gravity

The Jeans Mass, or: How are stars born?

No, this has nothing to do with pants. The Jeans mass is a concept used in astrophysics and its unlikely name comes from the British physicist Sir James Jeans, who researched the conditions of star formation. The question at the core is: under what circumstances will a dark and lonely gas cloud floating somewhere in the depth of space turn into a shining star? To answer this, we have to understand what forces are at work.

One obvious factor is gravitation. It will always work towards contracting the gas cloud. If no other forces were present, it would lead the cloud to collapse into a single point. The temperature of the cloud however provides an opposite push. It “equips” the molecules of the cloud with kinetic energy (energy of motion) and given a high enough temperature, the kinetic energy would be sufficient for the molecules to simply fly off into space, never to be seen again.

It is clear that no star will form if the cloud expands and falls apart. Only when gravity wins this battle of inward and outward push can a stable star result. Sir James Jeans did the math and found that it all boils down to one parameter, the Jeans mass. If the actual mass of the interstellar cloud is larger than this critical mass, it will contract and stellar formation occurs. If on the other hand the actual mass is smaller, the cloud will simply dissipate.

The Jeans mass depends mainly on the temperature T (in K) and density D (in kg/m³) of the cloud. The higher the temperature, the larger the Jeans mass will be. This is in line with our previous discussion. When the temperature is high, a larger amount of mass is necessary to overcome the thermal outward push. The value of the Jeans mass M (in kg) can be estimated from this equation:

M ≈ 1020 · sqrt(T³ / D)

Typical values for the temperature and density of interstellar clouds are T = 10 K and D = 10-22 kg/m³. This leads to a Jeans mass of M = 1.4 · 1032 kg. Note that the critical mass turns out to be much greater than the mass of a typical star, indicating that stars generally form in clusters. Rather than the cloud contracting into a single star, which is the picture you probably had in your mind during this discussion, it will fragment at some point during the contraction and form multiple stars. So stars always have brothers and sisters.

(This was an excerpt from the Kindle book Physics! In Quantities and Examples)

The Difference Between Mass and Weight

In general, it is acceptable to use weight as a synonym for mass. However, in a very strict physical sense this is incorrect. Weight is the gravitational force experienced by an object and accordingly measured in Newtons and not kilograms. An object of mass m has the weight F:

F = m · g

with the gravitational acceleration g. On Earth the value of the gravitational acceleration at the surface is g = 9.81 m/s². So a typical adult with a mass of m = 75 kg has a weight of:

F = 75 kg · 9.81 m/s² = 735.75 N

On the moon (or any other point of the universe), the mass would remain at m = 75 kg. But since the gravitational acceleration on the moon is much lower (g = 1.62 m/s²), the weight changes to:

F = 75 kg · 1.62 m/s² = 121.5 N

Keep this distinction in mind. Mass is a fundamental property of an object that does not depend on the conditions outside the object, while weight is a variable that changes with the strength of surrounding gravitational field.

(This was an excerpt from Physics! In Quantities and Examples)

What is Mass? A Short and Simple Explanation

Mass is such a fundamental property of matter that it is hard to define without drifting into philosophical realms. Newton’s Second Law provides a great way to understand mass from a physical point of view. The law states that force F (in N) is the product of mass m (in kg) and acceleration a (in m/s²):

F = m · a

So according to this, mass is a measure of an object’s resistance to a change in speed. If the mass is small, a small force is sufficient to produce a noticeable acceleration. However, much more force is necessary to produce the same acceleration for a massive object.

Another way of looking at mass is provided by Newton’s Law of Gravitation. Newton found that the attracting gravitational force between two objects is proportional to the product of their masses m and M:

F ~ m · M

So additionally to creating resistance to changes in state of motion, mass is also the source of gravitational attraction. It seems obvious that in both cases we are talking about the same quantity. But is this actually the case? Is the inertial mass, the mass responsible for opposing changes in velocity, really the same as the gravitational mass, that gives rise to gravity?

This question has led to heated debates among physicist for centuries. All experiments conducted so far, with ever increasing accuracy, have shown that indeed the inertial mass is identical to the gravitational mass. Today, almost all physicists have accepted this equivalence as reality.

The SI unit of mass is kilograms. Ever since 1889, one kilogram has been defined as the mass of the international prototype kilogram (IPK) that is stored in the International Bureau of Weights and Measures in Paris. However, during the 24th General Conference on Weights and Measures that took place in 2011, physicists have agreed to redefine this unit by connecting it to the Planck constant.

Other units that are commonly used for mass are grams (1/1000 of a kilogram), the pound (equal to about 0.45 kilograms) and the tonne (equal to 1000 kilograms). For atoms and molecules scientists use the atomic mass unit u. One u is equivalent to 1.66 · 10-27 kg, which is roughly the mass of a neutron or proton.

(This was an excerpt from Physics! In Quantities and Examples)

Law Of The Lever – Explanation and Examples

Imagine a beam sitting on a fulcrum. We apply one force F'(1) = 20 N on the left side at a distance of r(1) = 0.1 m from the fulcrum and another force F'(2) = 5 N on the right side at a distance of r(2) = 0.2 m. In which direction, clockwise or anti-clockwise, will the beam move?

