Two small particles of mass 20 gm and 30 gm are connected with a rigid massless rod of length of 10 cm. The system's center of mass is suspended by a steel wire of torsional spring constant \(c=1.2\times 10^{-8}\) N.m/rad. If the system is slightly rotated in its plane and the angular frequency of its oscillations in rad/sec is \(n\times 10^{-3}\), then write the value of n.
Two small particles of mass 20 gm and 30 gm are connected with a rigid massless rod of length of 10 cm. The system's center of mass is suspended by a steel wire of torsional spring constant \(c=1.2\times 10^{-8}\) N.m/rad. If the system is slightly rotated in its plane and the angular frequency of its oscillations in rad/sec is \(n\times 10^{-3}\), then write the value of n.
Correct Answer: 10
Solution and Explanation
Moment of Inertia and Angular Frequency Calculation
The reduced mass \( m_{eq} \) is given by: \[ \frac{m_1 m_2}{m_1 + m_2} = \frac{(20)(30)}{20 + 30} = 12 \, \text{gm} = 12 \times 10^{-3} \, \text{kg} \]
The moment of inertia \( I_{cm} \) is: \[ I_{cm} = m_{eq} r^2 = (12 \times 10^{-3}) (0.1)^2 = 12 \times 10^{-5} \, \text{kg} \cdot \text{m}^2 \]
The time period \( T \) is given by: \[ T = 2\pi \sqrt{\frac{I_{cm}}{C}} = \omega_n = \sqrt{\frac{C}{I_{cm}}} = \sqrt{\frac{1.2 \times 10^{-8}}{12 \times 10^{-5}}} \]
Solving for \( \omega_n \): \[ \omega_n = 10 \times 10^{-3} \, \text{rad/sec} \]
The angular frequency of its oscillations is \( n \times 10^{-3} \, \text{rad/sec} \), where: \[ n = 10 \]
Answer: The value of \( n \) is \( \boxed{10} \).
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Concepts Used:
Oscillations
Oscillation is a process of repeating variations of any quantity or measure from its equilibrium value in time . Another definition of oscillation is a periodic variation of a matter between two values or about its central value.
The term vibration is used to describe the mechanical oscillations of an object. However, oscillations also occur in dynamic systems or more accurately in every field of science. Even our heartbeats also creates oscillations. Meanwhile, objects that move to and fro from its equilibrium position are known as oscillators.
Read More: Simple Harmonic Motion
Oscillation- Examples
The tides in the sea and the movement of a simple pendulum of the clock are some of the most common examples of oscillations. Some of examples of oscillations are vibrations caused by the guitar strings or the other instruments having strings are also and etc. The movements caused by oscillations are known as oscillating movements. For example, oscillating movements in a sine wave or a spring when it moves up and down.
The maximum distance covered while taking oscillations is known as the amplitude. The time taken to complete one cycle is known as the time period of the oscillation. The number of oscillating cycles completed in one second is referred to as the frequency which is the reciprocal of the time period.