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Reading

Fowles and Cassidy 1.3 - 1.5, 1.9 - 1.10, 4.1 - 4.4

Recall

 Summary of the SHO:

• Spring-like force: F = -kx
• Energy conservation, potential energy V = 1/2 kx²
• Undamped or natural frequency w ₀ = Sqrt(k/m)
• General solution x = Asin(w ₀t + f )
• Addition of a constant force changes the equilibrium position, but not results of the general solution.
• Any potential with a "parabolic" minimum has a SHO solution for small oscillations (for example, the simple pendulum or the Morse potential examples).

One More Example

What is the frequency for small oscillations of a mass m about the minimum of the potential V(r) = a/r² - b/r ?

First, find the minimum. dV/dr = b/r² - 2a/r³ = 0 thus r_min = 2a/b. And V(r_min) = b²/4a - b²/2a = -b²/4a. To determine the frequency for small oscillations, we need the second derivative of V at the minimum. d²V/dr² = 6a/r⁴ - 2b/r³ = b³/8a³(3b - 2b) = b⁴/8a³. In the small oscillation approximation, this is the equivalent of k, so w 0 = Sqrt(b⁴/8a³m) = b²/[2a Sqrt(2am)] .

Vectors

In preparation for problems in 2 and 3 dimensions, let's review the properties of vectors.

Vectors will be indicated by a bold face letter in typeset text, like A, and by the letter with an arrow above it in handwritten work. A vector has a magnitude and direction, relative to some chosen coordinate system. Notice that a vector doesn't have a fixed beginning or end coordinate, just the magnitude and direction. Often we will represent a vector by it's components on the x, y, and z axes: A = (A_x, A_y, A_z). If A points from P1 at (x1, y1, z1) to P2 at (x2, y2, z2), then Ax = x2 - x1, Ay = y2 - y1, and Az = z2 - z1. In many instances we can arrange for all relevant motion to lie in a plane. In such cases we can use 2 dimensional vectors rather than 3 dimensionsal.

1. Equality of vectors
2. Vector addition, multiplication by a scalar, vector subtraction
3. The null vector
4. The commutative and associative laws of addition
5. The distributive law
6. Magnitude of a vector
7. Unit coordinate vectors
8. The scalar product
9. The vector product or cross product
10. Derivative of a vector, sum of 2 vectors, scalar multiple of a vector, dot product, and cross product.

Position Vector of a Particle, Velocity and Acceleration

Suppose we have a reference frame (coordinate system), and in that frame we know the position of a particle is:

r = ix + jy + kz

If x, y, and z are functions of time, then we know the position as a function of time, and r as a function of time. We find the velocity as a function of time by taking the time derivative of r,   v = dr/dt = i[xdot] + j[ydot] + k[zdot] . The speed v is the absolute value of the velocity and is given by v = |v| = Sqrt([xdot]² + [ydot]² + [zdot]²).

The acceleration is given by the time derivative of the velocity, a = dv/dt .

Newton's Laws

In 3 dimensions we will rewrite Newton's 2^nd and 3^rd laws in vector form:

F = dp/dt = ma

F₁₂ = -F₂₁

Each of these stands for 3 component equations, equality of the x, y, and z components. This compactness is the reason that we use vector notation! If the equation of motion can be separated into its 3 components giving 3 equations of the types studied previously (i.e. force is constant, depends only on time, or each component depends only on the corresponding component of position) then the equations can be solved with the techniques used previously. Only now there are 2 or 3 equations to solve instead of only 1! The simplest example of this case is the motion of a projectile in uniform gravity and without air resistance.