Math Problem Statement
A spring has a natural length of 40 cm. If a 60-N force is required to keep the spring compressed 10 cm, how much work is done during this compression? How much work is required to compress the spring to a length of 25 cm?
Solution
To solve these problems, we use Hooke's Law and the work integral:
Step 1: Find the Spring Constant
Hooke's Law states that the force required to compress or stretch a spring is:
where:
- is the force applied,
- is the spring constant,
- is the displacement from the natural length.
Given:
- The natural length of the spring is 40 cm.
- A 60 N force is required to compress it by 10 cm.
Thus,
or in meters:
Step 2: Calculate Work Done for 10 cm Compression
The work done to compress a spring is given by the integral:
For compression from to cm (0.1 m):
Solving:
Step 3: Calculate Work Done for Compression to 25 cm
A length of 25 cm means the displacement is:
Now, we calculate:
Final Answers:
- Work required for 10 cm compression: 3 Joules.
- Work required for 15 cm compression (to 25 cm length): 6.75 Joules.
Would you like a more detailed explanation of any step?
Related Questions
- How much work is required to stretch the spring by 15 cm instead of compressing it?
- If we double the spring constant, how much work would be required for the same compression?
- What is the force required to compress the spring to 25 cm?
- What would be the natural frequency of oscillation for a mass attached to this spring?
- How would the work required change if the spring obeyed a nonlinear force law?
Tip: The work done in compressing a spring follows a quadratic relationship with displacement, meaning doubling the compression distance quadruples the required work!
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Math Problem Analysis
Mathematical Concepts
Physics
Work and Energy
Spring Mechanics
Calculus
Formulas
F = kx
W = ∫ kx dx
Theorems
Hooke's Law
Work Integral
Suitable Grade Level
Grades 11-12
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