Question

На тело действуют 2 силы:F1 и F2.Определите модуль и направление равнодействующей этих сил если F1=F2=3H,а угол между ними составляет а)90° ,б)60°.Выполните рисунок в определенном масштабе,и укажите масштаб

Answer 1

А) т.к угол 90 градусов
То по теореме Пифагора: F=квадратный корень из (F1^2+F2^2)=3 корня из 2

б) т.к угол 60 градусов
F^2=F1^2+F2^2-2F1F2cos60F=3 H 

Answer 2

Problem Analysis

We are given two forces, F1 and F2, acting on a body. We need to determine the magnitude and direction of the resultant force (F_res) when F1 = F2 = 3H and the angle between them is given.

Solution

To find the magnitude and direction of the resultant force, we can use the concept of vector addition. The resultant force is the vector sum of F1 and F2.

Case a) Angle = 90°

When the angle between F1 and F2 is 90°, the forces are perpendicular to each other. In this case, we can use the Pythagorean theorem to find the magnitude of the resultant force.

The magnitude of the resultant force (F_res) can be calculated using the formula:

F_res = sqrt(F1^2 + F2^2)

Substituting the given values, where F1 = F2 = 3H, we get:

F_res = sqrt(3^2 + 3^2) = sqrt(18) = 3√2 H

The direction of the resultant force can be determined using trigonometry. Since F1 and F2 are perpendicular to each other, the angle between F_res and F1 is 45°.

Therefore, the magnitude of the resultant force is 3√2 H and the direction is 45° with respect to F1.

Case b) Angle = 60°

When the angle between F1 and F2 is 60°, we can use the law of cosines to find the magnitude of the resultant force.

The magnitude of the resultant force (F_res) can be calculated using the formula:

F_res = sqrt(F1^2 + F2^2 + 2 * F1 * F2 * cos(angle))

Substituting the given values, where F1 = F2 = 3H and the angle = 60°, we get:

F_res = sqrt(3^2 + 3^2 + 2 * 3 * 3 * cos(60°)) = sqrt(18 + 18 + 18) = sqrt(54) = 3√6 H

The direction of the resultant force can be determined using the law of sines. The angle between F_res and F1 can be calculated using the formula:

sin(angle) / F_res = sin(angle_F1) / F1

Substituting the given values, where F1 = F2 = 3H and the angle = 60°, we get:

sin(angle_F1) / 3 = sin(60°) / (3√6)

Simplifying the equation, we find:

sin(angle_F1) = (3√6 * sin(60°)) / 3

sin(angle_F1) = √6 / 2

Taking the inverse sine of both sides, we find:

angle_F1 = sin^(-1)(√6 / 2) ≈ 45.57°

Therefore, the magnitude of the resultant force is 3√6 H and the direction is approximately 45.57° with respect to F1.

Conclusion

In conclusion, the magnitude and direction of the resultant force depend on the angle between F1 and F2. When the angle is 90°, the magnitude of the resultant force is 3√2 H and the direction is 45° with respect to F1. When the angle is 60°, the magnitude of the resultant force is 3√6 H and the direction is approximately 45.57° with respect to F1.

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