A small sphere of mass \( 0.1\text{ kg} \) suspended from a ceiling by a light string is pulled horizontally by a force \( F \) until the string makes an angle of \( 45^\circ \) with the vertical at equilibrium. Taking \( g = 10\text{ m/s}^2 \), the magnitudes of horizontal force \( F \) and string tension \( T \) are:

In equilibrium, the forces acting on the sphere are the gravitational force \( mg \) acting downwards, the horizontal force \( F \) acting horizontally, and the tension \( T \) in the string acting at an angle of \( 45^\circ \) to the vertical.
Resolving forces vertically: \( T \cos 45^\circ = mg \)
Given \( m = 0.1\text{ kg} \) and \( g = 10\text{ m/s}^2 \), we have:
\( T \left(\frac{1}{\sqrt{2}}\right) = (0.1)(10) = 1\text{ N} \implies T = \sqrt{2}\text{ N} \)
Resolving forces horizontally: \( F = T \sin 45^\circ \)
\( F = (\sqrt{2}) \left(\frac{1}{\sqrt{2}}\right) = 1\text{ N} \)
Therefore, \( F = 1\text{ N} \) and \( T = \sqrt{2}\text{ N} \).