91Ó°ÊÓ

Elasticity is a fundamental property of materials that allows them to return to their original shape or length after being stretched or compressed. In our exercise, the elastic string exhibits this property. The string’s natural length is the length it retains when no external forces act upon it. When forces such as tension are applied, the string stretches according to its modulus of elasticity. This modulus is a measure of the string’s stiffness, determining how much it stretches under a given force. For this string, the modulus of elasticity is given as 4g N, indicating how resistant it is to deformation.

Hooke's Law is a principle that describes the behavior of elastic materials. It states that the force needed to extend or compress a spring by some distance is proportional to that distance. Mathematically, it is expressed as: \[ F = k \times x \] where:

• F is the force exerted by the spring in newtons (N).
• k is the spring constant or modulus of elasticity, which describes the stiffness of the spring.
• x is the extension or compression of the spring from its natural length in meters (m).

In our context, the extension x of each section of the string is the vertical displacement y since the string’s natural length is 0.5 m. By applying Hooke's Law, we see that the tension T in each string section is given by: \[ T = k \times y \] Given our string's modulus of elasticity, the tension is calculated accordingly to balance the particle's weight at equilibrium.

The period of oscillation refers to the time it takes for an oscillating particle, such as the one in our problem, to complete one full cycle of its motion. This is an important concept in understanding the dynamics of oscillatory systems. The period (T) of small oscillations for our setup can be determined using the formula: \[ T = 2\pi \sqrt{\frac{m}{k_{\text{effective}}}} \] Here, m is the mass of the particle, and \(k_{\text{effective}}\) is the effective spring constant of the system. In this exercise, \(k_{\text{effective}}\) is derived from the combined effect of both stretched string sections. Calculations show that our system can be simplified to have an effective spring constant, leading us to determine the oscillation period relative to this combined spring constant.

Vertical equilibrium occurs when all the forces acting on a particle in the vertical direction sum to zero, meaning the particle is at rest and there is no net force acting on it. In this exercise, the particle's weight and the tensions in the upper and lower string sections must balance each other out. We first calculate the stretched lengths of the string sections using the displacement y from the equilibrium point. This balance can be mathematically expressed as: \( \text{Upward force (tension)} = \text{Downward force (weight)} \) Using the equations of equilibrium, we can solve for y, finding that the equilibrium height above point Q is 0.125 meters.

Particle mechanics involves the study of forces and motion acting on a particle. In this problem, we're looking at a particle of mass 1 kg attached to an elastic string and how it behaves under various forces. When examining such systems:

• Consider the forces: weight (gravity) and tension need to be balanced for equilibrium.
• Using principles like Hooke’s Law helps understand tension in the string.
• Calculating equilibrium positions ensures we identify the rest points in such systems.
• Period of oscillation provides insights into the dynamics and stability of small disturbances.

Detailed understanding of these concepts enables us to predict and analyze the particle's behavior under various physical conditions.