Picture 6.02
Angular momentum is actually the product of mass (m), velocity (v) and radius (r) from the centre.
Angular momentum is conserved and not changing. Therefore, the product or m, v and r is constant.
Velocity comes from angular velocity. Angular velocity is ratio of velocity (v) /radius (r). From there, we can work out that velocity is the product of angular velocity and radius. When we replace v in mvr product with angular velocity times r, we get a final equation for angular momentum which is:
mass x angular velocity x radius on squared
This result is constant, does not change as it represents conservation of angular momentum.
Angular momentum = m v r
Therefore: m v r = constant
Velocity (v) is coming from angular velocity (ω) Angular velocity is:
ω = v (velocity) / r (radius)
From this equation, velocity is:
v = ω r
It is the same as:
3(ω)=6(v)/2(r) where
6(v)= 3(ω) times2(r)
So in angular momentum, v can be replaced by ωr:
Therefore, angular momentum = m v r is:
Angular momentum= m ωr r
(r multiplied by r = r2 )
So the final equation for angular momentum is:
angular momentum = m ω r 2
m = mass
ω = angular velocity
r2 = radius
This means that if the distance or radius from the centre is reduced, then the angular velocity or speed of spinning has to increase for the product to remain constant. Equally, spinning or angular velocity will be reduced if the radius from the centre is increased, as the product has to remain constant because angular momentum is conserved. So when the Moon was closer to the Earth, radius or distance of the Moon to the centre of Earth was smaller. Consequently, the Moon was spinning around the Earth faster and with it, Earth rotating faster. Here, there is also another effect which the Moon exerted to the Earth, and that is the gravitational effect on the oceans causing tides. I will not go into this now.
Picture 6.03
The illustration above is just my attempt to demonstrate that an ice dancer spins faster when her hands are brought together close to her body (chest). If she spins with her arm and legs outstretched, then she is not able to spin so fast.
In modern science and physics in particular there is one very important entity, which is tightly related to laws of conservation. It is called invariance.
Invariance is a property, or quantity, which does not change and is always measured having exactly the same value regardless of the conditions under which measurement is taking place. Speed of light is an example of invariance. Conservation of baryonic or lepton numbers is another example.
If the closed system, with the internal energy of a certain value, gets larger or space increases within the system, the quantity of internal energy will remain the same. Roughly speaking, the number of energy units per unit of volume will be less in increased space compared to the time when the system was reduced. However, the total amount of internal energy will not change with a change of space and time. This is obviously when we are considering a closed system with no surroundings. The same applies to linear and angular momentum.
We know now that speed of light is constant and does not change regardless of the conditions under which the speed is measured. It is therefore an example of invariance.
Before establishing this as a fact, it was known that everything is in motion within the universe and whether some particular object moves or not depends on the relation this particular object or matter has with another object, which is not in this same frame of motion.
The situation where an object moves with constant velocity is called an inertial reference frame. This also refers to a situation when an object is at rest. An inertial reference frame refers to these two occasions where we do not have acceleration. In any other situation, where an object is moving so that its velocity changes and acceleration or deceleration is present, it is not an inertial reference frame. It is called non-inertial reference.
When we are in an inertial reference frame (with a constant speed) we are not able to know if we are moving or not unless we compare our reference frame with another. If ten people are in the same reference frame then there is no way they will know or they can test whether they are moving or at rest if they focus their attention on physical phenomena taking place in this same reference frame. The reason is that all objects move with the same speed in this inertial reference frame and therefore look as if they are at rest within this inertial frame of reference. If we are sitting in a train with another passenger in front of us and are moving at a speed of 20 miles per hour, we are not going to reach a passenger in front of us due to this speed. The distance between us and that passenger will remain the same because he moves at the same speed as he is in the same reference frame. We can also easily pour coffee into a cup without spilling it all over as all of these are in the same reference frame, moving at the same speed. It looks as if we do not move. There are two ways we can test and confirm that we are moving. One is to look outside and see trees moving away, for instance. The other way of finding out that we are moving is if the train accelerates or decelerates or changes its velocity. With the change of velocity, we are no longer in an inertial frame as explained above.
The fact that we cannot know if we are moving when we pay attention to events taking place in our reference frame was pointed out by Galileo. It is also called Galilean relativity.
Before going further, I would like to emphasise the difference between velocity and acceleration.
Velocity is the change of position of an object from place A to place B over a period of time. This change takes place in a particular direction such as left, right, up and down or backwards and forwards. Therefore, velocity has vector property and that characteristic is very important in defying velocity. The magnitude of velocity is measured by speed.
