Physics 1252.2: where are we in class?

February 12

Organization

Center of Mass: What is it? Why is it interesting? How can you figure out where it is for a given object? For simple objects and more complex ones which require use of integrals.

Center of Mass: Figuring out the position of COM, continued.
How is COM used in a sample problem (textbook problem 14).

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kinematics of rotation motion. It makes comlete parallel of translational motion as the table below indicates.

rotational motion	translational (regular) motion
angle	position
angular displacement	displacement
angular velocity	velocity
angular acceleration	acceleration
rotational kinetic energy	kinetic energy
angular momentum	momentum
moment of inertia	mass
torque	force

When a motion of an extended body is observed, the velocities of parts are not necessarily the same as that of the object as a whole. When the kinetic energy of parts are added for an object rotating around a fixed axis, it can be written in as

K=(1/2)Iw^2, where the moment of inertia I is the sum of m_i*r_i^2, and m_i is the mass of part i, and r_i is the distance of part i from the fixed rotation axis.
By adding Newton's 2nd Law equations for the same parts as above after multiplying them by r_i, one gets an equation similar to Newton's 2nd law equation, but it is for rotational motion:

torque=I*(angular acceleration, alpha)
where torque is the sum of r*force and "r" is the distance between the fixed rotation axis and where the force is applied
The summing is done to avoid having to know the internal forces between the parts, but also to eliminate the effect of the force from the rotation axis, since for that force, "r" = 0.

Application of rotational kinetic energy and torque-angular acceleration Law is shown.

More precise definition of torque="r"*Force*sin(phi), where phi is the angle between the two vectors, force vector and the lever-arm vector. As a vector, torque can be equated to the cross product of r and F, the previous two vectors.
Then we analysed the motion of a yo-yo. The acceleration is reduced from "g", one for a free fall, by a factor of (1+2*I/r^2), which can be reduced to (1+R^2/r^2), where R and r are radii of the yo-yo and its axel, where the string wraps around. I also model the yo-yo as a disk.

Work in rotation, more application of rotation: rolling

The same math can be used to show that the change in rotational kinetic energy (1/2)Iw^2 can be equated to work done to the object if one defines work as W=torque*angular displacement. In many cases, the same work can be canculated using force*displacement, too. We also discussed tricky point of deciding "distance" in an example.

Then we worked out an example of a cylinder rolling down an incline to study why certain objects roll down the same incline faster than others.

Rolling and energy. We analysed the rolling of an object in term of energies. The loss in the potential energy equals in magnitude the gain in the rotational and translational kinetic energies of the object. This means when there is no slippage, no energy is dissipated despite the friction between the object and the slope. i.e.static friction does not (negative) work.

We started on static.

Application of static laws: net torque=0, and net force = 0. Where should the reference point be w.r.t. which we calculate torques? It turns out that you can choose any point since it can be proven in general that if the net torque w.r.t. one arbitrary point is zero, and if the net force is zero, net torque w.r.t. ANY point is zero. This also implies that you should choose a point so that the torque equation contains the fewest number of unknowns to make algebra solution easier. This often means the reference point should be where many unknown force(s) act on.

Did a problem from last week's practice problem set.

Angular momentum plays very similar role as (linear momentum) in translational motion. Definition of L=Iw is analoguous to p=mv, and it is conserved when the net torque is zero. There are examples where even if the net FORCE is not zero, net torque is zero if the reference point for torque is properly chosen. Like the angular momentum of the earth is conserved since the force due to the sun creates no torque w.r.t. the sun itself.

Worked on qualitative estimate of rotational inertia comparing a ring, a disk and a sphere. Then went on to use integrals to calculate rotational inertia of a bar, and looked at how parallel axis theorem works. Proof is left to be done next week.

Also we did a few examples of angular momentum conservation using a spinning ice skater, and stunt by an iceskater, which will be continued next week.

Today, we reviewed our materials by looking at a few problems. One of the suggested problems, problem 11 of chapter 13, was solved. A simple static problem was also solved, and computation of rotational inertia of a ring when the axis is contained in the plane of the ring (as opposed to perpendicular to the plane) was carried out partially.

We finished calculating the rotational inertia of a ring around the axis parallel to (instead of perpendicular to) the right through its center.

We also finished the problem of last week.

Now we will study oscillation motion, which include a mass hanging off a spring and pendula.

Oscillation motion are described by the position of the oscillating object to be x(t)=x_m*cos(wt+phi). This can be shown to be consistent with F_net(t)=-kx(t) by differentiating x(t). The reverse can also be true, namely if F_net is propotional to x, the displacement, the object must oscillates like x_m*cos(wt+phi).

Here the one of the important thing is that "w" is determine by the proportionality constant between the force and the displacement, or more directly between the acceleration and the displacement, and in fact, w^2= prop constant. The other two constants, x_m and phi depend on individual cases. All of this is analogous to constant-acceleration motion, where x(t)=x_0+v_0*t-(1/2)g*t^2. "g" is determine by the value of the aceleration, and the same for any free falling objects, but x_0 and v_0 depend on how they start their fall. Some objects start falling from REST, meaning v_0=0 while others have finite v_0. Some have zero x_0 and some others don't.

"phi" is called the phase of oscillation sometimes, and is related to which part of the cosine curve a given oscillation starts.

we had a few exercises to figure out this relation between "phi" and actual oscillation "phase".

We also look at how oscillation frequency (or period, angular freq) can be calculated from basic properties of the oscillators like spring constant, mass, gravity, etc.

Another thing we discussed was about how spring constant change if a spring is halved, or what the "effective" spring constant would be if you connect two springs in various ways. "effective" means what the spring constant of a single spring would be if it were to show the same elongation properties as the two springs together.

More oscillation

More look at oscillating motion, this time using "physical pendulum".

Grandfather clock problem from the final exam two years ago was solved as an example of application.

Waves

Waves are descirbed by y(x,t)=Asin 2pi/lambda(x-vt), or in general superposition of many waves of this form of different wave lengths. What does this imply in terms of how a section of wave-carrying string moves in time, how the whole string looks at different instances, and how these things are related to amplitude, period, frequency, wavelength and the velocity of the wave were discussed.