How to Simplify Physics Problems

To make physics simpler, an object can be approximated to act like it has all its mass concentrated at a single point.

This single point is called the ‘center of mass’. If the density of the object is uniform, it is simple to compute.

For instance, if a metal rod is of uniform density, the center of mass is just the center of the rod.

Techniques of integration can be used for more complicated situations, but clever analysis of an object can allow you to understand if it is symmetric about an axis or be twice the amount of one side of it.

Observations like these can lessen the work necessary.

Physics of Freeway Entrances

On round freeway turnouts there are almost always signs that say to slow down to a lower speed.

The reason for this is that the frictional force of the tires on the road will not always be enough to keep a car under control at a higher speed.

Two devices that civil engineers use when constructing roads to make them safer are giving the turnout circle a bigger radius and putting it at an angle.

Making the circle have a bigger radius will make the turn less sharp (thus lowering the centripetal acceleration).

Putting the turnout at an angle will make the normal (perpendicular to the surface) force have a component that keeps the car from losing control.

Both these methods will make it technically safer to drive at higher speeds on such a turnout, but it’s definitely still advisable to slow down.

Momentum, Baseball Bats, & Superstition

In any given system the total momentum will remain constant.

For the case of Sammy Sosa, a corked bat is lighter and might therefore be swung faster, but momentum is mass multiplied by velocity, so the higher velocity will be cancelled out by a lesser mass.

Also the bat is only in contact with the ball for about 1/1000 second, so the “trampoline” effect of the cork will not be great at all– the time of contact is so short.

The cork center can have a negative effect too, since energy that makes the more springy bat change shape will not be available to propel the ball forward.

The benefits probably are mostly psychological, since many ball players are superstitious.

Moments of Inertia & Spinning

Different types of moments of inertia can be better suited to different purposes and situations.

If you wanted to store power in a spinning object, you could store more power in an object with a greater moment of inertia since it would be harder to both start and stop the spinning.

However, for something like a bicycle tire you don’t want it to be hard to spin and to stop spinning.  It will be more useful if you just concentrate all the energy into moving forward efficiently.

A disc with more mass toward the center will go faster than a hoop with the mass on the outside.

The disc that is easier to spin wouldn’t be efficient in storing power though.

Impulse and Momentum

Closely related to momentum is impulse (denoted by J), which is simply the initial momentum subtracted from the final momentum.

The idea of impulse is important in things like mechanical engineering in which objects crash into each other.

The late scientist Harold Edgerton was able to further examine impulse and show how bodies interact when they crash at high speeds by using his strobe light.

Things such as baseballs being hits by bats and bullets going through playing cards could then be analyzed.

Impulse has more important consequences in car safety and the force of impact can be found by knowing the impulse through the equation F=J/Δt.

Using Conservation of Energy for Motion Problems

In motion problems you can use the truth, but you can also sometimes use the principle of conservation of energy, and it is often simpler.

In a closed system, energy will not be created or destroyed, but it can change forms, ie from potential energy to kinetic energy.

In the carnival game where you try to roll a ball over a hill and up an incline so that it doesn’t return over the hill on the way back, it would be impossible if there were no friction, but since there is friction- it is very difficult to be precise enough.

The energy lost due to friction makes it possible to go over the first incline and still not quite have enough energy to make it back over.

The presence of friction may make a system seem like it loses energy, but the energy in friction becomes heat and is not really “lost”.

When manipulating equations for initial and final energy, it is not really necessary to memorize a negative sign to put in a certain place, but one should know that the initial will always equal the final and the sign, for something like friction, should be changed accordingly.

Friction

The less wise of a group of monks, after hearing the equation x-x0=v0xt and rushing away from the physics lecture, thought that if they gave building blocks a simple push they would continue moving until stopped by another push.

They believed that by utilizing this piece of knowledge they could easily build a grand monastery of marble blocks.

They gathered up a group of rather gullible monks, who had not taken much physics, and went out to the quarry.

Unfortunately when they gave a block of marble a push, it did not start moving. Bewildered, the monks did not understand what was happening.

Fortunately, a bishop/physicist came along and explained that horizontal movement actually is also affected by vertical components through friction.

He stated that when something is in contact with a surface, the normal force and friction coefficient comprise a force opposite to the direction that the object is being moved in, with the formula f=μN.

This wise man said that the normal force also is changed if there is a slope of an angle theta. One of the less knowledgeable monks asked about force triangles, but the wiser bishop/physicist said that they are unnecessary if the physics is truly understood.

Springs – Kinetic Energy & Potential Energy

If you pull on a spring like a slinky it’s going to be pretty easy to stretch it, but if the spring is something more like that from a car’s suspension, it will be more difficult– the difficulty in pulling the spring is denoted as K (the particular spring constant).

On the topic of springs you can also understand something about energy.

When the spring is stretched out as far as it can be the potential energy is said to be the highest since it has the greatest pull back to the starting position.

When the spring reaches this starting position on the way back the kinetic energy will be highest since it is moving the fastest.

Once it passes this point though the kinetic energy will start decreasing since the spring will always have the tendency to move back to the resting position.

Air resistance and Levels of Detail in Physics

Somewhat similarly to friction, air resistance also acts as a retarding force.

The real world is not the same as idealized beginning physics situations; for example, projectiles will not follow the path of a parabola, unless they are in a vacuum.

Also, this air resistance is not a constant value, but instead it is proportional to velocity.

In situations with air resistance the acceleration is not constant either, to deal with that you can express acceleration as dv/dt (the instantaneous acceleration).

Since derivatives are involved, the use of integration, and methods such as u-substitution, can be used to solve these more complicated motion problems.

Because integrals can be complicated, computers were invented for greater artillery accuracy in wartime.

Momentum and Momentum’s SI Units

There is something in physics that does not have a special name like a Newton or a Joule.

It is momentum and can be expressed as mass multiplied by velocity (kg*m/s).

Like energy and matter, momentum is conserved.

If you shoot a gun, there will be a recoil; the bullet goes forward and there is also a force backward.

The greater the force forward is, the greater the opposite force will be, so shooting a shotgun will have more recoil than James Bond’s ppk since the a shotgun shell has greater mass and about the same speed.

Also if you’re ever on a frictionless frozen lake you could throw your boot in the opposite direction of the direction you want to go, and you would end up moving to the edge of the lake (if you disregard friction).

In the real world though there is friction, and losing the toasty interior of the boot might not be the best idea.