Mechanics and motion problems are all about understanding how objects move and interact with forces. We'll dive into Newton's laws, , and 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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states that the net force F on an object is equal to the mass m of the object multiplied by its 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≈9.8m/s2 near Earth's surface
The for an object in free fall, with initial v0 and initial height y0, are:
Velocity as a function of time: v(t)=v0−gt
Position as a function of time: y(t)=y0+v0t−21gt2
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
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 vt=ρACD2mg, where m is the mass, g is the acceleration due to gravity, ρ is the density of the fluid (air), A is the cross-sectional area, and CD 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 v0, launch angle θ, and initial height y0, are:
Horizontal position: x(t)=(v0cosθ)t
Vertical position: y(t)=y0+(v0sinθ)t−21gt2
The time of flight for a projectile launched from ground level is t=g2v0sinθ
The range of a projectile launched from ground level is R=gv02sin2θ, 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 , resulting in periodic motion
The equation of motion for a simple harmonic oscillator is dt2d2x+ω2x=0, where ω=mk is the , k is the spring constant, and m is the mass
The solution to the equation of motion is x(t)=Acos(ωt+ϕ), where A is the amplitude and ϕ 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 dt2d2x+2γdtdx+ω02x=0, where γ is the and ω0 is the
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 dt2d2x+2γdtdx+ω02x=F0cos(ωt), where F0 is the amplitude of the driving force and ω is the angular frequency of the driving force
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)