2.1. Oscillations and Simple Harmonic Motion

Oscillations

• Oscillations are a periodic movement

Characteristics

• Movements repeats occur in equal time intervals
• Resting force always points to where the equilibrium position is
• The acceleration is not constant

Equations for Oscillations

• Position: $x(t) = A\cos(\omega t + \phi)$
• Velocity: $v(t) = -A\omega\sin(\omega t + \phi)$
• Acceleration: $a(t) = -A\omega^2\cos(\omega t + \phi)$
• $A$ stands for amplitude
• $\omega$ is the angular frequency
• $\phi$ is the initial phase

Relations

• $\omega = 2\pi f$
• $T = \frac{1}{f}$
• $T$ stands for period
• $f$ stands for frequency

Simple pendulum

• Simple pendulums are a physical system that perform Simple Harmonic Motion for small amplitudes

Simple Pendulum Period

• $T = 2\pi\sqrt{\frac{L}{g}}$
• $L$ is pendulum length
• $g$ is gravity acceleration

System Mass

• It's a system that performs Simple Harmonic Motion

System Mass Period

• $T = 2\pi\sqrt{\frac{m}{k}}$
• $m$ : mass
• $k$ : elastic constant

Energy in Simple Harmonic Motion

Elastic Potential Energy

• $E_p = \frac{1}{2} kx^2$

Kinetic Energy

• $E_c = \frac{1}{2} mv^2$

Total Mechanical Energy

• $E = E_p + E_c = \frac{1}{2} kA^2$

Energy

Work

• When a constant force is applied, work is the product of force and object displacement in force direction
• $W = F \cdot d \cdot cos(\theta)$
• W stands for work
• F for force
• d for displacement magnitude
• $\theta$ Angle between force and displacement

Work Qualities

• Work is scalar, it has positive/driving energy , negative/resistant or is null/zero

Kinetic Energy

• Objects in motion posses Kinetic energy based on their movement
• $K = \frac{1}{2} m v^2$
• $K$ stands for kinetic energy
• $m$ is the mass
• $v$ is the speed
• Total work is equal to the variation of its kinetic energy
• $W_{total} = \Delta K = K_f - K_i$

Potential Energy

• Potential energy is stored energy for object position/config
• Gravitational Potential energy
  • $U_g = mgh$
• $U_g$ represents the gravitational potential energy
• $m$ is the mass
• $g$ is the terrestrial gravity acceleration = 9.8m/s^2 on Earth's surface
• Elastic Potential Energy
• $U_e = \frac{1}{2} k x^2$
• $U_e$ Stands for elastic potential energy
• $k$ is the elastic spring constant
• $x$ is the shift from the movment

Energy Conservation

• In isolated system , total energy remains constant when time passes
• $E_{total} = K + U = constant$
• $E_{total}$ is total energy
• $K$ total kinetic energy
• $U$ total potential energy
• Energy conservation is a physics law that applies to a variety of situations

Power

• Power is speed at which work is made
• $P = \frac{W}{\Delta t} = \frac{\Delta E}{\Delta t}$
• $P$ stands for potency
• $W$ stands for work
• $\Delta t$ lapse of time
• $E$ stands for change in energy

Impulse , Movement amount , and collisions

Impulse

• Impulse is defined as the product of force by the lapse of time which acts
• $I = F \cdot \Delta t$
• $I$ stands for impulse
• $F$ stands for force
• $\Delta t$ is lapse of time

Qualities of impulse

• The impulse is vector with the same direction/sense of the force

Amount Of Movement

• Amount of movment is defined as the product of mass and speed $p = m \cdot v$
• $p$ is amount of movement
• $m$ stands for object's mass
• $v$ stands for speed
• Vector follows the same direction and sense of speed

Theorem of Impulse Quantity of Movement

• The resulting impulse is equal to change in its quantity of movement
• $I = \Delta p = p_f - p_i$

Quantity of Movement Conservation

• Quantity of Movment is constant in isolated systems(without external forces)
• $p_{total} = constante$
• Movement conservation is a physics law

Collisions

• Collisions are the interaction of 2 or more objcts exchanging energy and momentum
• Elastic (Kinetic Total Energy is conserved) , Inelastic( Kinetic energy is not conserved)
• Momentum is conserved in every collision
• Restitution Coefficient (e): $e = \frac{|v_{afastamento}|}{|v_{aproximação}|}$
• $e = 1$ elastic collision
• $0 < e < 1$ inelastic collision
• $e = 0$ perfectly inelastic collation

Static

Balance Condidtions

• To be in static balance 2 conditions need to be met
• The object must have a null resultant force $\sum F = 0$
• The resulting torque over the object must be null $\sum \tau = 0$

Torque

• Torque is a movement caused around an axis
• $\tau = r \cdot F \cdot sin(\theta)$
• $\tau$ is torque
• $r$ Distance from application point to rotation axis
• $F$ force amount
• $\theta$ Angle between force ans position

Qualities of Torque

• Torque is perpendicular to the plane formed by position vector and force

Gravity Center

• Gravity center is where we consider all gravitational force on the object
• Coincides with the geometric center in symmetric objects
• Stability depends on the center in relation to their support base