A mass m is suspended from two springs of spring constants...

A mass $m$ is suspended from two springs of spring constants $k_{1}$ and $k_{2}$ as shown. The time period of vertical oscillations of the mass will be (a) $2 \pi \sqrt{\left(\frac{k_{1}+k_{2}}{m}\right)}$ (b) $2 \pi \sqrt{\frac{m}{\left(k_{1}+k_{2}\right)}}$ (c) $2 \pi \sqrt{\frac{m k_{1} k_{2}}{\left(k_{1}+k_{2}\right)}}$ (d) $2 \pi \sqrt{\frac{m\left(k_{1}+k_{2}\right)}{\left(k_{1} k_{2}\right)}}$

A mass $m$ is suspended from two springs of spring constants $k_{1}$ and $k_{2}$ as shown. The time period of vertical oscillations of the mass will be
(a) $2 \pi \sqrt{\left(\frac{k_{1}+k_{2}}{m}\right)}$
(b) $2 \pi \sqrt{\frac{m}{\left(k_{1}+k_{2}\right)}}$
(c) $2 \pi \sqrt{\frac{m k_{1} k_{2}}{\left(k_{1}+k_{2}\right)}}$
(d) $2 \pi \sqrt{\frac{m\left(k_{1}+k_{2}\right)}{\left(k_{1} k_{2}\right)}}$

Key Concepts

Simple Harmonic Motion (SHM)

Simple harmonic motion describes the oscillatory behavior of systems where the restoring force is directly proportional to the displacement from equilibrium, typically resulting in sinusoidal motion. Its dynamics are governed by Newton’s second law and are characterized by a constant period that depends only on the properties of the system, such as mass and the stiffness or the spring constant.

Effective Spring Constant

When multiple springs are arranged so that they act simultaneously on the same mass, their combined stiffness can be represented by an effective spring constant. For springs connected in such a way that they support the same load (often referred to as a parallel configuration), the effective spring constant is the sum of the individual spring constants, thereby simplifying the analysis of the system's dynamic behavior.

Oscillation Period of a Mass-Spring System

The oscillation period of a mass-spring system in simple harmonic motion is determined by the mass of the object and the effective spring constant. It is commonly expressed by a formula that shows the period is directly proportional to the square root of the mass and inversely proportional to the square root of the effective stiffness. This period is a fundamental property that helps in understanding and predicting the system's response to disturbances.

Transcript

Please give Ace some feedback