Work, Energy and Power

Basic Work

W = \vec{F} \cdot \vec{S}

Work with angle

W = F S \cos\theta

Work in 3D path

$$W = \int{x1}^{x2} Fx \

Work by Spring

W = -\frac{1}{2} k (xf^2 - xi^2)

Potential Energy of Spring

W{\text{ext}} = \frac{1}{2} k (xf^2 - x_i^2)

Potential Energy

\text{Work by Agent}

Power

P = \frac{m g h}{t}

Power

P = \frac{W}{t}

Power

P = \frac{F S}{t}

Efficiency

\eta = \frac{P'}{P_{\text{total}}}

Work (Well/Tank)

W = m' g h'

Average height (Well/Tank)

h' = \frac{hi + hf}{2}

Work (Bricks)

W = m g n (n-1) \frac{d}{2}

Work to straighten lying rod

W = m g \frac{l}{2}

Work to straighten lying brick

W = m g \left(\frac{l}{2} - \frac{h}{2}\right)

Work to straighten lying cylinder

W = m g \left(\frac{l}{2} - \frac{d}{2}\right)

Work (Shackle lifting general)

W = m' g h'

Work (Shackle lifting specific)

W = \frac{m g l}{2 n^2}

Work by Bob

W = m g h

Bob height (horizontal displacement)

h = l - \sqrt{l^2 - x^2}

Bob height (angle)

h = l (1 - \cos\theta)

Bob height (half-angle formula)

h = 2 l \sin^2\frac{\theta}{2}

Work-Energy Theorem

\sum W = \Delta E = \frac{1}{2} m (v^2 - u^2)

Kinetic Energy from momentum

E_k = \frac{p^2}{2m}

Rain Work

W = \rho A d g h'

Potential Energy / Work by Agent (trapped)

PE = - (\text{trapped energy})

Conservation of Mechanical Energy

Ek + Ep = \text{constant}

Hammer & Nail Vertical Wall

F_x = \frac{1}{2} m v^2

Hammer & Nail Horizontal Wall

F_x = \frac{1}{2} m v^2 + m g x

Hammer & Nail Ceiling Wall

F_x = \frac{1}{2} m v^2 - m g x

Hammer & Nail Wall at angle

F_x = \frac{1}{2} m v^2 + m g x \cos\theta