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Leonid Afremov is a passionate painter from Mexico who paints with palette knife with oil on canvas. He loves to express the beauty, harmony and spirit of this world in his paintings, which are rich in different moods, colors and emotions.
We want to create an index of the lady scientists on Tumblr, much like shychemist did for all scientists. (Have we thanked you lately?!) However, it’s a bit more challenging as not all of you announce your gender on your blogs. So, if you identify as a woman/female,* let us know if you want to…
I am a female scientist with the “Physics in Drops" blog
Given two vectors in three dimensions, one can define their vector or cross product as new vector, perpendicular to both original vectors, and with magnitude proportional to the sine of the angle from the first to the second vector.
In this animation, the cross product of two vectors a (blue) and b (red) are used, and their cross-product (vertical, in purple) is shown varying as the angle between both vectors changes.
As you can see, when both vectors are separated by a right angle (90° = π/2 radians or τ/4 radians), their cross product’s vector reaches a maximum length, and when both vectors are parallel, their cross product is zero. (Here, both vectors a and b are unit vectors, so the magnitude of their cross product doesn’t grow beyond 1 either)
Similar to the way the dot product can be used to find if two vectors are perpendicular, one can use the cross product to find out if two vectors are parallel. You just have to check if these products are zero in each case.
In mathematical notation, we write |v| or ||v|| as the norm or magnitude of a vector v. With that notation, we can say that |a × b| = |a| |b| sin(θ), whereθ is the angle from a to b.
The cross product is said to be “anticommutative”, that is, the order used is important (so it is not commutative, in which case the order wouldn’t matter), and the “anti-” bit says that switching the order switches the sign of the product. Mathematically, a × b = -(b × a)
To be precise, the vector that results from the cross product is said to be a pseudovector, as it is not invariant through reflections: if you see your coordinate axes from a mirror, your right-handed coordinate system would become left-handed, so the definition of the vector breaks down. Vectors are extremely useful because they don’t depend on your system of coordinates, but pseudovectors do, and that’s why they are a caveat worthy of note. This is usually not a problem as long as you stick within the same handed-ness in different system of coordinates, by avoiding such reflections.
There are several ways to actually compute the value of the cross product between two vectors, but the most common one and easier to remember is by finding the determinant of a particular matrix.
Curiously enough, the definition of a vector product returning another vector is unique to 3 and 7 dimensions. More general but similar objects can be defined for other dimensions.
The cross product is very useful in physics for describing things such as torques (the rotational equivalent of a force), angular momentum (the rotational equivalent of linear momentum), and magnetic forces on charged particles (which act perpendicular to both the velocity of the particle and the magnetic field).
The physical nature of the vector quantity in such cases as torque or angular momentum can be tricky to understand without the proper insight, which is something that is rarely addressed by physics text books.
It is common for students to get stuck to the idea that a vector, as represented by the arrow, points to the direction the force or whatever it is acting or “going towards”. But for angular momentum and torque, this intuition breaks down.
The proper way to think about the vector for angular momentum is that the vector gives you an axis of rotation. The way the arrow is pointing tells you which of two possible ways the rotation is going (clockwise or counterclockwise, depending on your choice of coordinates and point of view). Using the right-hand rule (for a right-handed coordinate system), you can figure this out easily.
The magnitude of the vector is the actual magnitude of the momentum. So the vector is just a compact way to merge both bits of information on angular momentum in a single mathematical object. The consistency in all of the definitions is what makes it all work nicely, not coincidence.
The idea of a vector representing the axis and direction of rotation is the same here, but you can also, alternatively, consider it a plane in which the torque is acting on. The vector is normal to this plane.
But another tricky idea here is the dimensions of the torque vector. Remember that in physics, we say length, time and mass, for instance, are dimensions for a physical quantity. We say a meter, a second and a kilogram are units with the dimensions described before, respectively. This difference in terms (units vs. dimensions) is very important, and a lot of people don’t get it right the first time.
So, the dimensions of the torque vector are pretty weird: Newton-meter. A lot of students realize that this is the same thing as a Joule, which is a unit of energy. So why not say torque has the same dimension as energy, call it Joule, and get rid of the Newton-meter thing?
The answer is that while the dimensions match, the concepts don’t. Torque and energy are entirely different concepts, entirely different physical quantities, so they shouldn’t be treated the same even though their dimensions seem to match. But in my opinion, this difference is dogmatic if taken as Newton-meter vs. Joule, because it hides a very important detail.
