The two ends of a spring are displaced along the length of the spring. All displacements have equal magnitudes. In which case or cases the tension or compression in the spring will have as maximum magnitude?

Options

A. the right end is displaced towards right and the left end towards left
B. both ends are displaced towards right
C. both ends are displaced towards left
D. the right end is displaced towards left and the left end towards right.

Answer

To solve the problem of determining in which case the tension or compression in a spring will have the maximum magnitude when the two ends of the spring are displaced along its length, we will analyze each case step by step.

### Step 1: Understanding the Spring Force Formula
The force exerted by a spring (either tension or compression) is given by Hooke's Law:
\[ F = k \cdot \Delta x \]
where:
- \( F \) is the force (tension or compression),
- \( k \) is the spring constant (a measure of the stiffness of the spring),
- \( \Delta x \) is the change in length of the spring.

### Step 2: Analyzing Each Case
We will analyze four different cases based on how the ends of the spring are displaced.

#### Case 1: Right end displaced to the right and left end displaced to the left
- Displacement of right end = \( y \)
- Displacement of left end = \( y \)
- Initial length of the spring = \( L \)
- Final length = \( L + y + y = L + 2y \)
- Change in length, \( \Delta x = (L + 2y) - L = 2y \)
- Force in the spring:
\[ F = k \cdot 2y \]

#### Case 2: Both ends displaced to the right
- Displacement of both ends = \( y \)
- Final length = \( L + y + y = L + 2y \)
- Change in length, \( \Delta x = (L + 2y) - L = 2y \)
- Force in the spring:
\[ F = k \cdot 2y \]

#### Case 3: Both ends displaced to the left
- Displacement of both ends = \( y \)
- Final length = \( L - y - y = L - 2y \)
- Change in length, \( \Delta x = (L - 2y) - L = -2y \)
- Force in the spring:
\[ F = k \cdot (-2y) = -2ky \]
- The magnitude is \( 2ky \).

#### Case 4: Right end displaced to the left and left end displaced to the right
- Displacement of right end = \( y \) (to the left)
- Displacement of left end = \( y \) (to the right)
- Final length = \( L - y + y = L - 2y \)
- Change in length, \( \Delta x = (L - 2y) - L = -2y \)
- Force in the spring:
\[ F = k \cdot (-2y) = -2ky \]
- The magnitude is \( 2ky \).

### Step 3: Comparing the Magnitudes
From the analysis:
- Case 1: Magnitude of force = \( 2ky \)
- Case 2: Magnitude of force = \( 2ky \)
- Case 3: Magnitude of force = \( 2ky \)
- Case 4: Magnitude of force = \( 2ky \)

The maximum magnitude of tension or compression occurs in Case 1 and Case 4, where the spring is either stretched or compressed to the maximum extent.

### Conclusion
The cases in which the tension or compression in the spring will have the maximum magnitude are:
- **Case 1**: Right end displaced to the right and left end displaced to the left.
- **Case 4**: Right end displaced to the left and left end displaced to the right.