# Pendulum Question Homework: Resolving Tension Components

• Aninda
In summary, the author found the tension in the string for the three cases, but had some difficulty with the free body diagram of the pendulum in the third case. He found the tension in x and y-directions by solving the force equations in all directions.
Aninda

## 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

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
Alternatively, you can just use the fact that the net force needs to be zero and that a and g are orthogonal ... Then you never need to introduce a coordinate system or care about the components.

Aninda
There's no mistake, he just broke down the tension in x and y-direction. Try to draw a graph with two axes (x, y) and draw on it the decomposed tension in x and y-direction, the cage force (ma) and the weight force (mg), and you will see what the book did. I hope that someone more qualified helps you, but for now, think about it.
(Sorry, the message came too late, someone has already answered the question)

## 1. What is a pendulum and how does it work?

A pendulum is a simple device that consists of a weight suspended from a fixed point so that it can swing freely back and forth. This weight is known as the "bob" or "pendulum bob." The motion of a pendulum is based on the principles of gravity and conservation of energy.

## 2. How do you resolve tension components in a pendulum?

To resolve tension components in a pendulum, you must first identify the forces acting on the pendulum. These forces include the weight of the pendulum bob, the tension force from the string or rod holding the bob, and the force of gravity. You can then break down the tension force into its horizontal and vertical components using trigonometry.

## 3. What is the formula for resolving tension components?

The formula for resolving tension components is Tcosθ for the horizontal component and Tsinθ for the vertical component, where T is the tension force and θ is the angle between the tension force and the horizontal axis.

## 4. How do you calculate the tension force in a pendulum?

The tension force in a pendulum can be calculated using the formula T = (m * g) + (m * v^2) / L, where m is the mass of the pendulum bob, g is the acceleration due to gravity, v is the velocity of the pendulum bob, and L is the length of the string or rod holding the bob.

## 5. Can tension components affect the period of a pendulum?

Yes, tension components can affect the period of a pendulum. The tension force acts as a restoring force that helps maintain the pendulum's motion. As the tension force changes, it can affect the speed and frequency of the pendulum's swings, thus altering its period.

• Introductory Physics Homework Help
Replies
20
Views
1K
• Introductory Physics Homework Help
Replies
9
Views
1K
• Introductory Physics Homework Help
Replies
3
Views
771
• Introductory Physics Homework Help
Replies
3
Views
1K
• Introductory Physics Homework Help
Replies
1
Views
257
• Introductory Physics Homework Help
Replies
2
Views
985
• Introductory Physics Homework Help
Replies
21
Views
2K
• Introductory Physics Homework Help
Replies
4
Views
3K
• Introductory Physics Homework Help
Replies
18
Views
385
• Introductory Physics Homework Help
Replies
2
Views
1K