To tackle this problem, we need to analyze the forces acting on the triangular body formed by particles A, B, and C, as well as the effects of the applied force F. Let's break this down step by step.

Understanding the System

We have three particles A, B, and C, each with mass m, connected to form an equilateral triangle with side length l. The triangle is hinged at point A and can rotate about this point with a constant angular velocity ω on a frictionless table. The hinge will exert forces to maintain the equilibrium of the system.

Part (a): Magnitude of the Horizontal Force Exerted by the Hinge

When the triangular body rotates about point A, each particle experiences a centripetal force due to the rotation. The centripetal force required to keep a particle moving in a circle is given by:

• F_c = m * a_c

Where a_c is the centripetal acceleration, which can be expressed as:

• a_c = r * ω²

For particles B and C, the distance from A (the hinge) to each particle is:

• r = l

Thus, the centripetal force for each particle becomes:

• F_c = m * l * ω²

Since there are two particles (B and C) exerting a centripetal force directed towards A, the total horizontal force exerted by the hinge (F_hinge) must balance these forces. The horizontal component of the force exerted by the hinge is equal to the sum of the centripetal forces acting on B and C:

• F_hinge = 2 * F_c = 2 * m * l * ω²

Therefore, the magnitude of the horizontal force exerted by the hinge on the body is:

• F_hinge = 2 * m * l * ω²

Part (b): Force Components After Applying Force F

At time T, when side BC is parallel to the x-axis, we apply a force F on particle B along the line BC. This force will affect the hinge's reaction forces. We need to analyze the components of the forces acting on the system immediately after the force F is applied.

First, let's consider the components of the force F. Since it acts along BC, we can resolve it into x and y components. The angle θ between the line BC and the horizontal axis (x-axis) can be determined from the geometry of the equilateral triangle:

• θ = 60°

The components of the force F are:

• F_x = F * cos(θ) = F * (1/2)
• F_y = F * sin(θ) = F * (√3/2)

Next, we need to consider the forces acting on the system. The hinge will exert a reaction force to balance the forces acting on the triangle. Let’s denote the hinge reaction force as R, which has components R_x and R_y. The equations of motion in the x and y directions can be set up as follows:

In the x-direction:

• R_x - 2 * m * l * ω² - F_x = 0

In the y-direction:

• R_y - F_y = 0

From the x-direction equation, we can solve for R_x:

• R_x = 2 * m * l * ω² + F * (1/2)

From the y-direction equation, we can solve for R_y:

• R_y = F * (√3/2)

Final Results

To summarize, the components of the force exerted by the hinge on the body immediately after the force F is applied are:

• x-component: R_x = 2 * m * l * ω² + F * (1/2)
• y-component: R_y = F * (√3/2)

This analysis provides a comprehensive understanding of the forces acting on the triangular body in both scenarios, ensuring that we account for the effects of rotation and the applied force.