Irodov Problem 1.198

Problem Statement

A heavy sphere of mass m is suspended on a string of length l. A horizontal force F is applied to the sphere. Find the angle of deflection θ and the tension T in the string at equilibrium.

Given

m, l, F horizontal. Static equilibrium.

Concepts & Formulas

Equilibrium: T sinθ = F, T cosθ = mg. → tanθ = F/(mg).

Step-by-Step Solution

Step 1: Horizontal: T sinθ = F.
Step 2: Vertical: T cosθ = mg.
Step 3: tanθ = F/(mg) → θ = arctan(F/mg).
Step 4: T = √(F² + m²g²).

Worked Calculation

tanθ = F/(mg). T = √(F² + m²g²).

Boxed Answer

θ = arctan(F/mg); T = √(F² + m²g²)

Physical Interpretation

The string tension always exceeds mg — it must support both the weight and resist the horizontal pull, with the vector sum giving the resultant.