A 2.0 kg box rests on a plank that is inclined at an angle of 65° above the horizontal. The upper end of the box is attached to a spring with a force constant of 19 N/m, as shown in Figure 6-32. If the coefficient of static friction between the box and the plank is 0.21, what is the maximum amount the spring can be stretched and the box remain at rest?

Question

A 2.0 kg box rests on a plank that is inclined at an angle of 65° above the horizontal. The upper end of the box is attached to a spring with a force constant of 19 N/m, as shown in Figure 6-32. If the coefficient of static friction between the box and the plank is 0.21, what is the maximum amount the spring can be stretched and the box remain at rest?

Answer 1

To find the maximum amount the spring can be stretched and the box remain at rest, we need to consider the forces acting on the box and find the condition when the static friction just becomes maximum. Here's how we can approach this problem:

Step 1: Identify the forces acting on the box:
- Weight force (mg): acting vertically downward
- Normal force (N): perpendicular to the inclined plane
- Static friction force (fs): parallel to the inclined plane and opposes motion
- Force from the spring (Fspring): parallel to the inclined plane

Step 2: Resolve the weight force and normal force components:
- Weight force component parallel to the inclined plane: mg * sin(65°)
- Normal force component perpendicular to the inclined plane: mg * cos(65°)

Step 3: Determine the maximum static friction force:
The maximum static friction force can be calculated using the coefficient of static friction (μs) and the normal force component.
- Maximum static friction force (fs-max): μs * Normal force component

Step 4: Find the maximum stretch of the spring:
The spring force is opposing the static friction force. Therefore, the maximum stretch of the spring will occur when the spring force equals the maximum static friction force.
- Fspring = fs-max
- k * stretch = fs-max
- stretch = fs-max / k

Step 5: Calculate the maximum stretch of the spring:
- Substitute the values into the equation:
- stretch = (μs * Normal force component) / k
- stretch = (0.21 * mg * cos(65°)) / k

Step 6: Substitute the known values to find the final result:
- m = 2.0 kg (mass of the box)
- g = 9.8 m/s^2 (acceleration due to gravity)
- k = 19 N/m (force constant of the spring)

Now, plug in the values and calculate the maximum stretch of the spring:

stretch = (0.21 * 2.0 kg * 9.8 m/s^2 * cos(65°)) / 19 N/m

Converting the angle from degrees to radians:

stretch = (0.21 * 2.0 kg * 9.8 m/s^2 * cos(65 * π/180)) / 19 N/m

Finally, calculate the value of the maximum stretch.

To find the maximum amount the spring can be stretched and the box remain at rest, we need to consider the forces acting on the box and find the condition when the static friction just becomes maximum. Here's how we can approach this problem:

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