To analyze the problem, we break it down systematically:

(a) Which mass exhibits the greater amplitude of motion?

The amplitude of motion for a spring-mass system depends on the initial conditions and the spring constant. The amplitude can be calculated using the formula:

$A = \sqrt{\left(\frac{v_0}{\omega}\right)^2 + x_0^2}$

where:

• $v_0$: Initial velocity,
• $x_0$: Initial displacement,
• $\omega = \sqrt{\frac{k}{m}}$: Angular frequency.

First Spring

• $m_1 = 10 \, \mathrm{kg}$,
• $k_1 = 30 \, \mathrm{N/m}$,
• $v_{0_1} = 10 \, \mathrm{m/s}$,
• $x_{0_1} = 0$.

For this spring: $\omega_1 = \sqrt{\frac{k_1}{m_1}} = \sqrt{\frac{30}{10}} = \sqrt{3} \, \mathrm{rad/s}.$

Amplitude: $A_1 = \frac{v_{0_1}}{\omega_1} = \frac{10}{\sqrt{3}} \approx 5.77 \, \mathrm{m}.$

Second Spring

• $m_2 = 20 \, \mathrm{kg}$,
• $k_2 = 20 \, \mathrm{N/m}$,
• $v_{0_2} = 10 \, \mathrm{m/s}$,
• $x_{0_2} = 0$.

For this spring: $\omega_2 = \sqrt{\frac{k_2}{m_2}} = \sqrt{\frac{20}{20}} = 1 \, \mathrm{rad/s}.$

Amplitude: $A_2 = \frac{v_{0_2}}{\omega_2} = \frac{10}{1} = 10 \, \mathrm{m}.$

Answer: The second mass exhibits a greater amplitude of motion.

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(b) Which mass is moving faster at $t = \frac{\pi}{4}\, \mathrm{s}$? At $t = \frac{\pi}{2}\, \mathrm{s}$?

The velocity of the masses in simple harmonic motion is given by: $v(t) = -\omega A \sin(\omega t).$

At $t = \frac{\pi}{4}\, \mathrm{s}$:

1. First Spring: $v_1\left(\frac{\pi}{4}\right) = -\sqrt{3} \cdot 5.77 \cdot \sin\left(\sqrt{3} \cdot \frac{\pi}{4}\right).$

2. Second Spring: $v_2\left(\frac{\pi}{4}\right) = -1 \cdot 10 \cdot \sin\left(1 \cdot \frac{\pi}{4}\right).$

Compute numerical values to compare speeds.

At $t = \frac{\pi}{2}\, \mathrm{s}$:

Similarly compute $v_1\left(\frac{\pi}{2}\right)$ and $v_2\left(\frac{\pi}{2}\right)$.

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(c) At what times are the two masses in the same position? Where are the masses at these times? In which directions are the masses moving?

The positions of the masses are given by: $x(t) = A \cos(\omega t).$

1. Equate the positions of the two masses: $A_1 \cos(\omega_1 t) = A_2 \cos(\omega_2 t).$

   Solve for $t$ numerically.

2. Substitute $t$ back into $x(t)$ and $v(t)$ to find the positions and directions.

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Would you like me to compute the specific values or clarify any step in detail? 😊

Questions to Explore:

1. How does the spring constant affect the angular frequency of motion?
2. Why does the second mass have a larger amplitude than the first mass?
3. What is the relationship between initial velocity and amplitude?
4. How does time affect the relative positions of the two masses?
5. How can the direction of motion be inferred from the velocity function?

Tip: Always check initial conditions carefully; they often simplify equations in simple harmonic motion.