To start off this section, there are some keynotes to be made about the use of the word ‘acceleration’ in physics. Velocity is a vector measurement, meaning that velocity has both a magnitude and direction. Speed, however, does not have a direction and only has a magnitude; hence speed is a scalar measurement.
To put this simply, speed is how fast an object is moving in an arbitrary direction whilst velocity is how fast an object is traveling in a set or linear direction. If an object is traveling at a constant speed, but is traveling in a circular, zigzagging or sinusoidal path, the velocity is constantly changing direction; hence there is a change in velocity.
Acceleration is a measurement of change in velocity over time. This means that if a body is traveling at a constant speed, but again in a circular, zigzagging or sinusoidal path then the body is accelerating.
To say an object is decelerating is not strictly wrong in physics, though the term ‘deceleration’ is none-standard. When writing an equation, it would be impractical to have a different symbol for both acceleration and deceleration, so if an object appears to be decelerating then it is often referred to as accelerating by a negative magnitude. For instance, instead of accelerating by 5 ms-2 and then decelerating by 5 ms-2, it would usually be said that a body was accelerating by 5 ms-2 and then accelerating by -5 ms-2.
Acceleration also uses derivatives, which is a mathematical process for the measurement of how a function – in this case, an objects velocity – changes as an input – in this instance, time – changes. Acceleration can be written as ∂v/∂t, which translates to ‘delta v’ divided by ‘delta t’, or ‘change in velocity over ‘change in time’. The symbol ‘∂’ is simply an abbreviation of the phrase ‘change in’.
This can be used in graph theory whereby if we plotted a graph of speed against time, the area under the graph would indicate total distance traveled and the gradient of the graph would indicate ‘∂v/∂t’, or ‘acceleration’. The gradient of the graph is a measurement in how y changes as x changes. See Figure 1 for more information.
Figure 1
Alternatively, differentiation can also be written as ∆v/∆t or f’(v) (whilst f(v) is shorthand for ‘a function of v’). These all, however, have the same meaning; in fact, ‘∆’ is simply the capitalised Greek alphabetical symbol of ‘∂’.
The mathematical implications of differentiation are not needed the following segment of this article, so can be dismissed for the time being. I will be tackling differentiation at a later date, along with graph theory, logarithms and integration.
So, let us now observe Newton’s three laws of gravitational attraction. Newton originally proposed that.
1. A body, at rest or traveling at a constant speed, has a resultant force of zero.
2. The relationship between a force and a mass is acceleration, and the acceleration acts in the direction of the force. This can be equated as F=ma.
3. For every action, there is an equal and opposite reaction.
So, let us analyse these rules. The first rule tells us that if an object is at rest, i.e. the object is not in motion; the force on the object is zero. Whilst an object is moving, if the velocity of the object is constant then there is no force on the object. This is analogous to when a bowling ball is rolled down a lane. As the bowling ball is swung, the ball accelerates from its resting position behind a persons back and a force is exerted on the ball during the initial swing in order to “push” the ball forwards. Once the ball has left the player’s hand, however, the ball no longer has any force acting on it, and it travels down the lane at a constant velocity. In order to speed up or slow down the velocity of the ball, external forces would need to be used, but for the ball to roll at a constant velocity, no force is required. In reality, there is a force of friction slowing down the ball, but this force is so small as to be considered negligible in practice. It would be safe to assume that in this instance, there is not force acting on the ball.
However, an object can experience a force whilst traveling at a constant velocity. Take a skydiver for example. As the skydiver jumps from a plane, a force of gravity pulls the skydiver downwards. Initially, the skydiver accelerates towards the earth, but as the skydiver falls, air pushes up against the skydiver in a process called air resistance. As the skydiver descends at a faster and faster rate, the air resistance increases. When the skydiver reaches a certain speed, the air resistance on the skydiver is so great that the skydiver stops accelerating and remains traveling at a constant velocity; this is called a terminal velocity.
At this velocity, there is still a force pulling the skydiver down to earth, but there is equal force acting in the opposite direction that is resisting the skydivers fall. This means that there are forces acting on the skydiver, but the resultant force (the sum of the two forces) amount to zero. This law was derived heavily from Galileo’s Law of Inertia.
Newton’s second law was perhaps his most famous and certainly has the biggest implications, allowing calculations and equation derivatives. We have already seen how Newton’s second law can be applied, but Newton’s second law also paved the way for much more advances in mechanics and motion, both directly and indirectly.
