How to do a phasor calculation with radians using a principle of Eulers Formula

phasors problem

“How has this been calculated? I try to convert the 7.11 to radians and sum it up, then converting it back to degrees. However I do not get 1.61.”

-Quora question

My response:

The numbers are already in radians.

Dividing the coefficients does not seem to be the problem you had.

If you divide the e^x terms you would normally subtract the jπ/4 from the -j7.11.

That would get you

~-j7.8954

It’s best to save this in memory to not lose accuracy.

From there we can think about Euler’s formula. Shown on the right side in green. Much like the trigonometric functions we learn much before doing anything with phasors, you can add or subtract 2π to the input and get the same result for periodic functions. You can also do that with e^ix.

Eulers Formula imaginary calculation

For this problem we add j2π to the -j7.8954 to get -j1.61. The answer is an approximation.

What is the Moment of Inertia?

We looked at rotational moment and the moment of inertial as well as some connected ideas.

The Greek letter omega stands for angular velocity, alpha stands for angular acceleration. Capital L stands for angular momentum. Capital T stands for the period.

Focused some time on algebraic manipulation of equations, especially with reciprocals and ways to simplify the equations.

It would help to write notes about what the teacher is saying for some problems rather than just what is on the board.

The ‘moment of inertia’, for which we use the capital letter I is basically how difficult it is to rotate something. Similar to mass in that it takes more force to accelerate a larger mass, it takes more torque to accelerate something in an angular way. It does not have to do with ‘a moment in time’.

“A 2 kg ball is dropped from 25 m up. How fast is it going 5 meter above the ground, no air friction?”

One way to approach the problem is to use conservation of energy. Part of the gravitational potential energy is converted to kinetic energy. Given the directions, we can assume conversation of energy.

The energy converted is the change in height multiplied by gravity multiplied by the mass. That same amount of energy will become kinetic energy.

So you can set that amount of energy equal to the equation for kinetic energy and solve for the velocity.

ball dropping energy conversion

You could also use the kinematic equations. The initial velocity would be zero.

two linear motion equations

“How many US gallons of water are in one metric ton?”

You can input this question directly in Google verbatim and it will output your answer.

Google has not always been able to do this, but it has been able to do this for at least a few years. I could use this functionality as an undergrad and as a graduate student.

Google is aware of many physical quantities.

gallons in a ton

“How do you calculate the volume of a liquid? What’s the unit of measurement?”

Volume is basically the three dimensional space that something fills. So you can place the liquid in a container with standardized markings.

Volume can change based on temperature and pressure though.

The SI unit of volume is the cubic meter. Other units used can be cubic centimeters, liters, etc. Often we choose the units based on the situation. To use units in an equation though, it’s often better to start with SI units.

You can convert between units of volume using ratios. Another simple way to convert units is just to use Google,

You can input something like this:

“Two liters is how many gallons”

And Google will tell you.

Example:

volume liters gallons

You can also calculate the volume if you know the density and mass. (Again, temperature and pressure are also factors).

Density = mass/volume

Volume = mass/density

How are Decibels Measured?

You would need a device to measure the intensity of the sound.

Intensity is the power divided by area, it’s similar conceptually to pressure.

Once you have the intensity, you would use an equation to get decibels, which are often a convenient unit for measuring sound.

decibels sound intensity

I0 is the reference intensity, which is often 10^-12 W/m^2. That is the threshold for human hearing.

You can imagine the intensity as being the power of the sound propagating outward from a point like a sphere if the sound spreads equally in all directions. Speakers can also be more directional. The logarithm is base 10 and you multiply by ten in front because of the prefix deci.

A Song of Ice and Steam

0.6 kg of ice at zero degrees Celsius.

0.2 kg of steam at 100 degrees Celsius.

What will the final temperature be?

Using the latent heat of fusion, the latent heat of vaporization, and the specific heat. All of water.

How to calculate the weight, given a mass

Newton’s Second Law states that

F = m a

(Vectors in bold)

Weight = gravitational force (units of Newtons in SI units)

F = m g

g being the acceleration due to gravity.

On Earth, g ~ -9.8 m/s^2

Changes in weight

But that acceleration will change depending on position on the Earth and altitude and will also change on the moon, in space, and in other places.

You can find the acceleration due to gravity by dividing the gravitational force by the mass.

There is an acceleration due to gravity whenever there are two masses, they can be a person sized mass and a planet sized mass, two smaller masses, or any combination of two masses.

Therefore the weight is less or more depending on where it is.

There is less gravity on the moon, so the weight is less there (for the same mass) compared to on Earth.

Let’s say on a bigger planet ‘X’, the mass of the planet is more than Earth. Therefore the acceleration due to gravity is greater and weight would be greater (for the same mass) compared to Earth.

If there are no other masses nearby, in space for instance, things are weightless.

You can use the Law of Gravitation for various situations.

Here’s a diagram

weight given mass

Physics of Speeding Tickets & Skidmarks

If you know the length of a car’s tires skidmarks and the coefficient of friction, you can figure out the starting velocity.

Tutoring Physics – Conversion of Energy, Kinetic Energy into Work Done by Friction

We started by looking at some problems involving conservation of energy and conversion of energy between different types.

The problem we went through a few times involved a car stopping using the brakes and how kinetic energy was converted to heat by the mechanism of friction.

We looked at the units of joules and how equations with work, gravitational potential energy, elastic potential energy, kinetic energy, etc all have the same units.

We also used the equation to derive the units of a spring constant.