How to derive a formula for simple harmonic motion?

• Matej Pertot
In summary, the solutions for the undamped pendulum simple harmonic motion are s = s0 sin(2 pi f t) and s = A cos(2 pi f t) + B sin(2 pi f t), where A and B are constants determined by the initial conditions. Starting from the middle point, the equation would be s = s0 sin(2 pi f t), while starting from the extreme point, the equation would be s = A cos(2 pi f t) + B sin(2 pi f t). The more general solution includes the possibility of starting at a point other than 0 and with a non-zero initial velocity.
Matej Pertot

Homework Statement

You need to derive a formula for undamped pendulum simple harmonic motion;
1. Starting from the middle point
2. Starting from the extreme point

Homework Equations

The solutions are;
1. s = s0 sin(2 pi f t)
2. s = s0 cos(2 pi f t)

The Attempt at a Solution

I can derive the first answer -- you have to imagine that it is similar to circular motion. But I have difficulties with the second question; I arrive at solution 1-cos(...) because - doesn't it knly move for the difference between radius (full amplitude) and cosine?

Matej Pertot said:

Homework Statement

You need to derive a formula for undamped pendulum simple harmonic motion;
1. Starting from the middle point
2. Starting from the extreme point

Homework Equations

The solutions are;
1. s = s0 sin(2 pi f t)
2. s = s0 cos(2 pi f t)

The Attempt at a Solution

I can derive the first answer -- you have to imagine that it is similar to circular motion. But I have difficulties with the second question; I arrive at solution 1-cos(...) because - doesn't it knly move for the difference between radius (full amplitude) and cosine?

Your relevant equations are not completely general. The equation ##s = s_0 sin(2 \pi f t)## describes motion that starts at ##s = 0## at time ##t=0##. The more general solution is:

##s = A cos(2 \pi f t) + B sin(2 \pi f t) ##

where ##A## and ##B## are constants that have to be chosen based on the value of ##s## at time ##t=0##

1. What is simple harmonic motion?

Simple harmonic motion is a type of periodic motion in which an object moves back and forth around a central equilibrium point, with a constant frequency and amplitude.

2. How do you derive a formula for simple harmonic motion?

The formula for simple harmonic motion can be derived using basic principles of calculus and Newton's laws of motion. The resulting formula is x(t) = A*cos(ωt+φ), where x is the displacement, A is the amplitude, ω is the angular frequency, and φ is the phase constant.

3. What is the significance of the amplitude and frequency in the formula for simple harmonic motion?

The amplitude represents the maximum displacement of the object from its equilibrium point, while the frequency represents the number of complete oscillations per unit time. These values affect the behavior and characteristics of the motion, such as the speed and period.

4. How is the formula for simple harmonic motion applied in real-world scenarios?

The formula for simple harmonic motion can be applied to various real-world scenarios, such as the motion of a mass on a spring, the swinging of a pendulum, or the vibrations of a guitar string. It is also used in fields such as physics, engineering, and astronomy to model and analyze periodic phenomena.

5. Are there any limitations to the formula for simple harmonic motion?

While the formula for simple harmonic motion is a useful model for many systems, it has some limitations. It assumes ideal conditions, such as a frictionless environment, and it does not take into account external forces or factors that may affect the motion. As a result, it may not accurately represent all real-life situations.

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