How to solve simple harmonic motion problems using initial conditions

We have seen how the general solution for simple harmonic motion involves sines or cosines. These functions are periodic functions, and it makes sense that they are used to solve period motion problems. But what about problems where we are given initial conditions (i.e. at t = 0)? How do we find specific solutions to specific problems? Well, here is an example of how to do this. We will have a case where a mass oscillating on a spring has BOTH initial position and initial velocity. We introduce the general solution that involves a phase angle. Check it out.

How to find solutions for simple harmonic motion (SHM)

Simple harmonic motion is a subset of periodic motion, defined as motion that depends on displacement. Springs certainly follow this definition since Hooke's law gives us F = -kx. This picture of a mass oscillating at the end of a spring also makes use of energy, U = .5kx^2. But our goal is to understand the motion as a function of time, as well as a function of position. Bringing time into the picture is the issue, and it turns out we need a second order differential equation. This video outlines how to get a solution for position of the mass on a spring as functions of time. Turns out these require sine or cosine functions. Other videos will focus on other details and specific problems to show a general solution we can use.

How to handle rotations with Rolling without Slipping

This is a classic rotations problem, where a ball rolls down a hill without any slipping. Because both linear and rotational motions happen simultaneously, we need to solve both motions simultaneously with F = ma and t = I(alpha). This would be true for any sort of 'rolling without slipping' problem, as well. Check it out!

How to Apply Conservation of Angular Momentum to Rotational Motion and Collisions

Here is a sliding block and hanging rod, where the block collides and sticks to the rod. Angular momentum is needed, and here is a case where conservation of angular momentum is used to figure out the initial speed of the block before the collision occurs. Take a look to see the general, symbolic setup for such a problem, which will be similar to a ballistic pendulum from linear momentum days. I hope this helps.

How to Use the Parallel Axis Theorem to Find Moments of Inertia

The parallel axis theorem is a neat shortcut that allows us to find moments of inertia for objects when the axis of rotation is somewhere other than the center of mass of the object. If you know the inertia for objects when going through the center of mass, you can quickly find the new value of I for any axis that is parallel to the center of mass axis and displaced by some distance from the center of mass, d. The theorem says I_new = I_cm + Md^2. We do not have to use the integral to apply the theorem, which is why it is such a nice shortcut.

This video shows a couple quick examples of how to apply the theorem. Hope it helps!

How to do Rotational Motion for a rotating, falling bar - NON-constant acceleration

Check out an example of a NON-constant angular acceleration problem, where a bar starting in static equilibrium (up = down, cw = ccw) goes into non-equilibrium and accelerates. You can see how torque = I*(alpha) gives us the angular acceleration of the bar at any given angle it has rotated through, and also how to use rotational energy and energy conservation to determine the angular speed at any given angle.

One thing to keep in mind as far as linear acceleration and linear speed is that each point of the bar has different values for these quantities, as determined by a = R*alpha and v = R*omega. Check it out...

Rotational Motion - New Concepts

For the 3 Chem-Phys sections, we are into rotational motion. This is typically a challenging topic because it is brand new. Keep in mind what makes it new revolves (ha, ha) around 3 new concepts:
- Torque
- Moment of inertia
- Angular momentum

Torques are produced by forces, and specifically those forces that cause a change in rotational motion. In other words, torques produce angular accelerations (analogous to forces causing linear accelerations). Mathematically, individual forces cause a torque = F(r)[sin(theta)]. Torque is a cross product vector, t = r x F.

Moment of inertia is analogous to mass in linear motion. It is a 'resistance to a change in rotational motion.' The higher the inertia, the smaller the angular acceleration from the same torque. Together, torque, t, and moment of inertia, I, are related through the 2nd law for rotations:
t = I(alpha)

The moment of inertia has units of kg m^2, and numerically tells us about the distribution of mass about the axis of rotation of the object or system.

Angular momentum, L, is also a cross product vector, L = r x p. The direction is found with the curly RHR, as we do in class. Remember the conservation of linear momentum? Momentum is conserved for a system if no external forces act on the system. Here is the analogy: angular momentum is conserved for a system if there is no external torques acting on the system. Individual objects can have angular impulse in collisions, but for the system it is conserved. We will get into this in a big way, and I'll soon have some how to videos up for rotations.

Let's have some fun with it!

How to Apply Ampere's Law

Ampere's law is to magnetic fields as Gauss's law is to electric fields. We only use it in 3 cases, just like Gauss, and it even looks similar to Gauss's law, only it is a 1-D integral compared to a 2-D integral. A line integral, or to some a path integral, basically means we are looking for the magnetic field times the length of the path the B-field follows. This can work for us with long, straight wires with current, a solenoid, and a toroid. Check out how to apply Ampere's law in 2 of the 3 cases, those being a straight wire and toroid.

How to Find Magnetic Forces Between Current Carrying Wires

Many electronic devices have parallel wires with currents flowing. Now, each current produces magnetic fields that circulate around the current, and these magnetic fields interact with the other current to produce a force, due to F = Il x B. Check out this video to see how to combine a couple concepts - Ampere's law determines the strength of the magnetic field from one of the currents, and then this goes into the force equation to determine the strength of the force. The right hand rule will help determine the direction of the force. Net result is that currents in the same direction attract, and in opposite directions repel. Hope this helps!