# What is the equations of motion for a pendulum and spring

• swooshfactory
In summary, for a math-based physics class, the equations for a pendulum and a spring are needed. The equations are x=Asin(wt-phi)+B for the spring and theta=theta0cos(wt+phi) or theta=theta0cos(sqrt(g/l)sin(theta)) for the pendulum. For small displacements and ignoring friction, the equations of motion are those of simple harmonic motion. The differential equation is solved as \frac {d^2 x}{ dt^2} + \omega^2 x = 0, with a solution of x = A sin(\omega t + \phi). The parameters \omega, A, and \phi are determined from initial conditions and can be found in
swooshfactory

## Homework Statement

This is for a math based physics class. I need the equation ofa pendulum and of a spring.

## Homework Equations

Spring: x=Asin(wt-phi)+B
Pendulum: theta=theta0cos(wt+phi) or theta=theta0cos(sqrt(g/l)sin(theta))

## The Attempt at a Solution

I don't know which of these are correct. Please let me know the right equation for each; if it's a differential equation, please let me know what it is solved if possible. Thank you.

For small displacements of the pendulum when $$sin \theta \simeq \theta$$ and ignoring friction in both cases, the equations governing the motion of the mass-spring and the pendulum are those of simple harmonic motion.

The D.E. is $$\frac {d^2 x}{ dt^2} + \omega^2 x = 0$$

Which has a solution with two arbitrary constants (necessary because the D.E. is second order). There are various equivalent forms for the solution. One of which is

$$x = A sin(\omega t + \phi)$$

$$\omega$$ is called the angular frequency and its exact form in terms of the physical parameters in your problem can be determined when you set up the differential
equation. A is the amplitude and $$\phi$$ the phase. These two parameters are determined from initial conditions. All of this is covered in standard physics textbook.

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## 1. What is the equation of motion for a pendulum?

The equation of motion for a pendulum is given by T = 2π√(l/g), where T is the period of the pendulum, l is the length of the pendulum, and g is the acceleration due to gravity.

## 2. What is the equation of motion for a spring?

The equation of motion for a spring is given by F = -kx, where F is the force exerted by the spring, k is the spring constant, and x is the displacement from the equilibrium position.

## 3. What is the relationship between a pendulum and a spring?

A pendulum and a spring both exhibit simple harmonic motion, meaning that their motion can be described by sinusoidal equations. However, a pendulum's motion is affected by gravity, while a spring's motion is affected by the force exerted by the spring itself.

## 4. Can the equations of motion for a pendulum and spring be combined?

Yes, the equations of motion for a pendulum and spring can be combined to describe a pendulum attached to a spring. This system would exhibit a combination of both pendulum and spring motion.

## 5. Are there any factors that can affect the equations of motion for a pendulum and spring?

Yes, there are various factors that can affect the equations of motion for a pendulum and spring, such as air resistance, friction, and the mass of the pendulum or spring. These factors can alter the period, amplitude, and overall behavior of the system.

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