Acceleration is a change of velocity.
Mass multiplied with acceleration (change of velocity) is a force.
force = mass x acceleration
Acceleration is a change of velocity so
force = mass x change of velocity
Whenever there is a change of velocity we have a force applied on a particular motion, which was at a constant velocity (inertial reference frame) or at rest before force changed it. That is in agreement with Newton’s first and second laws of motion which state:
First law: an object is either at rest or moves at a constant velocity unless acted on by external forces.
Second law: it basically refers to a change of velocity or change of momentum when an object is subject to external forces.
We have a change of velocity when velocity:
•Increases
•Decreases
•Changes direction
An increase or decrease in speed is obviously due to force being applied to a particular motion, which is at constant velocity or an inertial reference frame before the force was applied.
However, when the direction of constant velocity changes then we have acceleration even if we have a constant speed. An example is orbital movement or circular movement with a constant speed of 20 miles per hour all the time around. Yet we have change of velocity or acceleration, which we can feel very much.
How can we have acceleration there when speed is constant?
Change of speed can be another way of describing acceleration.
So how can we have acceleration when we have a constant speed?
The answer is in the fact that velocity is a vector and refers to a specific direction.
If an object travels at a constant speed of 20 miles per hour in the direction of left to right then it remains in an inertial reference frame with constant
velocity only for this direction. The constant velocity of this object in the direction of up to down is zero while it is in an inertial reference frame of moving left to right. If this object now changes direction of movement from left to right to direction from up to down, remaining at the same speed, the acceleration will be noticed as the object suddenly moves in a direction which until that moment has zero velocity. So, although the speed does not change, the acceleration will happen upon entering a new direction with zero speed for that direction before entering it. Therefore, in any circular motion there is acceleration, which is very much noticeable in everyday life, when we drive a car or motorbike on a bendy road, for example. As change of velocity or acceleration implies force then we have presence of force, which really is not exerted from outside. This force is experienced when an object is moving in a circular motion although it is not a force applied from outside and is only the result of acceleration due to change of velocity as the consequence of a change of direction (not taking here into account gravitational force). That is why centrifugal force is referred to as a fictitious force.
Any other motion, which is not at a constant velocity, is referred as a non-inertial reference frame. Rotational movement of the object is one of these frames as velocity constantly changes due to constant change of direction.
Before it was known that speed of light is a constant, it was believed that time and space can be regarded as invariance, or the separation in the time between two events or separation in a space between two places is a constant and not changeable. Speed, including the speed of light, was considered to be relative. Speed, i.e. how fast an object is moving, depends on the reference frame from where the measurement of this speed is taken.
If we are on a train moving at 80 miles per hour and throw the ball in the direction of the train’s movement at a speed of 5 miles per hour, we will measure this speed to be 5 miles/hour as we, and the ball, are in the same reference frame. Our inertial reference frame is 80 miles per hour.
However, for a man who is standing outside the train and is in a rest reference frame compared with our reference frame, the ball’s speed will be 85 miles per hour, which is the sum of the speed of the train and ball moving in the same direction. If the ball is thrown in the opposite direction of the train’s movement then the ball’s speed for the man outside the train will be 75 miles per hour. For the man on the train it will remain the same or at a speed of 5 miles per hour. This is logical and does not require more detailed explanation. These calculations are needed when we relate two different frames. They are called Galilean transformations. It refers to Galilean relativity and time when space and time were regarded as a constant and unchangeable value. It was Galilean relativity where also the speed of light was regarded to be such in relation to so-called ether. Basically, it was believed that the speed of light moves at such a speed only in relation to ether or another reference frame.
Galilean transformations are still valid and applicable to reference frames where speed is not significant but become less reliable with increased speed particularly going towards the speed of light. In this case, it is Lorentz transformations that are precise and relevant.
Once more, clearly to outline that observations which are done within the same inertial frame of reference cannot prove that we are moving, as all objects in this frame of reference move at the same constant speed. This is referred to as Galilean relativity.
Once more, to repeat that close to the end of the 19th century, it was believed that light moves through space at such a speed only in relation to ether or another reference frame.
Having in mind Galilean relativity and the possibility that light moves at such a speed according to ether, Einstein did a so-called mind experiment when he was only sixteen years old in 1886.
I will outline this experiment in a way which helps me to understand it.
