Problem 95 - A Complete Resource Book in Physics for JEE Main

Question

Four forces act on a point object. The object will be in equilibrium if (A) all of them are in the same plane (B) they are opposite to each other in pair (C) the sum of \(x, y\) and \(z\) components of all the force is zero separately (D) they are from a closed figure of four sides

Step-by-Step Solution

Answer

The object will be in equilibrium if the sum of the \(x, y\), and \(z\) components of all the forces is zero separately (Option C). This condition guarantees the net force and net torque acting on the object is zero.

Step 1: (Option A Analysis)

(Check the condition that all the forces are in the same plane) All the forces being in the same plane does not guarantee that the net force acting on the body will be zero. It is possible for the forces in the same plane to have a non-zero net force acting on the object. Hence, it is not sufficient for equilibrium.

Step 2: (Option B Analysis)

(Check the condition that the forces are opposite to each other in pairs) When the forces acting on the object are opposite to each other in pairs, the net force acting on the body will be zero. This means the resultant of one pair will exactly cancel out the resultant of the other pair. However, this condition does not guarantee the net torque acting on an extended body will be zero. So, this condition is necessary but not sufficient for equilibrium.

Step 3: (Option C Analysis)

(Check the condition that the sum of \(x, y\) and \(z\) components of all the forces is zero separately) When the sum of the \(x, y\) and \(z\) components of all the forces is zero separately, it means that the net force acting on the object is zero in each direction. Additionally, the net torque acting on the body is directly related to the net force, so when the net force is zero, the net torque will be zero as well. Hence, this condition guarantees that the object will be in equilibrium.

Step 4: (Option D Analysis)

(Check the condition that the forces form a closed figure of four sides) If the forces form a closed figure of four sides, this would mean that all forces are connected tip-to-tail and form a loop. This implies that the net force acting on the object is zero. However, similar to Option B, this condition does not guarantee the net torque acting on an extended body will be zero. So, this condition is necessary but not sufficient for equilibrium. From the analysis, it is clear that Option C is the correct choice for the object to be in equilibrium, as it satisfies both the conditions for equilibrium: net force and net torque should be zero.