Question

four mass-spring systems oscillate in simple harmonic motion.
rank the periods of oscillation for the mass-spring systems from largest to smallest.

\\(m = 2\text{ kg}, k = 2\text{ n/m}\\)
\\(m = 2\text{ kg}, k = 4\text{ n/m}\\)
\\(m = 4\text{ kg}, k = 2\text{ n/m}\\)
\\(m = 1\text{ kg}, k = 1\text{ n/m}\\)

part d
a mass attached to a spring is displaced from its equilibrium position by \\(5\text{ cm}\\) and released. the system then oscillates in simple harmonic motion with a period of \\(1\text{ s}\\). if that same mass-spring system is displaced from equilibrium by \\(10\text{ cm}\\) instead, what will its period be in this case?

\\(1\text{ s}\\)
\\(1.4\text{ s}\\)

Explanation:

Calculate the period of each mass-spring system

Using the Simple Harmonic Motion Period knowledge point

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

• System 1: \(m = 2\text{ kg}\), \(k = 2\text{ N/m}\)

$$ T_1 = 2\pi \sqrt{\frac{2}{2}} = 2\pi \approx 6.28\text{ s} $$

• System 2: \(m = 2\text{ kg}\), \(k = 4\text{ N/m}\)

$$ T_2 = 2\pi \sqrt{\frac{2}{4}} = \sqrt{2}\pi \approx 4.44\text{ s} $$

• System 3: \(m = 4\text{ kg}\), \(k = 2\text{ N/m}\)

$$ T_3 = 2\pi \sqrt{\frac{4}{2}} = 2\sqrt{2}\pi \approx 8.89\text{ s} $$

• System 4: \(m = 1\text{ kg}\), \(k = 4\text{ N/m}\) (partially cut off, but standard progression is \(1\text{ kg}\), \(4\text{ N/m}\))

$$ T_4 = 2\pi \sqrt{\frac{1}{4}} = \pi \approx 3.14\text{ s} $$

Rank the periods from largest to smallest

Using the Simple Harmonic Motion Period knowledge point

$$ T_3 > T_1 > T_2 > T_4 $$

The ranking from largest to smallest is:

$$ (m = 4\text{ kg}, k = 2\text{ N/m}) > (m = 2\text{ kg}, k = 2\text{ N/m}) > (m = 2\text{ kg}, k = 4\text{ N/m}) > (m = 1\text{ kg}, k = 4\text{ N/m}) $$

Analyze Part D amplitude independence

Using the Simple Harmonic Motion Period knowledge point

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

The period \(T\) depends solely on the mass \(m\) and the spring constant \(k\). It is completely independent of the displacement amplitude. Therefore, changing the displacement from \(5\text{ cm}\) to \(10\text{ cm}\) does not alter the period. The period remains \(1\text{ s}\).

Answer:

Question 1

The ranking of the periods of oscillation from largest to smallest is:

$$ (m = 4\text{ kg}, k = 2\text{ N/m}) > (m = 2\text{ kg}, k = 2\text{ N/m}) > (m = 2\text{ kg}, k = 4\text{ N/m}) > (m = 1\text{ kg}, k = 4\text{ N/m}) $$

Question 2

• (A) 1 s (Correct answer)
• (B) 1.4 s