Problem 59 - University Physics with Modern Physics

Question

An unhappy \(0.300-\mathrm{kg}\) rodent, moving on the end of a spring with force constant \(k=2.50 \mathrm{N} / \mathrm{m},\) is acted on by a damping force \(F_{x}=-b v_{x}\) (a) If the constant \(b\) has the value \(0.900 \mathrm{kg} / \mathrm{s},\) what is the frequency of oscillation of the rodent? (b) For what value of the constant \(b\) will the motion be critically damped?

Step-by-Step Solution

Answer

(a) Frequency of oscillation: 0.393 Hz; (b) Critical damping constant: 1.732 kg/s.

Step 1: Determine the natural frequency

The natural frequency of an undamped system is given by the formula:\[ \omega_0 = \sqrt{\frac{k}{m}} \]Substituting the given values, \( k = 2.50 \) N/m and \( m = 0.300 \) kg:\[ \omega_0 = \sqrt{\frac{2.50}{0.300}} = \sqrt{8.33} \approx 2.89 \text{ rad/s} \]

Step 2: Calculate damped frequency of oscillation

The damped frequency of oscillation \( \omega_d \) is related to the natural frequency and the damping coefficient by:\[ \omega_d = \sqrt{\omega_0^2 - \left(\frac{b}{2m}\right)^2} \]Substitute \( \omega_0 = 2.89 \) rad/s, \( b = 0.900 \) kg/s, and \( m = 0.300 \) kg:\[ \frac{b}{2m} = \frac{0.900}{2 \times 0.300} = 1.5 \text{ s}^{-1} \]\[ \omega_d = \sqrt{2.89^2 - 1.5^2} = \sqrt{8.35 - 2.25} = \sqrt{6.1} \approx 2.47 \text{ rad/s} \]

Step 3: Determine the frequency of oscillation

The frequency \( f \) can be found using:\[ f = \frac{\omega_d}{2\pi} \]Substitute \( \omega_d \approx 2.47 \text{ rad/s} \):\[ f = \frac{2.47}{2\pi} \approx 0.393 \text{ Hz} \]

Step 4: Check condition for critical damping

For critical damping, the condition is:\[ b_c = 2\sqrt{km} \]Calculate \( b_c \) with \( k = 2.50 \) N/m and \( m = 0.300 \) kg:\[ b_c = 2\sqrt{2.50 \times 0.300} = 2\sqrt{0.75} = 2\times 0.866 = 1.732 \text{ kg/s} \]

Key Concepts

Damping Force

When objects like our rodent on a spring oscillate, they might not just keep bouncing forever. That’s where the damping force comes into play. It’s an opposing force that reduces the amplitude of oscillations over time. The damping force can be mathematically expressed as \( F_{x} = -b v_{x} \), where \( b \) is the damping coefficient and \( v_{x} \) is the velocity of the object.

Here’s what happens:

• As the rodent swings back and forth, the damping force acts in the opposite direction of the movement.
• It effectively slows down the oscillations, causing the rodent’s bouncing to eventually come to a stop.
• The damping coefficient \( b \) determines how quickly this stop occurs. Greater values mean faster damping.

Understanding the damping force helps us see why objects don’t oscillate indefinitely in real-life scenarios. This phenomenon prevents everlasting motion, a concept vital in mechanical and engineering applications.

Natural Frequency

The natural frequency of an object is its inherent oscillation speed when it’s not subject to any external forces other than the system’s own restoring force. For a rodent on a spring, natural frequency, noted as \( \omega_0 \), is determined by how the spring constant \( k \) and mass \( m \) interact. The formula we use is:

• \[ \omega_0 = \sqrt{\frac{k}{m}} \]
• From our exercise: \( k = 2.50 \, \text{N/m} \) and \( m = 0.300 \, \text{kg} \)
• This gives \( \omega_0 \) about \( 2.89 \text{ rad/s} \)

The natural frequency tells us how fast the rodent would oscillate without resistance or external influence. It represents the system's pure frequency and is a critical component in understanding how any oscillatory system behaves and is used in various fields, including engineering and physics.

Critical Damping

Critical damping is a special condition in oscillatory systems where the system returns to equilibrium without oscillating. This is achieved with the right amount of damping force, preventing the system from overshooting equilibrium. For the rodent’s spring system, when the damping coefficient \( b \) hits a critical value \( b_c \), we achieve critical damping. Here’s how we calculate \( b_c \):

• \[ b_c = 2\sqrt{km} \]
• Given \( k = 2.50 \, \text{N/m} \) and \( m = 0.300 \, \text{kg} \)
• Calculated \( b_c \) equals \( 1.732 \, \text{kg/s} \)

By reaching this critical damping value, the rodent would come to rest as quickly as possible without making any additional bounces or oscillations. This concept is extremely useful in designing systems like car suspensions or building structures, where stabilization without oscillation is desired.