A student places her 490 g physics book on a frictionless table. She pushes the book against a spring, compressing the spring by x = 9.80 cm, then releases the book. What is the book's speed as it slides away? The spring constant is 1550  N/m .

To find the speed of the book as it slides away, we can use the principle of energy conservation. When the student compresses the spring, it stores potential energy, which is then transferred to the book as kinetic energy when the spring is released. 

### Step-by-Step Solution

1. **Convert the Mass and Compression to SI Units:**
   - Mass of the book: \( m = 490 \ \text{g} = 0.490 \ \text{kg} \)
   - Compression of the spring: \( x = 9.80 \ \text{cm} = 0.098 \ \text{m} \)

2. **Calculate the Potential Energy Stored in the Spring:**
   The potential energy \( U \) stored in a compressed (or stretched) spring is given by:
   \[
   U = \frac{1}{2} k x^2
   \]
   where \( k = 1550 \ \text{N/m} \) is the spring constant, and \( x = 0.098 \ \text{m} \) is the compression.

   Plugging in the values:
   \[
   U = \frac{1}{2} \times 1550 \times (0.098)^2
   \]
   \[
   U = \frac{1}{2} \times 1550 \times 0.009604
   \]
   \[
   U = 7.4426 \ \text{J}
   \]

3. **Apply Energy Conservation:**
   The potential energy in the spring is converted to the kinetic energy of the book as it slides away:
   \[
   U = \frac{1}{2} m v^2
   \]
   where \( v \) is the speed of the book.

4. **Solve for the Speed \( v \):**
   \[
   v = \sqrt{\frac{2U}{m}}
   \]
   Substituting \( U = 7.4426 \ \text{J} \) and \( m = 0.490 \ \text{kg} \):
   \[
   v = \sqrt{\frac{2 \times 7.4426}{0.490}}
   \]
   \[
   v = \sqrt{\frac{14.8852}{0.490}}
   \]
   \[
   v = \sqrt{30.3782}
   \]
   \[
   v \approx 5.51 \ \text{m/s}
   \]

### Final Answer:
The speed of the book as it slides away is approximately \( \mathbf{5.51 \ m/s} \).

Would you like further details on any part of this solution?

---

1. How is energy conserved in spring and kinetic energy scenarios?
2. How does spring compression affect potential energy?
3. What changes if friction is added to this system?
4. How does changing the spring constant affect speed?
5. What role does mass play in determining speed?

**Tip:** In spring problems, always verify units, especially for displacement and mass, to ensure correct calculations.

A student places her 490 g physics book on a frictionless table. She pushes the book against a spring, compressing the spring by x = 9.80 cm, then releases the book. What is the book's speed as it slides away? The spring constant is 1550 N/m.

Find the speed of a 490 g book sliding on a frictionless surface after being released from a compressed spring. Learn to apply the principles of energy conservation, kinetic energy, and potential energy in springs using real-world values: spring constant (1550 N/m) and compression (9.80 cm).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation