- Determine the elastic potential energy from the area under a force–extension graph within the limit of proportionality
- Recall and apply (E_P = tfrac{1}{2} F x) and (E_P = tfrac{1}{2} k x^2) for elastic materials
- Interpret and sketch force–extension graphs for springs and wires
- Calculate spring constant (k = frac{F}{x}) and energy stored for given extensions
- Use terms “elastic potential energy,” “limit of proportionality,” “force–extension graph” correctly in English
- Describe graph shapes (“linear,” “curved,” “area under the curve”) accurately
- Explain calculations and reasoning in clear, academic English
- Interpret problem statements and express results using precise physics language
| English Term | Russian | Kazakh |
|---|---|---|
| Elastic potential energy | Упругая потенциальная энергия | Серпімді потенциалдық энергия |
| Limit of proportionality | Предел пропорциональности | Пропорционалдылық шегі |
| Force–extension graph | График сила–удлинение | Күш–ұзарту графигі |
| Spring constant ((k)) | Жёсткость пружины | Пружина тұрақтылығы |
| Extension ((x)) | Удлинение | Ұзарту |
| Load ((F)) | Нагрузка | Жүк |
Elastic potential energy
Definition: Energy stored when a material is deformed elastically
Formula: (E_P = tfrac{1}{2}Fx = tfrac{1}{2}kx^2)
Limit of proportionality
Definition: Maximum load where (F propto x) holds
Note: Beyond this, material yields permanently
Force–extension graph
Use: Graphical representation of (F) vs (x)
Key: Area under line = energy stored
Spring constant ((k))
Definition: (k = frac{F}{x}), stiffness measure
Units: N/m
Elastic potential energy
Energy stored in a material when it is deformed elastically, equal to the work done to deform it.
Russian: Энергия, накопленная при упругой деформации, равная работе, проделанной для деформации.Kazakh: Серпімді деформация кезінде жинақталған энергия, оны деформациялау үшін жасалған жұмысқа тең.
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Limit of proportionality
The maximum force for which the extension of the material remains directly proportional to the force applied.
Russian: Максимальная сила, при которой удлинение остается прямо пропорциональным приложенной силе.Kazakh: Қолданылатын күшке ұзарту тура пропорционалды болатын ең үлкен күш.
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Area under Force–Extension Graph
For an elastic material within its , the graph of force (F) versus extension (x) is a straight line from the origin to ((x, F)). The elastic potential energy (E_P) stored is the area under this line:
(E_P = text{area} = tfrac{1}{2} times F times x)
Using Hooke’s Law
Because (F = k x) in the linear region, substitute to get:
(E_P = tfrac{1}{2} k x^2)
Match the Formula
- Energy from area under graph: ______
- Energy using Hooke’s law: ______
- Spring constant: ______
- Limit beyond which Hooke’s law fails: ______
Example 1: Area Method
A wire is extended by 0.08 m under 10 N. Find (E_P).
Example 2: Using (k)
A spring has (k=200) N/m and is stretched 0.05 m. Calculate (E_P).
Use the PhET “Hooke’s Law” simulation to measure energy stored:
In pairs, design and carry out an experiment to verify (E_P = tfrac12 k x^2). Share results and discuss deviations.
Solve these structured questions:
- Derive (E_P = tfrac12 k x^2) by integrating (F(x)) from 0 to (x).
- A spring with constant 150 N/m is compressed 0.04 m. Find energy stored.
- Explain why area under a non-linear graph requires calculus beyond proportionality limit.
- Compare energy stored in two springs of different (k) but same extension.
- Design a real-world application where elastic potential energy is crucial; calculate required parameters.
- Which method (area vs formula) did you find clearer for calculating (E_P)?
- What challenges arise when the graph is non-linear?
- How will you ensure accurate measurements of (k) and (x) in experiments?
- Describe an everyday device that stores elastic potential energy.