91Ó°ÊÓ

A \(3.0\) -g particle at the end of a spring moves according to the equation \(y=0.75 \sin 63 t\), where \(y\) is in centimeters and \(t\) is in seconds. Find the amplitude and frequency of its motion, its position at \(t=0.020 \mathrm{~s}\), and the spring constant. The equation of motion is \(y=y_{0} \sin 2 \pi f t\). By comparison, we see that the amplitude is \(y_{0}=0.75 \mathrm{~cm}\). Also, $$ 2 \pi f=63 \mathrm{~s}^{-1} \quad \text { from which } \quad f=10 \mathrm{~Hz} $$ (Note that the argument of the sine must be dimensionless; because \(t\) is in seconds, \(2 \pi f\) must have the unit \(1 /\) s.) When \(t=0.020 \mathrm{~s}\), we have $$ y=0.75 \sin (1.26 \mathrm{rad})=(0.75)(0.952)=0.71 \mathrm{~cm} $$ Notice that the argument of the sine is in radians, not degrees. To find the spring constant, use \(f=(1 / 2 \pi) \sqrt{k / m}\) to get $$ k=4 \pi^{2} f^{2} m=11.9 \mathrm{~N} / \mathrm{m}=12 \mathrm{~N} / \mathrm{m} $$

Step by step solution

01

Identify Amplitude

From the equation of motion, \( y = 0.75 \sin 63t \), we can directly identify the amplitude. The amplitude \( y_0 \) is the coefficient of the sine function, which is \( 0.75 \) cm.

02

Determine Frequency

The equation \( y = y_0 \sin 2 \pi f t \) shows the term \( 2 \pi f \) in comparison with \( 63 \) rad/s from the given equation. Hence, \( 2 \pi f = 63 \). Solving for \( f \) gives \( f = \frac{63}{2 \pi} \approx 10 \, \text{Hz} \).

03

Calculate Position at Specific Time

To find the position at \( t = 0.020 \) s, plug \( t \) into the equation: \( y = 0.75 \sin(63 \times 0.020) \). This becomes \( y = 0.75 \sin(1.26 \text{ rad}) \). Calculating the sine: \( \sin(1.26) \approx 0.952 \), so \( y = 0.75 \times 0.952 = 0.714 \) cm.

04

Calculate Spring Constant

Using the frequency formula \( f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \), rearrange to solve for \( k \): \( k = (2\pi f)^2 m \). With \( f = 10 \) Hz and mass \( m = 0.003 \) kg, \( k = 4\pi^2 \times 10^2 \times 0.003 \approx 11.9 \, \text{N/m} \). Rounding gives \( k = 12 \, \text{N/m} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Amplitude

In the context of simple harmonic motion, the amplitude refers to the maximum displacement from the equilibrium position. It represents how far the particle moves from its center position over the course of the motion.

When looking at the equation of motion, like the one given here, \( y = 0.75 \sin 63t \), the amplitude is indicated by the coefficient of the sine function. This means that the amplitude, \( y_0 \), is \( 0.75 \) cm.

• This value tells us that, at its peak, the object will be \( 0.75 \) cm from the equilibrium position.
• A higher amplitude would imply a larger range of motion, whereas a lower amplitude would imply a smaller movement range.

Frequency

The frequency of simple harmonic motion describes how many oscillations occur per second. It is a measure of how quick or slow the motion is.

Within the given equation, \( y = 0.75 \sin 63t \), the frequency \( f \) can be calculated by setting \( 2\pi f = 63 \). Solving this yields \( f = \frac{63}{2\pi} \approx 10 \) Hz.

• Frequency is measured in hertz (Hz), where 1 Hz equates to one oscillation per second.
• Thus, in this case, the particle completes about 10 full oscillations in one second.

This knowledge allows us to understand how frequently the system returns to its peak points, or cycles through one complete motion sequence.

Spring Constant

The spring constant, denoted by \( k \), is a parameter reflecting the stiffness of a spring in simple harmonic motion. The stiffer the spring, the higher the spring constant.

To find \( k \), we can use the formula related to frequency: \( f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \). From here, it is rearranged to solve for \( k \):\[ k = (2\pi f)^2 m \]Given that \( f = 10 \) Hz and \( m = 0.003 \) kg, the spring constant calculates to:\[ k = 4\pi^2 \times 10^2 \times 0.003 \approx 11.9 \, \text{N/m} \]Rounding it off, the spring constant is about \( 12 \, \text{N/m} \).

• A higher spring constant means a stronger force is required for the same amount of stretch or compression.
• Understanding \( k \) is crucial for designing systems that require precise spring dynamics, like shock absorbers or timing devices.

Equation of Motion

The equation of motion in simple harmonic movement describes the oscillatory behavior of a particle attached to a spring.

The standard form of this equation is \( y = y_0 \sin (2\pi f t) \), where:

• \( y \) is the displacement at any time \( t \).
• \( y_0 \) is the amplitude of motion.
• \( f \) is the frequency of oscillation.

Our specific equation, \( y = 0.75 \sin 63t \), includes terms that align with this format, where we can clearly identify both amplitude and frequency details.

The equation encapsulates how displacement varies with time and provides a complete description of the particle's behavior throughout its motion cycle. By understanding this equation, you can find out the position of the particle at any given time and predict future movements. This makes it a fundamental part of understanding and utilizing simple harmonic motion in various fields of physics.