The natural angular frequency of a simple harmonic oscillator (spring mass system) of
mass 2 gm is 0.8 Hz, if it undergoes critically damped motion when taken to a viscous medium,
the damping force on the oscillator is given by (speed of the oscillator is 2 mm/sec) in cgs

The problem involves a spring-mass system that undergoes critically damped motion. We need to find the damping force on the oscillator in the CGS (centimeter-gram-second) system, given the mass, natural angular frequency, and the velocity of the oscillator.

### Step 1: Identify the given information
- **Mass of the oscillator (m):** 2 gm = 2 grams
- **Natural angular frequency (ω₀):** 0.8 Hz (note that frequency in Hz can be converted to angular frequency in radians per second)
- **Speed of the oscillator (v):** 2 mm/sec = 0.2 cm/sec (in CGS)

### Step 2: Natural frequency formula
In simple harmonic motion, the natural angular frequency \(\omega_0\) is related to the spring constant \(k\) and mass \(m\) by the formula:
\[
\omega_0 = \sqrt{\frac{k}{m}}
\]
where:
- \( \omega_0 = 2\pi f \)
- \( f = 0.8 \text{ Hz} \)

So,
\[
\omega_0 = 2\pi(0.8) = 5.026 \, \text{radians/sec}
\]

### Step 3: Damping coefficient (critical damping)
For critical damping, the damping coefficient \(b_c\) is given by:
\[
b_c = 2m\omega_0
\]
where \(m\) is the mass in grams and \(\omega_0\) is the angular frequency. Substituting the values:
\[
b_c = 2(2)(5.026) = 20.104 \, \text{g/sec}
\]

### Step 4: Damping force
The damping force \(F_d\) is given by:
\[
F_d = b_c v
\]
Substituting the values for \(b_c = 20.104 \, \text{g/sec}\) and \(v = 0.2 \, \text{cm/sec}\):
\[
F_d = 20.104 \times 0.2 = 4.0208 \, \text{dynes}
\]

### Final Answer:
The damping force on the oscillator is **4.0208 dynes** in the CGS system.

Would you like more details or explanations?

---

**Relative Questions:**
1. What is the formula for natural angular frequency in terms of mass and spring constant?
2. How is critically damped motion different from underdamped or overdamped motion?
3. Why is the damping force proportional to velocity in a viscous medium?
4. How do we convert velocity from mm/sec to cm/sec in the CGS system?
5. What is the significance of critical damping in oscillatory systems?

**Tip:** Always convert units to the same system (like CGS or SI) when performing calculations to avoid errors.

The natural angular frequency of a simple harmonic oscillator (spring mass system) of mass 2 gm is 0.8 Hz, if it undergoes critically damped motion when taken to a viscous medium, the damping force on the oscillator is given by (speed of the oscillator is 2 mm/sec) in cgs.

Learn how to calculate the damping force for a critically damped spring-mass system in a viscous medium. This problem covers simple harmonic motion, damping force formulas, and converting units to CGS. Suitable for undergraduate students studying physics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation