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Pythagorean Theorem:

 

c² = b² + a²

 

Kinematics:

Dx = x_f – x_i

v_avg = Dx/Dt = (x_f – x_i)/( t_f – t_i)

a_avg = Dv/Dt = (v_f – v_i)/( t_f – t_i)

Dx = ½(v_f – v_i)Dt

v_f = v_i + aDt

Dx = v_iDt + ½a(Dt)²

v_f² = v_i² + 2aDx

 

 

Projectile problems: (v_x,i always constant)

Dy = -½g(Dt)²

Dx = v_x,iDt

Dx = v_i(cosq)Dt

Dy = v_i(sinq) – ½g(Dt)²

v_y,f = v_i(sinq) – gDt

v_y,f² = v_i² (sinq)² – 2gDy

 

Forces:

SF = Sm a_net                                           D

F_w = mg                                           D

F_n = mg cosq

F_{f, static} = -F_w,x(kinetic)                                            D

F_{f, static} = m_static mg cosq

F_{f, kinetic} = m_kinetic mg cosq

m_static = (Max.)F_{f, static} / F_n = tanq

m_kinetic = F_k / F_n

 

Work and Energy:

W_net = F_net d (cosq)

KE = ½mv²                                             D

PE_g = mgh                                              D

PE_elastic = ½kx²                          D

ME = KE + SPE

ME_i = ME_f                                                       D

½mv_i² + mgh_i = ½mv_f² + mgh_f                         D    

W_net = DKE

(power) P = W/Dt                                                             

(power) P = Fv                                                    D    

 

Momentum and Collisions:

p = mv                                              D

Dp = FDt = mv_i – mv_f

 

Conservation of Momentum:

m₁v_1,i + m₂v_2,i = m₁v_1,f + m₂v_2,f                                        D

 

Perfectly Inelastic Collision:

m₁v_1,i + m₂v_2,i = (m₁ + m₂)v_f                                                                      D

 

Momentum and Kinetic Energy remain constant in an elastic collision:

½m₁v_1,i² + ½m₂v_2,i² = ½m₁v_1,f² + ½m₂v_2,f²                               D

 

Rotational Motion and the Law of Gravity:

Dq = Ds/r

w_avg = Dq/Dt

α_avg = Dw/Dt = (w_f –w_i)/( t_f – t_i)

v_t = rw

a_t = rα                                              D

a_c = v_t² / r                                   D

a_c = w²r

F_c = (mv_t²)/ r                             D

F_c = mw²r                                  D

F_g = (Gm₁m₂)/r²                              D

G = 6.673 x 10^-11 (Nm²)/kg²                             D

w_f = w_i + αDt

Dq = w_iDt + ½α(Dt)²

w_f² = w_i² – 2αDq

 

Rotational Equilibrium and Dynamics

τ = F d (sinq)

 

Moment of Inertia:

 

Thin hoop about symmetry axis ( rotating around the center point of the circle)

I = mr²

 

Thin hoop about diameter ( rotating around a line axis, which is one of the diameter of the circle)

I = ½mr²

 

Point mass about axis ( a point of mass orbit around another point of mass)

I = mr²

 

Disk or cylinder about symmetry axis ( rotating around the center point of the circle)

I = ½mr²

 

Thin rod about perpendicular axis through center(axis of rotation at the midpoint)

I = (ml²)/(12)

 

Thin rod about perpendicular axis through end (axis of rotation at the endpoint)

I = (ml²)/(3)

 

Solid sphere about diameter

I = (2mr²)/(5)

 

Thin spherical shell about diameter

I = (2mr²)/(3)

 

τ_net =Iα                                              D

 

Angular Momentum:

 L= Iw

 

Rotational kinetic energy

KE_rotational = ½Iw²

 

 

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