Chapter 1: Example Equations of Oscillating Objects

• Mass on a spring:

  • Restoring force is -kx

  • Equation of motion: d^(2)x/dt^(2) = -k/m * x

  • Solution: x = A cos(ωt - φ), where A is amplitude and φ is phase

• Bar suspended by a cable:

  • Restoring torque is -κθ

  • Equation of motion: -κθ = I * d^(2)θ/dt^(2)

  • Solution: θ = A cos(ωt - φ), where ω^2 = κ/I

• Finding κ:

  • For a spring, κ can be found from the given force or displacement

  • For a pendulum, κ is found by analyzing the torque due to the disturbance

  • κ is specific to each problem and depends on the system's properties

• Frequency of vibration:

  • Frequency ω is related to the angular frequency and time period

  • ω = 2π/T

• Example with irregularly shaped object:

  • Center of mass lies on the vertical line passing through the pivot point

  • Torque equation: -κθ = I * d^(2)θ/dt^(2)

  • Moment of inertia includes parallel axis theorem

  • Frequency of oscillation can be calculated using κ and moment of inertia

Chapter 1: Restoring Torque in Twisting Example

• Student asks about how to describe the restoring torque in a twisting example

• Professor explains that the calculation of restoring torque is given in such problems

  • It cannot be calculated from first principles

  • The torque is given based on the material and torsional properties of the cable

• In cases where the torque needs to be found, the student is expected to figure it out

Chapter 2: Behavior of the Cable

• When the cable is left alone, it will go to a position where there is no torque

• If the cable is moved off that position, a torque will be present

• The torque is always proportional to the angle by which the cable is displaced

• The proportionality constant is denoted as κ

Chapter 3: Forces Acting on the Body

• When the body is hanging in its rest position, there are two forces acting on it

  • The nail pushing up

  • The weight of the body pushing down

• These forces cancel each other out, preventing the body from falling

• The nail cannot prevent the body from swinging because it cannot exert a torque at the pivot point

• However, when the body is rotated, the weight of the body (mg) can exert a torque

• This is why the body starts to rattle back and forth when twisted

Chapter 2: Superposition of Solutions to Linear (Harmonic) Equations

• Introduction to more complicated oscillations using techniques learned previously

• Introduction of the formula: e to the ix, or θ, is cos θ + I sin θ

  • Taking the complex conjugate of both sides gives e to the minus iθ is cos θ - i sin θ

• Importance of memorizing the formula

• Explanation of complex conjugation and its application to complex numbers

• Inversion of the formula to find cos θ and sin θ

• The exponential function as a replacement for trigonometric functions

• Unification of trigonometric functions and exponential functions through complex numbers

• Representation of complex numbers in Cartesian and polar forms

  • Complex number z can be written as its absolute value times some phase

  • Cartesian form: x + iy

  • Polar form: re to the iφ, where r is the radial length and φ is the angle

Chapter 1: Introduction

• Understanding and asking questions is important

• Equation: d^(2)x/dt^(2) = -ω^(2)x

• ω_0 is the frequency of vibration of the system

• Solving the equation using different methods

Chapter 2: Solving the Equation

• Trying to find a function x(t) that satisfies the equation

• Using sines and cosines as a solution

• Trying a different approach using a function that repeats itself

• Introducing the ansatz, a tentative guess for the solution

• Putting the guess into the equation and simplifying

• Grouping terms and finding the condition for the solution to vanish

Chapter 1: Solving the Differential Equation

• Solving a differential equation requires making a guess and testing it

• The equation must be satisfied at all times

• The equation α^2 + ω_0^2 = 0 is the only non-trivial solution

• The equation can be rewritten as α^2 = ω_0^2, or α = ±iω_0

Chapter 2: Multiple Solutions

• There are two solutions: x_1(t) = Ae^(iω_0t) and x_2(t) = Be^(-iω_0t)

• A and B can be any arbitrary numbers

• Both solutions satisfy the differential equation

• There is no need to choose between the two solutions, both can be used

Linear Equations and Superposition

• Linear equations are important in disciplines like economics, engineering, and chemistry

• Linear equations obey a property called superposition

• Superposition means that in a linear equation, the function or its derivatives appear, but not higher powers or derivatives

