To solve for the period of oscillation $T$ of a mass undergoing simple harmonic motion, we’ll start by applying the equations for displacement $x(t)$ and velocity $v(t)$ of a simple harmonic oscillator.

Given Data:

• Amplitude, $A = 0.382 \, \text{m}$
• Displacement from equilibrium, $x = 0.14134 \, \text{m}$
• Speed at this displacement, $v = 0.487 \, \text{m/s}$

Step 1: Set up the equations

For simple harmonic motion:

1. The displacement $x(t)$ is given by: $x(t) = A \cos(\omega t)$
2. The velocity $v(t)$ is given by: $v(t) = -A \omega \sin(\omega t)$ where $\omega$ is the angular frequency.

Step 2: Use the identity $\sin^2(\omega t) + \cos^2(\omega t) = 1$

From $x(t) = A \cos(\omega t)$, we have: $\cos(\omega t) = \frac{x}{A}$ Substituting the values: $\cos(\omega t) = \frac{0.14134}{0.382}$ Calculating this gives: $\cos(\omega t) \approx 0.37$

Using $\sin^2(\omega t) + \cos^2(\omega t) = 1$, we find $\sin(\omega t)$: $\sin(\omega t) = \sqrt{1 - \cos^2(\omega t)} = \sqrt{1 - (0.37)^2}$ $\sin(\omega t) \approx 0.9295$

Step 3: Solve for $\omega$ using the velocity equation

We have $v(t) = -A \omega \sin(\omega t)$, so: $\omega = \frac{v}{A \sin(\omega t)}$ Substitute the known values: $\omega = \frac{0.487}{0.382 \times 0.9295}$ Calculating this gives: $\omega \approx 1.37 \, \text{rad/s}$

Step 4: Find the period $T$

The period $T$ is given by: $T = \frac{2\pi}{\omega}$ Substituting $\omega \approx 1.37 \, \text{rad/s}$: $T \approx \frac{2 \pi}{1.37} \approx 4.59 \, \text{s}$

Final Answer:

The period of oscillation $T$ is approximately $4.59 \, \text{s}$.

Would you like more details on any part of this solution?

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Related Questions:

1. What is the maximum speed of the mass during its motion?
2. How does the amplitude of oscillation affect the period?
3. What is the acceleration of the mass at a displacement of 0.14134 m?
4. If the amplitude were doubled, how would the period change?
5. How is the energy distributed between kinetic and potential energy at a displacement of 0.14134 m?

Tip: In simple harmonic motion, the period depends only on mass and spring constants (for spring systems) or length and gravity (for pendulums), not on amplitude.