Great Formulas II_html_m2eb595ec

(Before reading on, please make sure that you understand the concept of torque)

To find that out we can take a look at the corresponding torques. The torque on the left side is:

T(1) = 0.1 m · 20 N = 2 Nm

For the right side we get:

T(2) = 0.2 · 5 N = 1 Nm

So the rotational push caused by force 1 (left side) exceeds that of force 2 (right side). Hence, the beam will turn anti-clockwise. If we don’t want that to happen and instead want to achieve equilibrium, we need to increase force 2 to F'(2) = 10 N. In this case the torques would be equal and the opposite rotational pushes would cancel each other. So in general, this equation needs to be satisfied to achieve a state of equilibrium:

r(1) · F'(1) = r(2) · F'(2)

This is the law of the lever in its simplest form. Let’s see how and where we can apply it.

—————————

A great example for the usefulness of the law of the lever is provided by cranes. On one side, let’s set r(1) = 30 m, it lifts objects. Since we don’t want it to fall over, we stabilize the crane using a 20,000 kg concrete block at a distance of r(2) = 2 m from the axis. What is the maximum mass we can lift with this crane?

First we need to compute the gravitational force of the concrete block.

F'(2) = 20,000 kg · 9.81 m/s² = 196,200 N

Now we can use the law of the lever to find out what maximum force we can apply on the opposite site:

r(1) · F'(1) = r(2) · F'(2)

30 m · F'(1) = 2 m · 196,200 N

30 m · F'(1) = 392,400 Nm

Divide by 30 m:

F'(1) = 13,080 N

As long as we don’t exceed this, the torque caused by the concrete block will exceed that of the lifted object and the crane will not fall over. The maximum mass we can lift is now easy to find. We use the formula for the gravitational force one more time:

13,080 N = m · 9.81 m/s²

Divide by 9.81:

m ≈ 1330 kg

To lift even heavier objects, we need to use either a heavier concrete block or put it at a larger distance from the axis.

—————————

The law of the lever shows why we can interpret a lever as a tool to amplify forces. Suppose you want use a force of F'(1) = 100 N to lift a heavy object with the gravitational pull F'(2) = 2000 N. Not possible you say? With a lever you can do this by applying the smaller force at a larger distance to the axis and the larger force at a shorter distance.

Suppose the heavy object sits at a distance r(2) = 0.1 m to the axis. At what what distance r(1) should we apply the 100 N to be able to lift it? We can use the law of the lever to find the minimum distance required.

r(1) · 100 N = 0.1 m · 2000 N

r(1) · 100 N = 200 Nm

r(1) = 2 m

So as long as we apply the force at a distance of over 2 m, we can lift the object. We effectively amplified the force by a factor of 20. Scientists believe that the principle of force amplification using levers was already used by the Egyptians to build the pyramids. Given a long enough lever, we could lift basically anything even with a moderate force.

—————————

This was an excerpt from More Great Formulas Explained.

Check out my BEST OF for more interesting physics articles.

Comets: Visitors From Beyond

The one thing we love the most in the world of astronomy is a good mystery. And if there was ever a mysterious and yet very powerful force of nature that we witness in the night skies, it is the coming of the mighty comet.

The arrival of a comet within view of Earth is an event of international importance. Witness the huge media attention that the Haley or Hale-Bopp have had when they have come within view The sight of these amazing space objects is simultaneously frightening and awe inspiring.

Image

Above all, it is during these comet viewings that the astronomer comes out in all of us. But what is a comet? Where did it come from? And how does it get that magnificent tail?

We should never confuse comets with asteroids. Asteroids are small space rocks that come from an asteroid belt between Mars and Jupiter. While still quite stunning to see, they pale in comparison to the arrival of a comet. Asteroids also have received considerable study by the scientific community.

Not as much is known about comets. As a rule, comets are considerably larger than asteroids. The composition of a comet is a mixture of nebulous, gasses, ice, dust and space debris. One scientist called the composition of a comet as similar to a “dirty snowball” because the composition is so diverse and changeable. The center or nucleus of a comet is usually quiet solid but the “snowball” materials often create a “cloud” around that nucleus that can become quite large and that extends at great lengths behind the comet as it moves through space. That trailing plume is what makes up the comet’s magnificent tail that makes it so exciting to watch when a comet comes within view of Earth.

The origins of comets is similarly mysterious. There are a number of theories about where they come from but it is clear that they originate from outside our solar system, somewhere in deep space. Some have speculated they are fragments left over from the organization of planets that get loose from whatever gravitational pull and are sent flying across space to eventually get caught up in the gravity of our sun bringing them into our solar system.

Another theory is that they come from a gaseous cloud called the Oort cloud which is cooling out there after the organization of the sun. As this space debris cools, it gets organized into one body which then gathers sufficient mass to be attracted into the gravity of our solar system turning into a fast moving comet plummeting toward our sun. However, because of the strong gravitational orbits of the many planets in our solar system, the comet does not always immediately collide with the sun and often takes on an orbit of its own.