• Non-linear equations involve higher powers or derivatives of the function

Adding Linear Equations

• Given two linear equations, x_1 double dot + ω_0 ^(2)x_1 = 0 and x_2 double dot + ω_0^(2)x _2 = 0

• Adding the two equations results in a new equation: x_1 + x_2 double dot + ω_0^(2)(x_1 + x_2) = 0

• The sum of the derivatives is the derivative of the sum, which proves that x_1 + x_2 is also a solution to the equation

Generalizing the Addition

• Multiplying the first equation by A and the second equation by B still results in a valid equation

• Adding the two multiplied equations gives a new equation: Ax_1 + Bx_2 double dot + ω_0^(2)(Ax_1 + Bx_2) = 0

• This shows that any linear combination of the two solutions, Ax_1 + Bx_2, is also a solution to the equation

Infinite Solutions in Linear Equations

• Linear equations typically have an infinite number of solutions

• Solutions can be built by combining a few building blocks, similar to unit vectors

• By choosing different coefficients A and B, an infinite number of solutions can be generated

Non-Linear Equations

• Non-linear equations involve higher powers or derivatives of the unknown function

• Adding solutions to non-linear equations does not result in a valid solution

• The sum of squares is not equal to the square of the sum in non-linear equations

Conclusion

• Linear equations have the property of superposition, allowing for the addition and combination of solutions

• Non-linear equations do not have this property and cannot be solved by adding solutions

Chapter 3: Conditions for Solutions to Harmonic Equations

• Harmonic oscillating equation is a linear equation

  • Solution: x(t) = Ae^(iω0) + B e^(-iω0t)

    • A and B are arbitrary constants

    • ω_0 is the original root of k/m

• Concerns with the solution

  • Need to choose the right A and B

    • To fit initial coordinate and velocity at t = 0

  • Solution contains imaginary numbers

    • x should be a real function, not complex

• Demanding a real solution

  • Real numbers are their own complex conjugates

  • Complex conjugate of x(t) is denoted as x star (t)

  • Complex conjugate of A is A star

  • Complex conjugate of e^(iωt) is e^(-iω0t)

  • Complex conjugate of B is B star

  • Equate x(t) and x star (t) to find conditions

    • A = B star

    • B = A star

• Writing the solution with the extra condition

  • x(t) = Ae^(iω0t) + A star e^(-iω0t)

  • B is not independent, it must be the complex conjugate of A

• x(t) in terms of absolute value and phase

  • x(t) = |A|e^(iφ) e^(iω0t) + |A|e^(-iφ) e^(-iω0t)

  • |A| is the length of the complex number A

  • φ is the phase of the complex number A

• Recognizing the familiar form of x(t)

Chapter 4: Exponential Functions as Generic Solutions

• The identity e^(iθ) + e^(-iθ) is 2 times cos θ

  • This can be written as 2 times absolute value of A cos ωt + φ

  • It can also be written as C cos ω_0t + φ, where C = 2A

• Exponential functions can be used to solve problems that cannot be solved using word problems

• The equation mx double dot + bx dot + kx = 0 represents a mass with friction

• The equation can be rewritten as x double dot + γx dot + ω_0 squared x = 0, where γ = b/m

• The equation can be solved using the guess x = Ae^(αt)

• The quadratic equation α^(2) + αγ + ω_0^(2) = 0 gives the values of α

• The solutions to the equation are x(t) = Ae^(-α plus t) + Be^(-α minus t), where α plus and α minus are the roots of the quadratic equation

• Both α plus and α minus are falling exponentials, ensuring that the solution eventually vanishes

Finding A and B

• To find A and B, extra data is needed, such as the initial position (x of 0)

• The sum of A and B must be equal to the initial position

• The initial velocity can be found by taking the derivative of x with respect to t

• The derivative of x at t = 0 gives the initial velocity (x dot of 0)

• Two simultaneous equations can be formed using the initial position and initial velocity to solve for A and B

Example of simultaneous equations

• Equation 1: A + B = initial position

• Equation 2: α plus A + α minus B = x dot of 0

Complex Numbers

• A and B may be complex numbers

• When taking the complex conjugate of the function, A and B were related by complex conjugation

• If the function remains real when taking the complex conjugate, A and B must be equal to their complex conjugates (A = A star and B = B star)

• The exponential term remains unchanged when taking the complex conjugate

Chapter 5: Undamped, Under-damped and Over-damped Oscillations

• Professor Shankar discusses the solutions to the harmonic oscillator with and without friction.

