Mechanics: Statics and Dynamics

Symbols and Formulas

(1.) Basic Formulas

$ W = mg \\[3ex] where \\[3ex] W = weight\;(N) \\[3ex] g = \text{acceleration due to gravity}\; (ms^{-2}) \\[3ex] m = \text{mass}\;(kg) \\[3ex] $ (2.) Basic Equations of Motion

$ (a.)\;\; v = u + at \\[3ex] (b.)\;\; s = ut + \dfrac{1}{2}at^2 \\[5ex] (c.)\;\; v^2 = u^2 + 2as \\[5ex] \underline{\text{Working towards gravity}} \\[3ex] a = g \\[5ex] \underline{\text{Working aganist gravity}} \\[3ex] a = -g \\[5ex] $ where:
u = initial velocity
v = final velocity
a = acceleration
t = time
s = distance
g = acceleration due to gravity

(3.) Total Momentum

$ m_1 = \text{mass of first body} \\[4ex] v_1 = \text{velocity of first body} \\[4ex] m_2 = \text{mass of second body} \\[4ex] v_2 = \text{velocity of second body} \\[5ex] \underline{\text{Before Collision: Moving in Same Direction}} \\[4ex] \text{momentum of first body} = m_1v_1 \\[4ex] \text{momentum of second body} = m_2v_2 \\[4ex] \text{total momentum} = m_1v_1 + m_2v_2 \\[5ex] \underline{\text{After Collision: If Stuck Together}} \\[3ex] \text{mass of: first body and second body} = m_1 + m_2 \\[4ex] \text{common velocity} = v \\[3ex] \text{total momentum} = v(m_1 + m_2) \\[5ex] \underline{\text{Based on the Principle of Conservation of Momentum}} \\[3ex] \text{total momentum before collision = total momentum after collision} \\[3ex] m_1v_1 + m_2v_2 = v(m_1 + m_2) \\[5ex] \underline{\text{Before Collision: Moving in Opposite Directions}} \\[4ex] \text{momentum of first body} = m_1v_1 \\[4ex] \text{momentum of second body} = m_2 * -v_2 = -m_2v_2 \\[4ex] \text{The negative value of the velocity is because the second body is moving in opposite direction} \\[3ex] \text{total momentum} = m_1v_1 - m_2v_2 \\[5ex] \underline{\text{After Collision: If Stuck Together}} \\[3ex] \text{mass of: first body and second body} = m_1 + m_2 \\[4ex] \text{common velocity} = v \\[3ex] \text{total momentum} = v(m_1 + m_2) \\[5ex] \underline{\text{Based on the Principle of Conservation of Momentum}} \\[3ex] \text{total momentum before collision = total momentum after collision} \\[3ex] m_1v_1 - m_2v_2 = v(m_1 + m_2) \\[5ex] $ (4.) Forces acting on a body

$ F_x = F \cos\theta \\[3ex] F_y = F \sin\theta \\[3ex] $ where:
$F$ = magnitude of the force
$F_x$ = horizontal component (x-component) of the force
$F_y$ = vertical component (y-component) of the force
θ = angle the force makes with the positive x-axis (measured counterclockwise).

$ F = \sqrt{F_x^2 + F_y^2} \\[3ex] \theta = \tan^{-1}\left(\dfrac{F_y}{F_x}\right) \\[5ex] $ (5.) A Body in Equilibrium

$ \Sigma F_x = 0 \\[3ex] \Sigma F_y = 0 \\[5ex] $ (6.) Gravitational Force

$ \underline{\text{Newton's Law of Universal Gravitation}} \\[3ex] F = \dfrac{Gm_1m_2}{r^2} \\[5ex] \text{where} \\[3ex] \text{F is the gravitational force between two bodies} \\[3ex] \text{G is the gravitational constant} \\[3ex] m_1 \text{ is the mass of the first body} \\[3ex] m_2 \text{ is the mass of the second body} \\[3ex] \text{r is the distance between the centers of mass of the two bodies.} \\[5ex] \underline{\text{Escape Velocity}} \\[3ex] v_e = \sqrt{\dfrac{2GM}{R}} \\[5ex] v_e = \sqrt{2gR} \\[3ex] g = \dfrac{GM}{R^2} \\[5ex] \text{where} \\[3ex] v_e \text{ is the escape velocity} \\[3ex] \text{G is the gravitational constant} \\[3ex] \text{M is the mass of the celestial body} \\[3ex] \text{R is the radius of the celestial body} \\[3ex] \text{g is acceleration due to gravity on the surface of the celestial body} \\[5ex] $