module 5 work and energy

work done by a constant force

W = F→ • Δr→ = F|Δr|cosθ

no work done by a normal force because

θ=90, no displacement

work can be done by forces not causing the _

motion

dot product

A • B = ABcosθ = B • A

A • (B + C) =

A • B + A • C

if A ⊥ B, A • B =

0

if A || B, A • B =

-AB

extrapolate dot product component form to work

W = F • Δx = F_xΔx + F_yΔy + F_zΔz

work done by a varying force

W_tot = ∫xi to xf of F(x)dx

any of unit vectors * any other unit vector that’s not itself =

0

any unit vector²

1

A→ • A→ = 

A²

work done by a spring

W_s|x_i to x_f = 1/2kx_i² - 1/2kx_f²

work done by bigger object in gravity e.g. sun (smaller object against it would be negative of this value)

C(1/x_f - 1/x_i), where c is a constant in the gravitational eqution (Gm1m2 or wtv)

work done related to kinetic energy

1/2m(v_f² - v_i²)

work-KE theorem

W_{done on system by Fnet} = ΔKE - in case where only change in system is it’s speed

gravitational potential energy =

mgh

work to move object vertically

mgy_b - mgy_a

gravitational potential energy is independent of

the path taken to the current position

for spring, U = _  and W_app =

1/2kx²; ΔU

conservative force

work done by it on a particle moving between 2 points is independent of the path taken; closed path = zero work

for a system with a conservative force acting between its members there is a corresponding

potential energy

non conservative forces

change the mechanical energy of a system; ΔE ≠ ΔK + ΔU

E_mech =

K + U

if no non-con forces, _ is constant and _ = 0

E_mech; ΔE

internal energy

portion of the kinetic energy that is converted into atomic thermal motion, manifesting as heat

relation between potential energy and forces

W_spr/grav = -ΔU aka F_x = -dU/dx

stable equilibrium on u vs x graph

upward parabola

metastable equilibrium on u vs x

cubic with left down and right up

unstable equilibrium on u vs x

downward parabola