  • Without friction (γ = 0), the solution is a simple harmonic oscillator with a cosine function.

  • With friction, the solution is an exponentially decaying function.

• The real situation is when there is some friction, but not too much.

  • The solution for this case can be obtained by considering the roots of the equation α = -γ/2 ± √(γ/2)^2 - ω_0^2.

  • When γ/2 < ω_0, the roots become complex, and the solution involves complex exponentials.

• The solutions for the damped oscillator with moderate friction are of the form x = Ce^(-γ/2t)cos(ω' t + φ), where C is a constant and ω' is a modified angular frequency.

Chapter 1: Introduction

• The graph represents an oscillation with a damping factor

• The oscillation is described by the function cos ωt

• The exponential term in front of the oscillation represents the damping factor γ

Chapter 2: No Friction

• If γ is zero, the exponential term vanishes

• The oscillation is a simple cosine function with constant amplitude

• The mass oscillates indefinitely without any damping

Chapter 3: Small Friction

• If γ is small (e.g., one part in 10,000), the exponential term is negligible for the first second

• The amplitude of the oscillation gradually decreases over time due to the damping factor

• The graph of the oscillation shows a gradual decrease in amplitude

Chapter 4: Overdamped

• If γ is larger than ω_0/2, the oscillation is overdamped

• There are no oscillations in the system, only exponential decay

• The mass slowly comes to rest without any oscillatory motion

Chapter 5: Real-life Examples

• Shock absorbers in cars are designed to dampen vibrations

• When hitting a bump, the shocks absorb the vibrations and gradually bring the car to a rest

• The ideal damping scenario is when the shocks are in the regime of small friction

Chapter 6: Clarification on Solutions

• When γ/2 equals ω_0, there is only one solution to the equation

• In this case, a new function, t times an exponential, can be used as the second solution

• The value of γ is not independent of ω, it is equal to 2ω_0

Chapter 7: Conjugate Solutions

• In the case of real solutions, the coefficients of the exponential terms must remain invariant when taking the complex conjugate

• The complex conjugate of a real exponential is the same as the original exponential

• The coefficients of the exponential terms must match on both sides of the equation

Chapter 6: Driving Harmonic Force on Oscillator

• Introduction to the problem of a driven oscillator

  • The oscillator is actively driven by an external force

  • The equation to solve is x double dot + γ x dot + ω_0 square x = F/m cos ωt

Solving the problem using a clever trick

• The problem cannot be solved directly due to the presence of cos ωt

• Introduce a new problem with sin ωt as the driving force

• Multiply the equation by i to manipulate it

• Combine the equations to obtain z double dot + γz dot + ω_0 square z = F/m e to the iωt

Finding the solution for z

• Assume z takes the form z = z_0 e to the iωt

• Substitute the assumed form into the equation and simplify

• Obtain the condition for z_0: z_0 = F/m / (ω_0 square - ω square + iωγ)

Impedance and the final solution

• The complex number I is defined as impedance: I = ω_0 square - ω square + iωγ

• The solution for z is z = F_0/m e to the iωt / I

• The impedance I has a real part (ω_0 square - ω square) and an imaginary part (iωγ)

Chapter 1: Introduction

• The problem is to find the number z and take its real part

• Complex numbers can be written as an absolute value times e to the iφ

Chapter 2: Finding z

• z = F_0 / m * e^(iωt - φ)

• φ is the angle

• The real part of z is x

Chapter 3: Finding x

• x = F_0 / m * cos(ωt - φ)

Chapter 4: Solution to the problem

• The answer is the magnitude of the applied force and the phase φ

• The absolute value of I is given by ω_0^2 - ω^2 + ω^2γ^2

• The phase φ is given by tan φ = ωγ / (ω_0^2 - ω^2γ^2)

Chapter 5: Free parameters

• Every equation should have two free numbers

• The solution without the driving force can be added to the solution with the driving force

• The complementary function is the solution of the equation with no driving force

Chapter 6: Full answer

• The full answer includes the solution with no driving force and the solution with the driving force

• The numbers A and B are chosen to match initial conditions

Chapter 7: Conclusion

• The missing part of the notes will be posted on the website

• Students can read it and do one or two problems

• The missing part will be taught again on Wednesday