Solution: Stick-Slip Oscillation
Diagram: Phase Space (v vs x)
(a) Qualitative Mechanism
1. Stick: Initially, friction holds the block to the belt. It moves at constant speed \( u \). The spring force \( kx \) grows linearly.
2. Break: When \( kx = \mu_s mg \), the spring force overcomes static friction. The block slips.
3. Slip: Kinetic friction \( \mu_k mg \) acts. The block performs SHM around a new equilibrium \( x_0 = \mu_k mg / k \).
4. Re-stick: The block slows, reverses, and accelerates forward. When its speed hits \( u \) again (at a lower extension), static friction re-engages.
(b) Deformations
The motion is SHM centered at \( x_0 = \frac{\mu_k mg}{k} \).
The oscillation starts at \( x_{break} = \frac{\mu_s mg}{k} \) with velocity \( u \).
The amplitude \( A’ \) is:
\[ A’ = \sqrt{ (x_{break} – x_0)^2 + \left(\frac{u}{\omega}\right)^2 } \]
\[ x_{max} = x_0 + A’, \quad x_{min} = x_0 – A’ \]
(c) Period Calculation
The total period \( T = t_{stick} + t_{slip} \).
- Stick time: Distance covered is \( 2(x_{break} – x_0) \). Speed is \( u \).
\[ t_{stick} = \frac{2(\mu_s – \mu_k)mg}{uk} \]
- Slip time: Time spent in SHM arc.
\[ t_{slip} = \frac{2}{\omega} \left[ \pi – \tan^{-1}\left( \frac{(\mu_s – \mu_k)g}{u\omega} \right) \right] \]
Total Period T is the sum of linear stick time and harmonic slip time.
Diagram: Phase Space (v vs x)
(a) Qualitative Mechanism
1. Stick: Initially, friction holds the block to the belt. It moves at constant speed \( u \). The spring force \( kx \) grows linearly.
2. Break: When \( kx = \mu_s mg \), the spring force overcomes static friction. The block slips.
3. Slip: Kinetic friction \( \mu_k mg \) acts. The block performs SHM around a new equilibrium \( x_0 = \mu_k mg / k \).
4. Re-stick: The block slows, reverses, and accelerates forward. When its speed hits \( u \) again (at a lower extension), static friction re-engages.
(b) Deformations
The motion is SHM centered at \( x_0 = \frac{\mu_k mg}{k} \).
The oscillation starts at \( x_{break} = \frac{\mu_s mg}{k} \) with velocity \( u \).
The amplitude \( A’ \) is:
\[ A’ = \sqrt{ (x_{break} – x_0)^2 + \left(\frac{u}{\omega}\right)^2 } \]
\[ x_{max} = x_0 + A’, \quad x_{min} = x_0 – A’ \]
(c) Period Calculation
The total period \( T = t_{stick} + t_{slip} \).
- Stick time: Distance covered is \( 2(x_{break} – x_0) \). Speed is \( u \). \[ t_{stick} = \frac{2(\mu_s – \mu_k)mg}{uk} \]
- Slip time: Time spent in SHM arc. \[ t_{slip} = \frac{2}{\omega} \left[ \pi – \tan^{-1}\left( \frac{(\mu_s – \mu_k)g}{u\omega} \right) \right] \]
Total Period T is the sum of linear stick time and harmonic slip time.