To find the period $T$ of the oscillating mass-spring system, we can use the formula for the period of a simple harmonic oscillator involving a spring:

$T = 2 \pi \sqrt{\frac{m}{k}}$

where:

• $T$ is the period of the system,
• $m = 0.4 \, \text{kg}$ is the mass attached to the spring,
• $k$ is the spring constant, which we can find using Hooke's Law and the initial conditions given.

Step 1: Determine the spring constant $k$

From Hooke's Law:

$F = k \Delta y$

Since the mass is in equilibrium initially, the force due to gravity is balanced by the spring force:

$mg = k \Delta y_0$

Solving for $k$:

$k = \frac{mg}{\Delta y_0}$

Given:

• $m = 0.4 \, \text{kg}$,
• $g = 9.8 \, \text{m/s}^2$ (acceleration due to gravity),
• $\Delta y_0 = 0.1 \, \text{m}$ (initial displacement due to mass),

we find:

$k = \frac{(0.4)(9.8)}{0.1} = 39.2 \, \text{N/m}$

Step 2: Calculate the period $T$

Now that we know $k = 39.2 \, \text{N/m}$, we can substitute back into the period formula:

$T = 2 \pi \sqrt{\frac{0.4}{39.2}}$

Calculating the inside of the square root:

$\frac{0.4}{39.2} \approx 0.0102$

Then,

$T = 2 \pi \sqrt{0.0102} \approx 2 \pi \times 0.101 \approx 0.635 \, \text{s}$

Answer

The period $T$ of this system is approximately:

$T \approx 0.635 \, \text{s}$

Would you like more details on the steps, or do you have any questions?

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Related Questions:

1. How would the period change if the mass were doubled?
2. How would the period change if the spring constant were halved?
3. What is the maximum velocity of the mass during its oscillation?
4. How would the period change if the system were on the moon with a different gravitational constant?
5. How would damping affect the period and amplitude over time?

Tip: For a mass-spring system, the period depends only on the mass and the spring constant, not on the amplitude of oscillation, as long as it remains within the elastic limit of the spring.