JNTU B.Tech I Year ENGINEERING MECHANICS Previous Question Paper (July - 2021 - R16)

To find the single force (equilibrant) that keeps a body in equilibrium under a system of coplanar forces, we must determine the resultant force of the system and then apply an equal and opposite force at the correct position.

The steps to solve this are:

1. Resolve all given forces into their horizontal (ΣF_x) and vertical (ΣF_y) components.
2. Calculate the magnitude of the resultant force (R) using the formula:
   R = √(ΣF_x² + ΣF_y²)
3. Determine the angle (θ) of the resultant force with respect to the x-axis:
   θ = tan^-1(|ΣF_y / ΣF_x|)
4. Calculate the sum of moments (ΣM_O) of all forces about an arbitrary point (usually the origin) to find the position of the resultant.
5. Determine the perpendicular distance (d) of the resultant from the reference point using:
   d = |ΣM_O / R|
6. The equilibrant force (E) has the same magnitude (R) and the same line of action as the resultant, but acts in the opposite direction (180° + θ).

By applying this equilibrant force at the calculated distance from the reference point, the net force and net moment acting on the body become zero, satisfying the conditions for static equilibrium.

To determine the forces P and Q required to maintain the man of mass 100 kg in equilibrium, we apply the conditions of static equilibrium. First, calculate the weight of the man (W):

Assuming the cables are attached to the man at a common point, the system is in equilibrium when the sum of vertical and horizontal forces equals zero. Let the angles of the cables with the vertical be θ₁ and θ₂ respectively. Resolving the forces into components:

1. Sum of horizontal forces: P sin(θ₁) - Q sin(θ₂) = 0, which implies P sin(θ₁) = Q sin(θ₂)
2. Sum of vertical forces: P cos(θ₁) + Q cos(θ₂) = W = 981 N

By solving this system of two equations, the tension in each cable is determined. If the angles are provided in the figure (e.g., θ₁ = 30° and θ₂ = 45°), substitute these values into the equations:

Substituting P = 1.414Q into the vertical equilibrium equation:

The forces P and Q depend directly on the specific geometry (angles) shown in the figure; using the equilibrium equations above allows for the precise calculation once those angles are identified.