I think torque makes more sense in units of Joules per radian. The radian is a dimensionless unit, which means it was hidden in there all along in our dimensional analysis. We were not comparing Joules with Joules, but Joules per radians with Joules! The radians bit comes from the fact torques act along an arc.
This is easy to see if we consider the work done by a torque τ: W = τθ, where θ is the angle the torque acted around, rotating an object. In this case, if you consider the dimensions of radians as non-disposable, you can easily see that it all works out.
The cross product is a very handy tool for defining some more complicated physical quantities. It may seem arbitrary at first, but the reasoning behind its definition is mathematically sound and extremely useful in practice.
In order to fully appreciate it, one must first get rid of a few intuitions on what vectors represent in physics. Vectors can represent a lot of things that are not explicitly directional, as you first start getting used to them, so the sooner you abandon that intuition the better.
Wave behavior, representing Schrodinger’s equation of quantum mechanics.
- The first [top] image shows three wave functions which satisfies the time-dependent Schrödinger equation for a harmonic oscillator.
- The second equation corresponds to a particle traveling freely through empty space
More on the TOP IMAGE
- Left: The real part (blue) and imaginary part (red) of the wave function.
- Right: The probability distribution of finding the particle with this wave function at a given position.
The top two rows are examples of stationary states, which correspond to standing waves.
The bottom row an example of a state which is not a stationary state.
The right column illustrates why stationary states are called “stationary”.
The interactions between Nonpolar molecules and Water molecules are not as favorable as interactions amongst just the water molecules, due to the inability of nonpolar molecules to form hydrogen bonding or electrostatic interactions.
When nonpolar molecules are introduced to the water molecules, the water molecules will initially surround the nonpolar molecules, forming a “cages” around the molecules. However, the tendency of nonpolar molecules to associate with one another will draw the nonpolar molecules together, forming a nonpolar aggregate.
Based on the second law of thermodynamics, the total entropy of the system plus its surrounding must always be INCREASING. Therefore, it is Favorable for the nonpolar molecules to associate without the interference of water. The water molecules that initially “caged” the nonpolar molecules are released from the nonpolar molecules’ surfaces, creating an Increase in Entropy in the surrounding (increase in disorder). The favorable release of water molecules from nonpolar surfaces is responsible for phenomenon of the hydrophobic effect.
When two nonpolar molecules come together, structured water molecules are released allowing them to interact freely with bulky water. The release of water from such cages is favorable. The result is that non-polar molecules show an increased tendency to associate with one another in water compared with others - less polar and less self-associating solvents. This tendency is called the hydrophobic effect and the associated interactions are called hydrophobic interaction.
The release from the cage-like clathrates is more favorable because it increases the entropy of the system.
Hydrophobic interactions can also be seen in the clustering of amphipathic/amphiphillic molecules such as phospholipids into bilayers and micelles. The hydrophobic areas of amphipathic molecules cluster together to avoid the ordered “cage” of water molecules that would surround them and orient the hydrophillic ends as a shield-like outer structure that interacts amicably with the polar water molecules.
Micelles occur when a spherical fatty acids structure is formed with a hydrophobic core and hydrophillic outer shell.
Bilayers can be commonly seen in cell membranes with hydrophillic outer (outside the cell) and inner (inside the cell) linings has hydrophobic (inside the membrane) center.
The Lipid bilayer is a more favored formation in nature due to the micelle formation may contain bulky fatty acids causing hindrance in its formation.
Most cell membranes are electrically polarized, such that the inside is negative [typically 260 millivolts (mV)]. Membrane potential plays a key role in transport, energy conversion, and excitability.
For example, membrane transport. Some molecules can pass through cell membranes because they dissolve in the lipid bilayer.
Additionally, most animal cells contain a high concentration of K1 and a low concentration of Na1 relative to the external medium. These ionic gradients are generated by a specific transport system, an enzyme that is called the Na1–K1 pump or the Na1–K1 ATPase.
The hydrolysis of ATP by the pump provides the energy needed for the active transport of Na1 out of the cell and K1 into the cell, generating the gradients.
The pump is called the Na1–K1 ATPase because the hydrolysis of ATP takes place only when Na+ and K+ are present. This ATPase, like all such enzymes, requires Mg2+