GRE 61, small oscillation problem

In summary, the question is asking for the ratio of the frequency of small oscillations of two different configurations of a long, straight, massless rod that pivots about one end in a vertical plane. In configuration I, two small identical masses are attached to the free end, while in configuration II, one mass is moved to the center of the rod. The correct answer is (6/5)^1/2. The formula used to solve this problem is (Mgd/I)^.5, where I is the moment of inertia and Mgd is the reduced mass times its distance from the pivot. The key thing to remember is to calculate I separately for each mass and find the center of mass and reduced mass when calculating Mgd.
  • #1
quantumworld
36
0
A long, straight, and massless rod pivots about one end in a vertical plane. In configuration I, two small identical masses are attached to the free end; in configuration II, one mass is moved to the center of the rod. What is the ratio of the frequency of small oscillations of configuration II to that of configuration I?
(A) (6/5)^1/2
(B) (3/2)^1/2
(C) 6/5
(D) 3/2
(E) 5/3

ok, here is my problem: now we have a pendulum that oscillates normally ( configuration I), and another one ( configuration II) that I have no idea how to calculate its frequency of oscillation. First I thought that the second configuration will have two normal modes, but I messed up, because this is a rod, so it is a rigid body problem .
the correct answer is A. :rolleyes:
thank u so much
 
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  • #2
Using the small angle approximation
[tex]I \frac {d^2 \theta}{dt^2} = -g \theta \Sigma_i m_i L_i[/tex]
where I is the moment of inertia.
 
  • #3
Thank u Tide
I did solve it! :biggrin:
I used the formula (Mgd/I)^.5
where I = I of the first mass + I of the second mass
and Mgd is like the reduced mass times its distance from the pivot.
so the key thing to remember, is that, when we calculate I, we do it
separately for each mass, and when we calculate Mgd, we find the center of mass of the system, and the reduced mass. :biggrin:
I am so happy, that it turned out to be easy.
 

Related to GRE 61, small oscillation problem

1. What is the GRE 61 small oscillation problem?

The GRE 61 small oscillation problem is a commonly used problem in physics and engineering that involves the analysis of the motion of a simple harmonic oscillator. It is often used as an example for studying the behavior of a system undergoing small oscillations around a stable equilibrium point.

2. How is the GRE 61 small oscillation problem solved?

The GRE 61 small oscillation problem is typically solved using the principles of classical mechanics and the equations of motion for a simple harmonic oscillator. This involves identifying the equilibrium point, determining the restoring force, and solving for the motion of the system using the appropriate equations.

3. What are the applications of the GRE 61 small oscillation problem?

The GRE 61 small oscillation problem has many applications in various fields such as physics, engineering, and mathematics. It can be used to analyze the behavior of pendulums, springs, and other oscillating systems. It is also used in the study of waves, vibrations, and resonance phenomena.

4. What are the assumptions made in the GRE 61 small oscillation problem?

The GRE 61 small oscillation problem makes several assumptions, including the absence of friction and damping forces, small amplitudes of oscillation, and a linear restoring force. These assumptions allow for a simplified analysis of the system and provide a good approximation for many real-world situations.

5. How is the GRE 61 small oscillation problem related to the concept of resonance?

The GRE 61 small oscillation problem is closely related to the concept of resonance, which occurs when a system is forced to oscillate at its natural frequency. In the case of the GRE 61 problem, the natural frequency is determined by the properties of the system and the equations of motion. Resonance can lead to large amplitudes of oscillation and can be observed in various systems, such as musical instruments and bridges.

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