Question

The ratio of force constants of two springs is \(1: 5\). The equal mass suspended at the free ends of both springs are performing S.H.M. If the maximum acceleration for both springs are equal, the ratio of amplitudes for both springs is \(\ldots \ldots\) (A) \((1 / \sqrt{5})\) (B) \((1 / 5)\) (C) \((5 / 1)\) (D) \((\sqrt{5} / 1)\)

Short answer

The ratio of amplitudes for both springs is (C) (5/1).

Step-by-step solution

Write down the known information

We know that the ratio of force constants is 1:5 and the masses on both springs are equal. We're given that their maximum accelerations are equal.

Write down the relation between force constant and maximum acceleration

In simple harmonic motion, the maximum acceleration (a_max) is given by: \(a_{max} = k \cdot A/m\) where \(k\) is the spring constant, \(A\) is the amplitude, and \(m\) is the mass.

Write down the ratio of amplitudes considering the equality of maximum acceleration

Let the force constants of the springs be \(k_1\) and \(k_2\), and their respective amplitudes be \(A_1\) and \(A_2\). Given that the maximum accelerations of both springs are equal, we can write: \( \frac{k_1 \cdot A_1}{m} = \frac{k_2 \cdot A_2}{m} \) Since the masses are equal, they will cancel out: \( k_1 \cdot A_1 = k_2 \cdot A_2\)

Substitute the ratio of force constants into the equation

Given that the ratio of the force constants is 1:5, we can write that \(k_2 = 5k_1\). Let's substitute this into the equation from step 3: \( k_1 \cdot A_1 = (5k_1) \cdot A_2\) Now, divide both sides of the equation by \(k_1\): \( A_1 = 5A_2\)

Find the ratio of amplitudes and the correct answer

The ratio of the amplitude of the first spring to the amplitude of the second spring is: \( \frac{A_1}{A_2} = 5 \) So, the correct answer is (C) (5/1).

Key concepts

Force Constant

When thinking about Simple Harmonic Motion (SHM), the force constant, often represented by the symbol \(k\), plays a crucial role. It defines how stiff or resistant a spring is to being compressed or stretched.

The force constant is directly connected to Hooke’s Law, which states:

• Force exerted by a spring, \(F = -k \cdot x\), where \(x\) is displacement from the equilibrium position.

This means that a larger force constant implies a stiffer spring. In SHM, the force constant is significant because it influences characteristics like frequency and period of the motion.

To compare two different springs, like in our exercise, we look at ratios—specifically, the ratio of their force constants. Here, the given ratio is 1:5, indicating that the second spring is five times "stiffer" than the first. Therefore, this spring will respond differently when a force is applied, affecting other SHM elements.

Amplitude

Amplitude is a key measure in any oscillating system, including Simple Harmonic Motion. It refers to the maximum extent of displacement from the equilibrium position that the object reaches during its oscillation.

In simple terms, amplitude tells us "how far" an object travels away from its central position in SHM. This distance directly affects how vigorous or "broad" the oscillation appears.

• Larger amplitudes imply a bigger swing in motion.
• Smaller amplitudes indicate tighter oscillations.

In our exercise, because the springs have equal maximum accelerations, amplitude differences reflect the springs' force constants. This balance is expressed mathematically as \(k_1 \cdot A_1 = k_2 \cdot A_2\). Hence, with a ratio of force constants as 1:5, the amplitudes must differ inversely. The resulting amplitude ratio \( \frac{A_1}{A_2} = 5 \) shows that the first spring must have a larger amplitude when counterbalancing the difference in stiffness to maintain equal acceleration.

Maximum Acceleration

Maximum acceleration in Simple Harmonic Motion is a critical factor because it determines the swiftest change in velocity that a mass will experience during its cycle. It is given by the formula:

• \(a_{max} = \frac{k \cdot A}{m}\)

Where:

• \(k\) is the spring constant,
• \(A\) is the amplitude,
• \(m\) is the mass.

Here, knowing that the maximum acceleration is the same for both springs, we understand they both reach their highest speed change at the same rate. This equivalence allows for solving amplitude differences by utilizing the force constant ratio.

Through the exercise, rearranging and simplifying the equality \( \frac{k_1 \cdot A_1}{m} = \frac{k_2 \cdot A_2}{m} \) clarified that differences in the force constant are compensated for by differences in amplitude. So, the maximum acceleration helps bridge variations in stiffness and amplitude, giving a complete picture of the SHM dynamics at play.