3.4 Mechanics and Motion Problems

Mechanics and motion problems are all about understanding how objects move and interact with forces. We'll dive into Newton's laws, free fall, and projectile motion to see how these concepts apply to real-world situations.

This topic builds on our knowledge of differential equations by applying them to physical systems. We'll explore how to model and solve problems involving gravity, air resistance, and oscillations using the tools we've learned so far.

Newton's Laws and Free Fall

Newton's Second Law and Free Fall

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• Newton's second law states that the net force $F$ on an object is equal to the mass $m$ of the object multiplied by its acceleration $a$: $F = ma$
• Free fall occurs when an object is only acted upon by the force of gravity, resulting in a downward acceleration of $g \approx 9.8 \, m/s^2$ near Earth's surface
• The equations of motion for an object in free fall, with initial velocity $v_0$ and initial height $y_0$, are:
  • Velocity as a function of time: $v(t) = v_0 - gt$
  • Position as a function of time: $y(t) = y_0 + v_0t - \frac{1}{2}gt^2$

Air Resistance and Terminal Velocity

• Air resistance is a force that opposes the motion of an object through the air and depends on factors such as the object's speed, shape, and size
• As an object falls through the air, the force of air resistance increases until it balances the force of gravity, resulting in a constant velocity called the terminal velocity
• The terminal velocity of an object depends on its mass, cross-sectional area, and the drag coefficient, which is determined by the object's shape (a streamlined shape like a raindrop has a lower drag coefficient than a flat shape like a piece of paper)
• The equation for terminal velocity is $v_t = \sqrt{\frac{2mg}{\rho AC_D}}$, where $m$ is the mass, $g$ is the acceleration due to gravity, $\rho$ is the density of the fluid (air), $A$ is the cross-sectional area, and $C_D$ is the drag coefficient

Projectile Motion

Equations of Motion for Projectile Motion

• Projectile motion is the motion of an object thrown or projected into the air, subject only to the acceleration of gravity
• The equations of motion for projectile motion, with initial velocity $v_0$, launch angle $\theta$, and initial height $y_0$, are:
  • Horizontal position: $x(t) = (v_0 \cos \theta)t$
  • Vertical position: $y(t) = y_0 + (v_0 \sin \theta)t - \frac{1}{2}gt^2$
• The time of flight for a projectile launched from ground level is $t = \frac{2v_0 \sin \theta}{g}$
• The range of a projectile launched from ground level is $R = \frac{v_0^2 \sin 2\theta}{g}$, with the maximum range achieved at a launch angle of 45°

Examples of Projectile Motion

• A football kicked at an angle of 30° with an initial velocity of 20 m/s will have a range of approximately 40 meters
• A cannon fired at an angle of 45° with an initial velocity of 100 m/s will have a range of about 1,020 meters and a time of flight of around 14.3 seconds
• In the absence of air resistance, a bullet fired horizontally and a bullet dropped from the same height will hit the ground at the same time, demonstrating the independence of vertical and horizontal motion in projectile motion

Oscillations

Harmonic Oscillator

• A harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement, resulting in periodic motion
• The equation of motion for a simple harmonic oscillator is $\frac{d^2x}{dt^2} + \omega^2x = 0$, where $\omega = \sqrt{\frac{k}{m}}$ is the angular frequency, $k$ is the spring constant, and $m$ is the mass
• The solution to the equation of motion is $x(t) = A \cos(\omega t + \phi)$, where $A$ is the amplitude and $\phi$ is the phase constant
• Examples of harmonic oscillators include a mass attached to a spring, a pendulum (for small angles), and an LC circuit

Damped and Forced Oscillations

• Damped oscillations occur when a harmonic oscillator experiences a damping force, such as friction or air resistance, that opposes its motion and causes the amplitude to decrease over time
• The equation of motion for a damped harmonic oscillator is $\frac{d^2x}{dt^2} + 2\gamma\frac{dx}{dt} + \omega_0^2x = 0$, where $\gamma$ is the damping coefficient and $\omega_0$ is the natural angular frequency
• Forced oscillations occur when an external driving force is applied to a harmonic oscillator, causing it to oscillate at the frequency of the driving force
• The equation of motion for a forced harmonic oscillator is $\frac{d^2x}{dt^2} + 2\gamma\frac{dx}{dt} + \omega_0^2x = F_0 \cos(\omega t)$, where $F_0$ is the amplitude of the driving force and $\omega$ is the angular frequency of the driving force

Resonance

• Resonance occurs when the frequency of the driving force in a forced oscillator matches the natural frequency of the oscillator, resulting in a large amplitude of oscillation
• At resonance, the amplitude of the oscillator is maximum and is limited only by the damping in the system
• Examples of resonance include a child pumping a swing at the natural frequency to achieve maximum height, a singer shattering a wine glass by singing at its resonant frequency, and a bridge collapsing due to wind-induced vibrations at its natural frequency (Tacoma Narrows Bridge, 1940)