Assume a sinusoidal function

Oct 13, 2025 by windailys.com 127 views

Introduction

In physics, the motion of a weight on a spring is a classic example of a simple harmonic motion (SHM). The weight oscillates about its equilibrium position, and its displacement from the equilibrium position is given by a sinusoidal function. In this article, we will discuss how to find a function that describes the motion of a weight on a spring, given its initial position and period.

Given Information

We are given the following information:

The initial position of the weight is $s(0) = 3 \, \text{in}$ .
The period of the motion is $P = 0.9 \, \text{sec}$ .

Finding the Function s(t)

The function $s(t)$ that describes the motion of the weight on a spring is given by:

s(t) = a \cos(\omega t)

where $a$ is the amplitude of the motion, and $\omega$ is the angular frequency.

To find the function $s(t)$ , we need to determine the values of $a$ and $\omega$ . We can use the given information to do this.

Finding the Amplitude a

The amplitude $a$ is the maximum displacement of the weight from its equilibrium position. Since the weight starts at its initial position $s(0) = 3 \, \text{in}$ , the amplitude $a$ is equal to the initial position:

a = s(0) = 3 \, \text{in}

Finding the Angular Frequency ω

The angular frequency $\omega$ is related to the period $P$ by the following equation:

\omega = \frac{2\pi}{P}

Substituting the given value of $P = 0.9 \, \text{sec}$ , we get:

\omega = \frac{2\pi}{0.9 \, \text{sec}} = \frac{2\pi}{9/10 \, \text{sec}} = \frac{20\pi}{9} \, \text{rad/sec}

Finding the Function s(t)

Now that we have found the values of $a$ and $\omega$ , we can substitute them into the equation for $s(t)$ :

s(t) = a \cos(\omega t)

s(t) = 3 \, \text{in} \cos\left(\frac{20\pi}{9} \, \text{rad/sec} \cdot t\right)

Conclusion

In this article, we have discussed how to find a function that describes the motion of a weight on a spring, given its initial position and period. We have used the given information to determine the values of the amplitude $a$ and the angular frequency $\omega$ , and then substituted these values into the equation for $s(t)$ . The resulting function $s(t)$ describes the motion of the weight on a spring.

Mathematical Derivations

Derivation of the Equation for s(t)

The equation for $s(t)$ is given by:

s(t) = a \cos(\omega t)

To derive this equation, we can use the following steps:

Assume a sinusoidal function: Assume that the displacement $s(t)$ is a sinusoidal function of time $t$ :

s(t) = A \cos(\omega t + \phi)

where $A$ is the amplitude, $\omega$ is the angular frequency, and $\phi$ is the phase angle.

Use the initial condition: Use the initial condition $s(0) = 3 \, \text{in}$ to determine the value of $A$ :

s(0) = A \cos(\omega \cdot 0 + \phi) = A \cos(\phi) = 3 \, \text{in}

Use the period: Use the period $P = 0.9 \, \text{sec}$ to determine the value of $\omega$ :

P = \frac{2\pi}{\omega}

\omega = \frac{2\pi}{P} = \frac{2\pi}{0.9 \, \text{sec}} = \frac{20\pi}{9} \, \text{rad/sec}

Substitute the values of A and ω: Substitute the values of $A$ and $\omega$ into the equation for $s(t)$ :

s(t) = A \cos(\omega t + \phi)

s(t) = 3 \, \text{in} \cos\left(\frac{20\pi}{9} \, \text{rad/sec} \cdot t + \phi\right)

Determine the phase angle φ: Determine the value of the phase angle $\phi$ by using the initial condition $s(0) = 3 \, \text{in}$ :

s(0) = 3 \, \text{in} \cos(\phi) = 3 \, \text{in}

\cos(\phi) = 1

\phi = 0

Substitute the value of φ: Substitute the value of $\phi$ into the equation for $s(t)$ :

s(t) = 3 \, \text{in} \cos\left(\frac{20\pi}{9} \, \text{rad/sec} \cdot t + 0\right)

s(t) = 3 \, \text{in} \cos\left(\frac{20\pi}{9} \, \text{rad/sec} \cdot t\right)

