탄성 위치 에너지 공식 (U = ½kx²): 계산기 & 예제

계산 spring 에너지 with U = ½kx². Link to F = kx, unit traps (cm vs m), 일ed examples, and free Hooke's law and PE calculators.

What 탄성 위치 에너지 Asks

탄성 potential 에너지 is the 에너지 stored in a stretched or compressed spring. Intro physics writes it as U = ½ k x² (also written PEₛ = ½kx² or Us = ½kx²).

k is the spring constant in N/m. x is the 변위 from the unstretched length in meters — not the full coil length. 에너지 comes out in joules.

Hooke's law F = kx gives the 힘; elastic PE stores the 일 done while stretching. 확인 힘 problems on the free Hooke's law calculator, then use U = ½kx² for 에너지.

The 공식

  • U = ½ k x² — elastic potential 에너지 in joules.
  • If you know U and k: |x| = √(2U / k).
  • If you know U and x: k = 2U / x².

    • Because x is squared, stretch and compression store the same 에너지 for the same |x|. Doubling the stretch multiplies stored 에너지 by four.

    How U Connects to F = kx

    The spring 힘 grows linearly with x. The 일 to stretch from 0 to x is the area under the F-vs-x line: a triangle with area ½ × base × height = ½ × x × (kx) = ½kx². That 일 becomes elastic potential 에너지 in an ideal spring.

    For gravitational PE = mgh instead of spring 에너지, see how to 계산 potential 에너지 (mgh).

    단위

  • k → N/m.
  • x → m (convert cm first: divide by 100).
  • U → J (1 J = 1 N·m).

    • 풀이 예제

    예제 1: find U

    k = 200 N/m, x = 0.15 m. U = ½(200)(0.15)² = 100 × 0.0225 = 2.25 J.

    예제 2: leave x in cm by 실수

    Same spring, x written as 15 instead of 0.15. Fake U = ½(200)(15)² = 22,500 J — wrong by a factor of 10,000. Always convert centimeters to meters.

    예제 3: find stretch from 에너지

    U = 0.80 J, k = 160 N/m. x² = 2U/k = 1.6/160 = 0.010. |x| = 0.10 m = 10 cm.

    예제 4: 비교 to Hooke 힘

    k = 200 N/m, x = 0.15 m. 힘 magnitude F = kx = 30 N, but 에너지 U = 2.25 J. Do not treat F and U as interchangeable.

    When the 공식 Fails

  • Past the elastic limit (permanent deformation) — F ≠ kx and U ≠ ½kx².
  • Non-ideal springs with friction or plasticity inside the coil.
  • Using gravitational mgh when the problem is about a spring, or the reverse.

    • 흔한 실수

  • Writing U = kx or U = ½kx instead of ½kx².
  • Leaving x in centimeters while k is in N/m.
  • Using total spring length instead of stretch from equilibrium.
  • Confusing elastic PE with kinetic 에너지 ½mv² or gravitational mgh.

    • 확인 Related Tools

    Hooke's law calculator — F = kx for 힘, k, or x.

    위치 에너지 calculator — gravitational PE = mgh.

    운동 에너지 calculator — KE = ½mv².

    SHM calculator — 질량-spring 주기 T = 2π√(m/k).

    학습 U = ½kx² and keep 힘 problems on F = kx.

    문서

    • 계산 방법 전기 전력 (P = VI, I²R, V²/R)
    • 계산 방법 파동 속도 (v = fλ): 주파수 × 파장
    • 계산 방법 무게 (W = mg): 질량 vs 힘
    • 계산 방법 밀도 (ρ = m/V): 질량 ÷ Volume
    • 계산 방법 가속도 (a = (v − u)/t)
    • 계산 방법 힘 (F = ma): Newton’s Second Law

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