The life expectancy of comets varies widely. Scientists refer to a comet that is expected to burn out or impact the sun within two hundred years as a short period comet whereas a long period comet has a life expectancy of over two hundred years. That may seem long to us as earth dwellers but in terms of stars and planets, this is a very short life as a space object indeed.

Scientists across the globe have put together some pretty impressive probes to learn more about comets to aid our understanding of these visitors from beyond. In 1985, for example, the United States put a probe into the path of the comet Giacobini-Zinner which passed through the comets tail gathering tremendous scientific knowledge about comets. Then in 1986, an international collation of scientists were able to launch a probe that was able to fly close to Haley’s comet as it passed near Earth and continue the research.

While science fiction writers and tabloid newspapers like to alarm us with the possibility of a comet impacting the earth, scientists who understand the orbits of comets and what changes their paths tell us this is unlikely. That is good because some comets reach sizes that are as big as a planet so that impact would be devastating. For now, we can enjoy the fun of seeing comets make their rare visits to our night sky and marvel at the spectacular shows that these visitors from beyond put on when they are visible in the cosmos.

Physics: Free Fall and Terminal Velocity

After a while of free fall, any object will reach and maintain a terminal velocity. To calculate it, we need a lot of inputs.

The necessary quantities are: the mass of the object (in kg), the gravitational acceleration (in m/s²), the density of air D (in kg/m³), the projected area of the object A (in m²) and the drag coefficient c (dimensionless). The latter two quantities need some explaining.

The projected area is the largest cross-section in the direction of fall. You can think of it as the shadow of the object on the ground when the sun’s rays hit the ground at a ninety degree angle. For example, if the falling object is a sphere, the projected area will be a circle with the same radius.

The drag coefficient is a dimensionless number that depends in a very complex way on the geometry of the object. There’s no simple way to compute it, usually it is determined in a wind tunnel. However, you can find the drag coefficients for common shapes in the picture below.

Now that we know all the inputs, let’s look at the formula for the terminal velocity v (in m/s). It will be valid for objects dropped from such a great heights that they manage to reach this limiting value, which is basically a result of the air resistance canceling out gravity.

v = sq root (2 * m * g / (c * D * A) )

Let’s do an example.

Skydivers are in free fall after leaving the plane, but soon reach the terminal velocity. We will set the mass to m = 75 kg, g = 9.81 (as usual) and D = 1.2 kg/m³. In a head-first position the skydiver has a drag coefficient of c = 0.8 and a projected area A = 0.3 m². What is the terminal velocity of the skydiver?

v = sq root (2 * 75 * 9.81 / (0.8 * 1.2 * 0.3) )

v ≈ 70 m/s ≈ 260 km/h ≈ 160 mph

Let’s take a look how changing the inputs varies the terminal velocity. Two bullet points will be sufficient here:

  • If you quadruple the mass (or the gravitational acceleration), the terminal velocity doubles. So a very heavy skydiver or a regular skydiver on a massive planet would fall much faster.
  • If you quadruple the drag coefficient (or the density or the projected area), the terminal velocity halves. This is why parachutes work. They have a higher drag coefficient and larger area, thus effectively reducing the terminal velocity.

This was an excerpt from the Kindle ebook: Great Formulas Explained – Physics. Mathematics, Economics. Check out my BEST OF for more interesting physics articles.

A tunnel through earth and a surprising result …

Recently I found an interesting problem: A straight tunnel is being drilled through the earth (see picture; tunnel is drawn with two lines) and rails are installed in the tunnel. A train travels, only driven by gravitation and frictionless, along the rails. How long does it take the train to travel through this earth tunnel of length l?

The calculation, shows a surprising result. The travel time is independent of the length l; the time it takes the train to travel through a 1 Km tunnel is the same as through a 5000 Km tunnel, about 2500 seconds or 42 minutes! Why is that?

Imagine a model train on rails. If you put the rails on flat ground, the train won’t move. The gravitational force is pulling on the train, but not in the direction of travel. If you incline the rails slighty, the train starts to move slowly, if you incline the rails strongly, it rapidly picks up speed.

Now lets imagine a tunnel through the earth! A 1 Km tunnel will only have a slight inclination and the train would accelerate slowly. It would be a pleasant trip for the entire family. But a 5000 Km train would go steeply into the ground, the train would accelerate with an amazing rate. It would be a hell of a ride! This explains how we always get the same travel time: the 1 Km tunnel is short and the velocity would remain low, the 5000 Km is long, but the velocity would become enormous.

Here is how the hell ride through the 5000 Km tunnel looks in detail:

The red, monotonous increasing curve, shows distance traveled (in Km) versus time (in seconds), the blue curve shows velocity (in Km/s) versus time. In the center of the tunnel the train reaches the maximum velocity of about 3 Km/s, which corresponds to an incredible 6700 mi/h!