# Pendulum Question

## Homework Statement

A pendulum is hanging from the ceiling of a cage. If the cage moves up with constant acceleration a, its tension is T1. If it moves down with the same acceleration a then the tension is T2. If the cage moves horizontally with the same acceleration a, then the tension is T. Now, 2T2 =
(a) T12 + T22
(b) T12 - T22
(c) 2T12 - T22
(d) T12 - 2T22

F = ma

## The Attempt at a Solution

I found the tension in the string for the above two cases from their respective Free Body Diagrams
i.e T1 = m(g+a) and T2 = m(g-a)
But I had some problems for the Free Body Diagram of the pendulum in the third case, i.e when the cage is accelerating sideways.
Here is my FBD

So from the FBD I get
T = mg cos θ ---(i) and tan θ = (a/g) ---(ii)
My confusion is that is the equation (i) correct ?
Since the equations in the book is given in the book is written differently
Instead of resolving the weight components they resolved the tension components i.e the equations are given as
T sin θ = ma --- (iii)
T cos θ = mg ---(iv)
As you can see both the equations (i) and (iv) looks different. I could not figure out where is the mistake

Orodruin
Staff Emeritus
Homework Helper
Gold Member
There are three forces acting on the pendulum. You are confusing yourself by introducing the components only of the gravitational force. Start by drawing the FBD without a coordinate system, then introduce a coordinate system and write down the force equations in all directions.

Aninda
Orodruin
Staff Emeritus