College Physics
11th Edition
ISBN: 9781305952300
Author: Raymond A. Serway, Chris Vuille
Publisher: Cengage Learning
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![**Problem:**
A mass of 0.78 kg is initially traveling across a horizontal surface. It then reaches a spring with a fixed end having a spring constant of 114 N/m that is positioned horizontally and initially at equilibrium. If the initial speed of the mass is 3.63 m/s as it contacts the spring, what is the speed of the mass after compressing the spring by 0.156 m, given that the coefficient of kinetic friction between the surfaces is 0.61?
**Solution:**
1. **Identify Given Data:**
- Mass (m): 0.78 kg
- Spring constant (k): 114 N/m
- Initial speed (v_i): 3.63 m/s
- Compression distance (x): 0.156 m
- Coefficient of kinetic friction (μ_k): 0.61
2. **Calculate Forces:**
- Friction force \( F_f = \mu_k \times m \times g \)
- Spring force \( F_s = \frac{1}{2} k x^2 \)
3. **Apply Work-Energy Principle:**
- Initial kinetic energy \( KE_i = \frac{1}{2} m v_i^2 \)
- Work done by friction \( W_f = F_f \times x \)
- Final kinetic energy \( KE_f = KE_i - F_s - W_f \)
4. **Solve for Final Speed:**
The final expression for the final speed \( v_f \) can be derived from the energy balance equation:
\[
\frac{1}{2} m v_f^2 = KE_i - W_f - F_s
\]
\[
v_f = \sqrt{\frac{2(KE_i - W_f - F_s)}{m}}
\]
Using computation, the final speed \( v_f \) is approximately 2.79 m/s.
**Conclusion:**
The speed of the mass after compressing the spring by 0.156 m, considering the friction, is approximately 2.79 m/s.](https://content.bartleby.com/qna-images/question/a32b3059-b4f0-4265-9dd3-abf2feacb616/9c5de7a2-7c9d-4f1c-bd2c-a3e9cb122e51/lyyxazw_processed.jpeg)
Transcribed Image Text:**Problem:**
A mass of 0.78 kg is initially traveling across a horizontal surface. It then reaches a spring with a fixed end having a spring constant of 114 N/m that is positioned horizontally and initially at equilibrium. If the initial speed of the mass is 3.63 m/s as it contacts the spring, what is the speed of the mass after compressing the spring by 0.156 m, given that the coefficient of kinetic friction between the surfaces is 0.61?
**Solution:**
1. **Identify Given Data:**
- Mass (m): 0.78 kg
- Spring constant (k): 114 N/m
- Initial speed (v_i): 3.63 m/s
- Compression distance (x): 0.156 m
- Coefficient of kinetic friction (μ_k): 0.61
2. **Calculate Forces:**
- Friction force \( F_f = \mu_k \times m \times g \)
- Spring force \( F_s = \frac{1}{2} k x^2 \)
3. **Apply Work-Energy Principle:**
- Initial kinetic energy \( KE_i = \frac{1}{2} m v_i^2 \)
- Work done by friction \( W_f = F_f \times x \)
- Final kinetic energy \( KE_f = KE_i - F_s - W_f \)
4. **Solve for Final Speed:**
The final expression for the final speed \( v_f \) can be derived from the energy balance equation:
\[
\frac{1}{2} m v_f^2 = KE_i - W_f - F_s
\]
\[
v_f = \sqrt{\frac{2(KE_i - W_f - F_s)}{m}}
\]
Using computation, the final speed \( v_f \) is approximately 2.79 m/s.
**Conclusion:**
The speed of the mass after compressing the spring by 0.156 m, considering the friction, is approximately 2.79 m/s.
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