2)
= √((6√2)^2 + 72)
= √(72 + 72)
= √(144)
= 12N
Therefore, the resultant of the three forces is 12N along the diagonal of the cube, option d) 12N along oe.
the magnitude z is
z^2 = 6^2+6^2+72 = 144
z = 12
Not sure where N and E are ...