Derivation of the Equation for ω

The equation for $\omega$ is given by:

\omega = \frac{2\pi}{P}

To derive this equation, we can use the following steps:

Use the definition of the period: Use the definition of the period $P$ to determine the value of $\omega$ :

P = \frac{1}{\omega}

Solve for ω: Solve for $\omega$ :

\omega = \frac{1}{P}

Substitute the value of P: Substitute the value of $P = 0.9 \, \text{sec}$ into the equation for $\omega$ :

\omega = \frac{1}{0.9 \, \text{sec}} = \frac{10}{9} \, \text{sec}^{-1}

Convert to radians per second: Convert the value of $\omega$ from seconds to radians per second:

\omega = \frac{10}{9} \, \text{sec}^{-1} \cdot \frac{1 \, \text{rad}}{1 \, \text{sec}} = \frac{10}{9} \, \text{rad/sec}

Multiply by 2π: Multiply the value of $\omega$ by $2\pi$ to get the final value:

\omega = \frac{10}{9} \, \text{rad/sec} \cdot 2\pi = \frac{20\pi}{9} \, \text{rad/sec}$$<br/> **Suppose that a Weight on a Spring has an Initial Position and Period** =========================================================== **Q&A** ----- **Q: What is the motion of a weight on a spring?** -------------------------------------------- A: The motion of a weight on a spring is a classic example of a simple harmonic motion (SHM). The weight oscillates about its equilibrium position, and its displacement from the equilibrium position is given by a sinusoidal function. **Q: What is the equation for the motion of a weight on a spring?** --------------------------------------------------------- A: The equation for the motion of a weight on a spring is given by: $$s(t) = a \cos(\omega t)

where $a$ is the amplitude of the motion, and $\omega$ is the angular frequency.

Q: How do I find the amplitude a?

A: To find the amplitude $a$ , you need to use the initial condition $s(0) = 3 \, \text{in}$ . The amplitude $a$ is equal to the initial position:

a = s(0) = 3 \, \text{in}

Q: How do I find the angular frequency ω?

A: To find the angular frequency $\omega$ , you need to use the period $P = 0.9 \, \text{sec}$ . The angular frequency $\omega$ is related to the period $P$ by the following equation:

\omega = \frac{2\pi}{P}

Substituting the given value of $P = 0.9 \, \text{sec}$ , we get:

\omega = \frac{2\pi}{0.9 \, \text{sec}} = \frac{20\pi}{9} \, \text{rad/sec}

Q: What is the significance of the phase angle φ?

A: The phase angle $\phi$ is the angle by which the sinusoidal function is shifted from the standard cosine function. In this case, the phase angle $\phi$ is equal to 0, which means that the sinusoidal function is not shifted.

Q: How do I determine the phase angle φ?

A: To determine the phase angle $\phi$ , you need to use the initial condition $s(0) = 3 \, \text{in}$ . The phase angle $\phi$ is equal to the angle by which the sinusoidal function is shifted from the standard cosine function. In this case, the phase angle $\phi$ is equal to 0.

Q: What is the relationship between the period P and the angular frequency ω?

A: The period $P$ and the angular frequency $\omega$ are related by the following equation:

P = \frac{2\pi}{\omega}

This equation shows that the period $P$ is equal to the reciprocal of the angular frequency $\omega$ .

Q: How do I convert the angular frequency ω from seconds to radians per second?

A: To convert the angular frequency $\omega$ from seconds to radians per second, you need to multiply the value of $\omega$ by $2\pi$ :

\omega = \frac{10}{9} \, \text{sec}^{-1} \cdot \frac{1 \, \text{rad}}{1 \, \text{sec}} = \frac{10}{9} \, \text{rad/sec}

Q: What is the final value of the angular frequency ω?

A: The final value of the angular frequency $\omega$ is:

\omega = \frac{20\pi}{9} \, \text{rad/sec}

Conclusion

In this article, we have discussed the motion of a weight on a spring, and how to find the equation for the motion of a weight on a spring. We have also answered some common questions related to the motion of a weight on a spring.