Light is composed of photons, which can be regarded as light particles. We can imagine a train travelling at a speed of 300 000 kilometres per second and that each passenger on the train is a photon. The train is a beam of light, which has photons (passengers) travelling at a speed of 300 000 kilometres per second.
We have to keep in mind that the train (beam of light) moves at the speed of light according to another reference frame. That is a train platform and people on it. All of these make ether according to which the light beam (train) moves at the speed of 300 000 kilometres per second.
Now let’s imagine a girl is standing on the platform looking in a mirror. She is in another reference frame to the train (light beam). She is part of ether. The photons will leave her face at 300 000 kilometres per second and bounce from the mirror towards her eyes at the same speed. She will see her reflection in the mirror instantly due to the light speed.
Now we can imagine that our girl has got on the train and is now moving at the same speed of 300 000 kilometres per second. She is not any more part of ether and is moving at an exact speed like photons or people on the train in relation to ether (platform station and people standing on the platform, the place where she was before getting on the train). Now when she sits behind a passenger, she cannot reach him, as he is moving at the same speed forward as she is. In other words, if she now looks in the mirror she would not be able to see herself in the mirror, as photons cannot move from her face to the mirror because both (her and photons on her face)travel at the same speed.
Einstein did exactly this experiment, imagining what would happen if he travelled at the speed of light and with a mirror in front of him. He would then travel at the speed of light in relation to the ether as light does. Light, therefore, cannot leave his face to reach the mirror and come back due to the fact that both him and lights or photons travel at the same speed in relation to ether or in relation to other frames of reference. This is, therefore, violation of the principle described by Galilean relativity, as it is now possible to know that he is travelling at the speed of light, looking at the events happening in his own inertial reference frame. That is because he would not be able to see himself in the mirror as he is travelling at the same speed as light or photons.
What would happen if photons did not move with such speed relative to other frames of reference?
What if they moved at such a speed relative to an observer regardless of the frame of reference the observer is in?
This would mean that photons leave the girl’s face towards the mirror and bounce back to her eyes at the speed of 300 000 km per second when she is at rest in the train station as well as on the train at a speed of 300 000 kilometres per second. It means that the speed of light moves at such a constant speed in any frame reference an observer is in. In such a situation, the speed of light does not change and remains the same in any reference frame and is, therefore, an example of invariance.
In this situation, however, we cannot apply Galilean relativity and Galilean transformations.
In the train and ball example, the speed of the ball was different for a man standing in a train speeding at 80 miles per hour (the ball for that man was travelling at a speed of 5 miles per hour) and the man standing on the ground. The man on the ground at rest measured a speed of 85 or 75 miles per hour depending whether the ball was thrown in the direction or opposite direction of the train’s movement.
The girl on the train moving at light speed can now see her face as light from her face moves in relation to her with the same speed to the mirror and back. But if we apply Galilean transformations and Galilean relativity then the man standing on the ground will see photons leaving her face at a speed of 600 000 kilometres per second as she herself is moving at a speed of 300 000 kilometres per second. It would be logical of Galilean relativity and transformations to make a sum of the speed of train 300 000 km/s and speed of photons leaving her face 300 000 km/s to get a speed of 600 000 km/s.
That is not possible because light is perceived from the man’s reference frame (at rest) as also t
ravelling at a speed of 300 000 km/s as it is for the girl’s reference frame on the train. The only way the man can perceive her seeing herself in the mirror is that with such speed, the distance between her and the mirror has reduced in the direction of the movement of the train. That is only perceived from his viewpoint or reference frame at rest. In this case, the distance between the mirror and the girl’s face has reduced to zero. The time passing by for photons leaving the girl’s face has also reduced to zero for the man who is at a rest reference frame in relation to the train. That means that we have space contraction and time dilatation to keep constant speed of light to be seen as such from any reference frame and only in relation to an observer.
To put it another way, light travels at the same speed of around 300 000 km per second for the girl on the train and the man on the platform. For the girl, photons leave her face, hit the surface of the mirror and bounce back to her eyes so she can see her reflection. For the man on the platform, photons reach the mirror as well and go back to her eyes. However, such things are possible only because her mirror is now fused with her face for the perspective of the man on the platform (distance is contracted to zero at the speed of light). Also, time has stopped at the speed of light from the point of view where the man is on the platform so photons cannot leave the girl’s face but photons again reach the mirror as the mirror is now fused with the girl’s face. Therefore, the speed of light remains constant at around 300 000 km per second for the man on the platform as well. It is because the girl is moving at that speed.
Journey Through Time Page 12
