Overview
This lecture covers the physics of springs, focusing on their elastic properties, uses, and the application of Hooke's Law to solve a range of conceptual and quantitative problems.
Introduction to Springs
- Springs are elastic materials that return to their original shape after being stretched or compressed (within limits).
- Springs are found in many devices, from pens to vehicles.
- They can break if stretched or compressed beyond their elastic limit.
Hooke's Law
- Hooke’s Law describes a linear relationship between force (F) applied to a spring and its displacement (x): ( F = kx ).
- ( x ) is the change in length from the spring’s natural equilibrium position, not total length.
- ( k ), the spring constant, measures stiffness; units are N/m (Newtons per meter).
- Sometimes written with an absolute value or negative sign, but in calculations, usually the magnitude is used.
Solving Hooke’s Law Problems
- To find the spring constant: ( k = F / x ).
- Example: A 25 N force stretches a spring by 0.03 m. ( k = 830 ) N/m (using sig figs).
- If force is multiplied, displacement is multiplied by the same amount (linear relationship).
- Ex: Tripling the force triples the stretch/compression.
- Hooke’s Law works for both stretching and compressing a spring.
Practice Problems
- Calculating Spring Constant: Use ( F = kx ), rearrange to solve for the unknown.
- Conceptual Relationships: If spring stretches 1 cm with force F, it stretches 3 cm with force 3F.
- Comparing Masses:
- Larger masses increase the force (due to gravity), thus increasing stretch by the same factor.
- Example: If a mass increases 350 times, so does the stretch.
- Spring Compression in Accelerating Systems:
- In an accelerating elevator, use ( F_{normal} = m(g+a) ) for normal force on spring.
- Plug this into Hooke’s Law to find compression.
- Spring Pulling on a Box:
- Use kinematics to find acceleration, then ( F = ma ), and then solve for spring stretch using ( F = kx ).
- Add unstretched length to stretch for total length.
- Spring on an Inclined Plane:
- Resolve gravitational force parallel to ramp: ( F_{gx} = mg \sin(\theta) ).
- Set this equal to spring force, solve for x.
Key Terms & Definitions
- Elastic Material — a material that returns to its original shape after deformation.
- Hooke’s Law — ( F = kx ); force on a spring is proportional to its displacement.
- Spring Constant (k) — a measure of a spring's stiffness (N/m).
- Equilibrium Position — the natural, unstressed length of a spring.
- Restoring Force — the force exerted by a spring to return to equilibrium.
Action Items / Next Steps
- Practice solving Hooke’s Law problems using different scenarios and varying given information.
- Understand when and how to use unit conversions (e.g., cm to m).
- Review concepts of forces in accelerating systems and inclined planes for exams.