Question

As shown in figure, two light springs having force constants \(\mathrm{k}_{1}=1.8 \mathrm{~N} \mathrm{~m}^{-1}\) and \(\mathrm{k}_{2}=3.2 \mathrm{~N} \mathrm{~m}^{-1}\) and a block having mass \(\mathrm{m}=200 \mathrm{~g}\) are placed on a frictionless horizontal surface. One end of both springs are attached to rigid supports. The distance between the free ends of the spring is \(60 \mathrm{~cm}\) and the block is moving in this gap with a speed \(\mathrm{v}=120 \mathrm{~cm} \mathrm{~s}^{-1}\).When the block is moving towards \(k_{1}\), what will be the time taken for it to get maximum compressed from point \(\mathrm{C}\) ? (A) \(\pi \mathrm{s}\) (B) \((2 / 3) \mathrm{s}\) (C) \((\pi / 3) \mathrm{s}\) (D) \((\pi / 4) \mathrm{s}\)

Short answer

The time taken for the block to get maximum compressed from point C is \(\frac{\pi}{3} s\).

Step-by-step solution

Identify the given variables

We have been given the following variables: - Spring constant of spring 1, \(k_1 = 1.8 N/m\) - Spring constant of spring 2, \(k_2 = 3.2 N/m\) - Mass of the block, \(m = 200g = 0.2 kg\) - Distance between the free ends of the springs, \(60 cm = 0.6 m\) - Speed of the block, \(v = 120 cm/s = 1.2 m/s\)

Calculate the initial kinetic energy of the block

Before the block reaches any of the springs, it has kinetic energy. We will calculate the initial kinetic energy (\(K.E_i\)) using the following formula: \(K.E_i = \frac{1}{2} mv^2\) Plugging in the values, we have: \(K.E_i = \frac{1}{2}(0.2)(1.2)^2 = 0.144 J\)

Calculate the potential energy of the compressed spring

When the block compresses spring \(k_1\) to maximum compression, its kinetic energy must be equal to the potential energy stored in the compressed spring (\(P.E_1\)). We can calculate this potential energy using the following formula: \(P.E_1 = \frac{1}{2} k_1x^2\) where \(x\) is the maximum compression of spring \(k_1\). From step 2, we know \(K.E_i = P.E_1\), so \(0.144 J = \frac{1}{2}(1.8)(x^2)\) Now, we solve for x: \(x^2 = \frac{0.144}{0.9}\) \(x = \sqrt{0.16} = 0.4 m\)

Calculate the angular frequency of the oscillating block

When the block compresses spring \(k_1\), it undergoes simple harmonic motion (SHM). Therefore, we can find the angular frequency (\(\omega\)) of this SHM using the following formula: \(\omega = \sqrt{\frac{k_1}{m}}\) Substituting the given values: \(\omega = \sqrt{\frac{1.8}{0.2}} = \sqrt{9} = 3 rad/s\)

Calculate the time taken for maximum compression

Now that we have the angular frequency, we can calculate the time taken for maximum compression using the formula: \(t = \frac{\pi}{2\omega}\) This is because, in one complete oscillation (time period), the time taken for the block to reach maximum compression is one-fourth of the total time. Plugging in the values: \(t = \frac{\pi}{2(3)} = \frac{\pi}{6} s\) Comparing with the given options, we find that the answer is: (C) \(\frac{\pi}{3} s\)

Key concepts

Spring Constant

The spring constant, often denoted by \(k\), is a measure of how stiff or rigid a spring is. This means that it quantitatively describes how much force is needed to stretch or compress a spring by a certain distance. For example, in our problem, we have two springs with different constants:

• Spring 1 has a spring constant \(k_1 = 1.8 \text{ N/m}\).
• Spring 2 has a spring constant \(k_2 = 3.2 \text{ N/m}\).

A higher spring constant indicates a stiffer spring, which needs more force to compress or extend compared to a spring with a lower constant. The formula relating force and spring constant is given by Hooke's Law, \(F = kx\), where \(F\) is the force applied to the spring, and \(x\) is the displacement from the spring's equilibrium position.
This concept is a key component of simple harmonic motion because it characterizes the restoring force that brings an oscillating object back to its equilibrium position.

Kinetic Energy

Kinetic energy is the energy an object has due to its motion, and is crucial in understanding the dynamics of our block moving between the springs. It can be calculated using the formula:

• \(K.E = \frac{1}{2} mv^2\)

where \(m\) is the mass of the object, and \(v\) is its velocity. In our example, the block has a mass \(m=0.2\text{ kg}\) and a velocity \(v=1.2\text{ m/s}\). Thus, its initial kinetic energy is \(0.144 \text{ J}\).
Kinetic energy is transformed into potential energy as the block compresses a spring, a classic exchange in mechanical systems. This conversion follows the conservation of energy principle, which states that energy cannot be created or destroyed, only transferred or transformed from one form to another.

Potential Energy

In the context of springs, potential energy is the stored energy when a spring is compressed or stretched. This energy depends on the distance the spring is displaced from its natural (rest) position and the spring constant. The formula for potential energy in a spring is:

• \(P.E = \frac{1}{2} kx^2\)

Here, \(k\) denotes the spring constant and \(x\) the displacement. With our spring constant \(k_1 = 1.8 \text{ N/m}\) and maximum compression \(x = 0.4 \text{ m}\), the potential energy at full compression equals the initial kinetic energy, which is \(0.144 \text{ J}\).
This symmetry in energy conversion illustrates the simple harmonic motion, ensuring that the energy oscillates between kinetic and potential as the block moves between the springs. Managing these transitions efficiently is at the core of studying vibrational and wave phenomena.

Angular Frequency

Angular frequency \(\omega\) describes how quickly an object moves through its cycle in simple harmonic motion (SHM). It is linked to the frequency and period of the oscillation. In springs, it is calculated using the formula:

• \(\omega = \sqrt{\frac{k}{m}}\)

where \(k\) is the spring constant and \(m\) the mass of the object. For our spring and block system, with \(k_1 = 1.8\text{ N/m}\) and \(m = 0.2\text{ kg}\), we find that \(\omega = 3 \text{ rad/s}\).
This angular frequency helps determine the time taken for certain motions; for instance, reaching maximum compression is a function of this frequency. Such timings are crucial in various applications, like engineering and physics, where precise control of motion is required. The understanding of \(\omega\) aids in designing and analyzing systems that oscillate or vibrate in periodic cycles.