풀이 방법 경사면 Problems (a = g sinθ): 공식s & 예제

풀기 inclined-plane 가속도, mg sinθ, normal 힘, and μ_min = tanθ with clear free-body steps. Verify every answer with a free inclined plane calculator.

What an 경사면 Problem Asks

An inclined plane (ramp) tilts the support surface by an angle θ above the horizontal. 무게 still points straight down, but the surface only pushes perpendicular to itself. Resolving mg into components parallel and perpendicular to the ramp is the key step in almost every home일 problem.

Once those components are clear, 가속도, friction thresholds, and normal 힘 follow from Newton’s laws along and into the plane.

If you already have angle, 질량, μ, and g and only need a 확인ed number, open the free inclined plane calculator first, then return here for free-body traps and 일ed examples.

Core 공식s

  • F∥ = mg sinθ — component of 무게 down the plane.
  • N = mg cosθ — normal 힘 into the plane (no other perpendicular 힘s).
  • a = g sinθ — 가속도 down a frictionless incline (질량 cancels).
  • a = g (sinθ − μ cosθ) — sliding down with kinetic friction μ.
  • μ_s,min = tanθ — smallest static coefficient that can hold the block at rest.

    • θ is measured from the horizontal. Swap sin and cos and every answer flips meaning.

    무료-Body Diagram 확인list

  • Draw 무게 mg straight down from the centre of 질량.
  • Draw N perpendicular out of the ramp surface.
  • Draw friction along the ramp, opposing the tendency to slide.
  • 선택 axes along the plane and perpendicular to the plane — not horizontal/vertical — so mg sinθ and mg cosθ appear cleanly.

    • 풀이 예제

    예제 1: frictionless 가속도 at 30°

    g = 9.8 m/s², θ = 30°. a = 9.8 sin(30°) = 9.8 × 0.5 = 4.9 m/s² down the plane. 질량 never 입력s when friction is zero.

    예제 2: parallel 힘 for 2.0 kg

    F∥ = (2.0)(9.8) sin(30°) = 9.8 N. 확인: a = 4.9 m/s² and F = ma = 9.8 N.

    예제 3: minimum μ at 30°

    μ_s,min = tan(30°) ≈ 0.577. If static friction is weaker, the block starts to slide.

    예제 4: kinetic friction

    θ = 30°, μ_k = 0.20, g = 9.8. a = 9.8 (sin30° − 0.20 cos30°) = 9.8 (0.5 − 0.20 × 0.866) ≈ 9.8 × 0.327 ≈ 3.20 m/s² down the plane.

    흔한 실수

  • Using cosθ for the down-plane component.
  • Leaving a calculator in radian mode while θ was given in degrees.
  • Thinking frictionless 가속도 depends on 질량.
  • Mixing static μ_min = tanθ with a kinetic-friction 가속도 problem.

    • For Ff = μN on flat surfaces, see the friction calculator.

    For N on flat or tilted supports, see the normal 힘 calculator.

    확인 With the 무료 계산기

    Inclined plane calculator — a, F∥, N, or μ_min.

    마찰 calculator — Ff = μN.

    힘 calculator — F = ma along a chosen axis.

    가속도 calculator — once a is known from the ramp.

    학습 mg sinθ and mg cosθ with clean axes, then confirm every answer on the calculator.

    문서

    • 탄성 위치 에너지 공식 (U = ½kx²): 계산기 & 예제
    • 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)

    모든 문서 · 모든 계산기 보기