91Ó°ÊÓ

A mass \(m\) is attached to one end of a massess spring with a force constant \(k\) and an unstretched length \(l_{0}\) . The other end of the spring is free to turn about a nail driven into a frictionless, horizontal surface (Fig. 13.44\() .\) The mass is made to revolve in a circle with an angular frequency of revolution \(\omega^{\prime}\) , (a) Calculate the length \(l\) of the spring as a function of \(\omega^{\prime} .\) (b) What happens to the result in part (a) when \(\omega^{\prime}\) approaches the natural frequency \(\omega=\sqrt{k} / m\) of the mass-spring system? (ff your result bothers you, remember that massless springs and frictionless surfaces don't exist as such, but are only approximate descriptions of real springs and surfaces. Also, Hooke's law is only an approximation of the way real springs behave; the greater the elongation of the spring, the greater the deviation from Hooke's law.)

Step by step solution

01

Analyze the Forces

When the mass rotates in a circle, it is subject to a centripetal force which is provided by the tension in the spring. The centripetal force is given by the formula \( F_c = m \omega'^2 l \), where \( l \) is the length of the spring and \( \omega' \) is the angular frequency of revolution.

02

Apply Hooke's Law

Hooke's Law states that the force exerted by a spring is proportional to its extension, which can be represented as \( F_s = k (l-l_0) \). Where \( l_0 \) is the natural length of the spring and \( k \) is the spring constant.

03

Equate the Forces

Since the tension in the spring provides the centripetal force, equate the centripetal force to the spring force: \( m \omega'^2 l = k (l-l_0) \).

04

Solve for \( l \)

Rearrange the equation from Step 3 to solve for \( l \). This results in \( l = \frac{k l_0}{k - m \omega'^2} \). This equation expresses the length of the spring as a function of \( \omega' \).

05

Analyze the Behavior as \( \omega' \to \omega \)

The natural frequency \( \omega \) is given by \( \omega = \sqrt{k/m} \). As \( \omega' \) approaches this natural frequency, \( m \omega'^2 \) approaches \( k \), making the denominator approach zero, causing \( l \) to grow very large, theoretically approaching infinity. This indicates an intense elongation which is not realistic and highlights the limitations of the model assumptions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Centripetal Force

Centripetal force plays a crucial role in the motion of objects moving in a circular path. When an object, like a mass attached to a spring, turns around a circle, it does not tend to fly away in a straight line. Instead, it continues its circular trajectory due to the presence of centripetal force.
The centripetal force is always directed towards the center of the circle and is necessary to keep the object moving in a circular path.
Let's look at the formula:

• The centripetal force is given by: \( F_c = m \omega'^2 l \)
• Where \( m \) is the mass, \( \omega' \) is the angular frequency, and \( l \) is the radius of the circle (in this case, the length of the spring).

This force is not a separate force but the resultant of other forces acting on the object. In our mass-spring system, the tension in the spring provides the necessary centripetal force.
This reveals the intrinsic connection between circular motion and forces, illustrating how the change in the length of the spring aligns with the centripetal requirement.

Hooke's Law

Hooke's Law is fundamental in understanding how springs behave. It is essential to remember that this law is an approximation and not absolute, yet it provides valuable insights into elastic behavior.
According to Hooke's Law:

• The force exerted by a spring is proportional to its extension: \( F_s = k (l - l_0) \)
• Here, \( k \) is the spring constant, a measure of spring stiffness, \( l \) is the extended length, and \( l_0 \) is the unstretched length of the spring.

This equation tells us that the farther a spring is stretched (or compressed), the more force it exerts to return to its original length.
In the system discussed, Hooke's Law helps equate the spring force with the centripetal force when the mass rotates, providing a relationship that calculates the spring's new length as angular frequency changes.
It shows how the spring's stiffness influences the entire system's behavior, drawing a direct line from elasticity principles to circular motion applications.

Angular Frequency

Angular frequency is a parameter that unveils the rhythm of rotation or oscillation in systems like the mass-spring model. It is a measure of how fast an object rotates in a circular motion or oscillates in simple harmonic motion.
Consider these points about angular frequency:

• Angular frequency \( \omega' \) relates to the velocity at which the object revolves around the center of the circle and follows: \( \omega' = \frac{2\pi}{T} \)
• Here, \( T \) is the period of one complete revolution.

In terms of our mass-spring system, angular frequency decides how much the spring is stretched.
When angular frequency approaches the natural frequency of the spring-mass system, \( \omega = \sqrt{k/m} \), interesting phenomena occur:

• The terms involved cause the length of the spring to tend towards infinity due to the denominator approaching zero.
• This reflects unrealistic physical conditions, underscoring theoretical boundaries where ideal conditions no longer hold.

Understanding these interactions reveals the complexities of rotational dynamics and how they tie to simple harmonic oscillations, enriching our comprehension of movement in physics.