# A pendulum oscillation problem

• hancock.yang@
In summary, we are considering a pendulum oscillation problem with a torsional spring that counteracts the pendulum motion. By adding the restoring forces of gravity and the spring together, we can determine the total force on the pendulum bob. This force is related to the acceleration of the bob, which in turn is related to the angular acceleration. Using this information, we can set up the differential equation of motion in the form \ddot{\theta} = -f(\theta).
hancock.yang@

## Homework Statement

Consider a pendulum oscillation problem, where pendulum oscillates around the vertical in the downward configuration.
Assume that there is no friction at the pivot point around which the pendulum rotates, and assume that there exists a torsional spring that counter acts the pendulum motion. Let the spring force Fs be a non-linear function of the displacement of the pendulum θ from the vertical configuration, that is,
F=K$$\theta^{3}$$
Considering the presence of gravitational forces, ignoring external torques on the pendulum

To find a the differential equation governing the pendulum dynamics.

Kind regards

## The Attempt at a Solution

J\ddot{\theta}=T-m*g*sin($$\theta$$)
I am tying to find T as a function of F

hancock.yang@ said:

## Homework Statement

Consider a pendulum oscillation problem, where pendulum oscillates around the vertical in the downward configuration.
Assume that there is no friction at the pivot point around which the pendulum rotates, and assume that there exists a torsional spring that counter acts the pendulum motion. Let the spring force Fs be a non-linear function of the displacement of the pendulum θ from the vertical configuration, that is,
F=K$$\theta^{3}$$
Considering the presence of gravitational forces, ignoring external torques on the pendulum

To find a the differential equation governing the pendulum dynamics.
...

## The Attempt at a Solution

J\ddot{\theta}=T-m*g*sin($$\theta$$)
I am tying to find T as a function of F
There are two restoring forces here: gravity and the spring force. Write out the expression for the restoring force of gravity as a function of angle $\theta$. Write out the expression for the restoring force of the spring as a function of $\theta$. Since the two forces are always the same direction, add them together to find the total force.

How is the total force on the pendulum bob related to its acceleration? How is this acceleration related to the angular acceleration (the rate of change of angular speed)?

Answer those questions and you will be able to set up the differential equation of motion.(hint: it is in the form:

$$\ddot{\theta} = -f(\theta)$$

AM

Andrew Mason said:
There are two restoring forces here: gravity and the spring force. Write out the expression for the restoring force of gravity as a function of angle $\theta$. Write out the expression for the restoring force of the spring as a function of $\theta$. Since the two forces are always the same direction, add them together to find the total force.

How is the total force on the pendulum bob related to its acceleration? How is this acceleration related to the angular acceleration (the rate of change of angular speed)?

Answer those questions and you will be able to set up the differential equation of motion.(hint: it is in the form:

$$\ddot{\theta} = -f(\theta)$$

AM
I have already worked it out .

Question 1:

## What is a pendulum oscillation problem?

A pendulum oscillation problem refers to a physical phenomenon in which a pendulum swings back and forth due to the force of gravity. It is a common problem in physics and engineering, and it can be used to study the principles of motion and energy.

Question 2:

## What factors affect the oscillation of a pendulum?

The oscillation of a pendulum can be affected by several factors, including the length of the pendulum, the mass of the pendulum bob, the amplitude of the swing, and the force of gravity. Friction and air resistance can also have an impact on the oscillation of a pendulum.

Question 3:

## What is the relationship between the length of a pendulum and its period of oscillation?

According to the law of simple pendulum, the period of oscillation of a pendulum is directly proportional to the square root of its length. This means that as the length of a pendulum increases, its period of oscillation also increases.

Question 4:

## How does the amplitude of a pendulum affect its oscillation?

The amplitude of a pendulum, which refers to the maximum angle of displacement from its equilibrium position, has a significant impact on its oscillation. As the amplitude increases, the period of oscillation also increases. However, if the amplitude becomes too large, the pendulum can lose its regular oscillation and become chaotic.

Question 5:

## What are some real-life applications of pendulum oscillation problems?

Pendulum oscillation problems have various real-life applications, such as clock mechanisms, seismometers for detecting earthquakes, and amusement park rides. They are also used in scientific experiments to study the principles of motion and energy.

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