A body is under the action of two forces 7N And 10N find the resultant of the two forces if
(a)the forces are parallel and act in the opposite direction
(b)the forces are parallel and act in the same direction
(c)the two forces are at 90° to each other
(d)the two forces are inclined at an angle of 60° to each other
To find the resultant of two forces, we need to use the concept of vector addition. The resultant is the vector sum of the two forces. We can find the resultant by using the Pythagorean theorem and trigonometric functions.
(a) If the two forces are parallel and act in the opposite direction:
In this case, the forces are collinear. The resultant can be found by subtracting the smaller force from the larger force.
Resultant = 10N - 7N = 3N
(b) If the two forces are parallel and act in the same direction:
In this case, the forces are also collinear. The resultant can be found by adding the forces together.
Resultant = 10N + 7N = 17N
(c) If the two forces are at 90° to each other:
In this case, the forces are perpendicular. To find the resultant, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b).
Resultant = √(7N² + 10N²) = √(49N² + 100N²) = √149N² ≈ 12.20N
(d) If the two forces are inclined at an angle of 60° to each other:
In this case, we can use trigonometric functions to find the resultant. We can use the law of cosines to find the magnitude (R) of the resultant and the law of sines to find the direction (θ) of the resultant.
R = √(7N² + 10N² - 2 * 7N * 10N * cos(60°)) = √(49N² + 100N² - 140N² * 0.5) ≈ √(149N² - 70N²) ≈ √79N² ≈ 8.89N
θ = sin^(-1)((sin(60°) * 10N) / R) = sin^(-1)((√3/2 * 10N) / 8.89N) ≈ sin^(-1)(0.866) ≈ 60.67°
Therefore, in case (a), the resultant is 3N in the opposite direction of the larger force.
In case (b), the resultant is 17N in the same direction as the forces.
In case (c), the resultant is approximately 12.20N perpendicular to the two forces.
In case (d), the resultant is approximately 8.89N at an angle of approximately 60.67° with respect to the larger force.