To find the amplitude of the motion, we can use the total energy formula for a simple harmonic oscillator:

$E = \frac{1}{2}kA^2$

Where:

• $E$ is the total energy ($1.4 \, \text{J}$),
• $k$ is the spring constant,
• $A$ is the amplitude.

Step 1: Relate the spring constant $k$ to the mass $m$ and the period $T$

The spring constant can be calculated using the formula:

$k = \frac{4\pi^2 m}{T^2}$

Given:

• $m = 0.389 \, \text{kg}$,
• $T = 0.818 \, \text{s}$,

Substitute the values:

$k = \frac{4\pi^2 (0.389)}{(0.818)^2}$

$k \approx \frac{15.327}{0.669} \approx 22.91 \, \text{N/m}$

Step 2: Solve for the amplitude $A$

Rearrange the total energy formula to solve for $A$:

$A = \sqrt{\frac{2E}{k}}$

Substitute $E = 1.4 \, \text{J}$ and $k = 22.91 \, \text{N/m}$:

$A = \sqrt{\frac{2(1.4)}{22.91}}$

$A = \sqrt{\frac{2.8}{22.91}} \approx \sqrt{0.1222} \approx 0.350 \, \text{m}$

Final Answer:

The amplitude of the motion is:

$\boxed{0.350 \, \text{m}}$

Let me know if you want further clarification or details! Here are five related questions you might find helpful:

1. How is the spring constant related to the motion's frequency?
2. How would the amplitude change if the total energy increased?
3. Can you derive the relationship between period and spring constant?
4. How does the mass affect the period of oscillation?
5. What happens to the energy distribution between kinetic and potential energy during oscillation?

Tip: Remember, the total energy in simple harmonic motion is conserved and distributed between potential and kinetic energy!