# [Physics] Tension in string and gravitational force on bob relative strength

forcesfree-body-diagramnewtonian-gravitynewtonian-mechanicsstring

When calculating time period of simple pendulum (an approximation of SHM at small amplitudes) we take gravitational force greater than tension in string and resolve gravitational force in two orthogonal components and equate one of them with tension. While in case of conical pendulum we take tension greater than gravitational force and resolve it into two orthogonal components and equate one of them with gravitational force on metallic bob. Similarly in banked road problem we take normal reaction greater than gravitational force.

Why in one case tension is more and in other gravitation is more?

For the simple pendulum, you see the the bob has zero radial acceleration. The only acceleration is angular. Based on that, it is convenient to choose a coordinate system which is tangential and transverse to the instantaneous velocity. Other coordinate systems are possible, but aren't as convenient. With this choice, you would resolve the weight vector into a component parallel to the string, which happens to also be parallel to the tension. Then the Newton's 2nd Law equations will be $$T-mg\cos \theta = ma_{radial} = 0.$$ $$-mg\sin \theta = ma_{angular} = m\ell\alpha$$ I have chosen $\theta$ to be measured with respect to the gravitational field direction. Because the string length is not changing, the radial velocity is constantly zero and the radial acceleration is zero.