Work and energy formulas

θ = angle between the force and the way it moves. Along the motion: θ = 0. Against it: θ = 180. Sideways (normal force): W = 0. Friction: F = μₖ×m×g, θ = 180. On a ramp: F = μₖ×m×g×cos(ramp angle). Gravity: F = m×g, d = height. Up θ = 180, down θ = 0. Stopping distance: W = −½mv₀² (e.g. −0.5×2×5²), F = friction, θ = 180. F vs x graph: W = area under it. F = c×x^n from x₁ to x₂: W = c×(x₂^(n+1) − x₁^(n+1))/(n+1).

Kinetic energy = energy of motion. v is the speed (always +, the direction does not matter). 2 times the speed = 4 times the K. Momentum p given instead of v: type v = p/m (e.g. 12/3).

Net work = change in kinetic energy (work-energy theorem). Starts from rest: v₀ = 0. Comes to a stop: v = 0. Wₙₑₜ = work of ALL forces added up (friction work is −). Only friction on flat ground: Wₙₑₜ = −μₖ×m×g×d (e.g. −0.3×2×9.8×5). Stopping distance: use W = Fd cosθ with W = −½mv₀².

h = height above the zero level you pick (often the ground). Below it, h is −. Change in U: type the height change as h (e.g. 12−3). Work done BY gravity = −ΔU. Work to lift something at steady speed = mgh.

Energy at start + Wnc = energy at end. No friction: leave Wnc = 0. No mass given: leave m = 1. Ramp of length d: h = d×sin(θ) (e.g. 4×sin(30)). Friction: Wnc = −μₖ×m×g×d. On a ramp: −μₖ×m×g×cos(θ)×d. Spring pushes it off: Wnc = ½kx² (e.g. 0.5×400×0.1²). It squeezes a spring: −½kx². Pendulum: h₀ = L−L×cos(θ). Zero height = lowest point. Energy lost = −Wnc. To find m or Wnc, press DEL on its box first.

Pendulum or swing: how high the bob is above its lowest point. θ = angle from straight down. Then use the energy formula with this h. "To what angle does it swing?": find h with the energy formula, then θ here.

Hooke's law: force to hold a spring stretched or squeezed by x. Find k here first, then use U = ½kx². Mass hanging at rest: F = m×g (e.g. 2×9.8). The spring pulls back the other way (F = −kx); type sizes here. cm to m: 5 cm = 0.05.

Energy stored in a stretched or squeezed spring. x = change from its natural length, in m (5 cm = 0.05). Work from x₁ to x₂: type x = √(x₂²−x₁²), e.g. √(0.1²−0.05²). Spring launches a mass: energy formula with Wnc = ½kx². How far it gets squeezed (flat floor): get Wnc from the energy formula (v = 0), type it here as U without the −. Dropped from h onto it: x = (m×g + √((m×g)² + 2×k×m×g×h))/k.

Power = how fast work is done (1 W = 1 J per second). W can be any energy change: lifting mgh (e.g. 70×9.8×5), speeding up ½mv². Energy used in a time: leave W empty. 1 hp = 746 W. 1 kW = 1000 W.

Power while moving at speed v. Lifting at steady speed: F = m×g (e.g. 500×9.8). Steady speed against friction or drag: F = that force. Up a slope at steady speed: F = m×g×sin(θ) + friction (e.g. 1200×9.8×sin(5)+500). Force at an angle: type F×cos(θ) as F.