We can see how this law was derived using the first law. The first law shows us that a body must have an acceleration in order to have a force. This would lead us to assume that force ‘F’ is proportional to acceleration ‘a’, which is indeed the case. This proportionality is the body’s mass, ‘m’. To accommodate for air resistance, the equation can equally be written as F1 - F2 = ma, where F1 is the pushing force, F2 is the opposing resistance and F = F1 - F2. As you can see from this, if F1 is equal to F2 then F1 - F2 = 0. Hence the resultant force ‘F’ is zero, and so ma = 0. As the mass is constant, only ‘a’ can change, so ‘a’ must also be equal to zero, hence there is no acceleration.
Newton’s third law tells us that for every action, there is an equal and opposite reaction. This essentially means that, if you push an object, the object you push “pushes back” against you. This is what resistance essentially is, an opposing force.
One way to consider this law is through gravity. Whilst you are standing on the earth, you are being pulled down and hence exerting a force on the earths surface. According to Newton’s third law, the earth must also be exerting a force upwards on you. This force is called the Normal Contact force, often written as a stylised R symbol.
As earlier mentioned, mass and weight are entirely different. Mass denotes the amount of a substance, whilst weight is the force of that mass from gravity. Taking the average adult male to be about 75 kg in mass, we could use the formula F=ma to work out the weight of an average human. However, whilst a person is not moving they are not accelerating, but we still have a weight otherwise we would float off into space. As with the skydiver traveling at terminal velocity, we do all have a force on us dragging us down, but there is a resistance that is repelling us, and the resultant force is zero.
Using this, we know we are accelerating downwards. This acceleration is caused by gravity, and is called – quite aptly – acceleration due to gravity. So, what is this acceleration’s value? There is not set value. Gravity changes depending on where you are, and your weight on the moon is about a tenth of your weight on earth, so to work out a person’s weight, we must first calculate the acceleration due to gravity.
On or close to the earths surface, the weight due to gravity is about 9.81 ms-2 (meters per second squared). This tells us that the weight of an average adult male is equal to 75 kg multiplied by 9.81 ms-2 which equals 736 Newtons, or kg m s-2. But how do we know that this acceleration is about 9.81? To do that, further calculations must be made.
A simultaneous equation is simply the process of merging two separate equations using a common variable. Starting with Newton’s second law of gravity (F=ma) and Newton’s equation of gravitational attraction (F=GmM/r2 we can deduce the following..
F=ma and F=MmG/r2
In this instance:
‘m’ is the mass of an average human,
And
‘M’ is the mass of the earth.
If F=ma and F=GmM/r2, then
ma = GmM/r2
We can then divide each side by ‘m’ to cancel these digits out. Remember, when dealing with equations such as 2x=y, whatever you do to one side of the equation you must do equally to the other side. If we tripled y, we would have to also triple 2x to become 6x. If we divided each side by x, we would get 2=y/x. By dividing by ‘m’, we can get the following equation:
a = GM/r2
Where ‘a’ is the acceleration due to gravity, ‘G’ is the universal gravitational constant (6.67x10-11); ‘M’ is the mass of the earth (in this instance at least) and ‘r’ is the distance between the two centers of mass.
The mass of the earth has been calculated to be approximately 6x1024 kg and the radius of the earth is approximately 6,400,000 meters. As we are working from the center of the two masses and we are on the earth’s surface, the radius of the earth is approximately the equivalent of the distance between the two centers of mass, so this distance is used. Now we can continue with the calculations:
a = GM/r2
a = (6.67x10-11)x(6x1024) / (6.4x106)2
a = 4.002x1014 / 4.09613
a = 9.7705 ms-2
Which is remarkably close to the 9.81 ms-2 as measured by Newton. It could also be noted that this example used approximate values and so will have a margin of error.
Using and rearranging the equations F=ma and F=GmM/r2 allows other possibilities too and, by incorporating other simple equations, we could equally calculate the mass or radius of the earth for ourselves. The gravitational constant ‘G’ is much more difficult to accurately measure as gravity is a very weak force compared to other forces such as subatomic or electromagnetic forces. Wikipedia has the following to say on the measurements of the universal gravitational constant:
History of the measurement of G
The gravitational constant appears in Newton's law of universal gravitation, but it was not measured until 1798 — 71 years after Newton's death — by Henry Cavendish (Philosophical Transactions, 1798). Cavendish measured G implicitly, using a torsion balance invented by the geologist Rev. John Michell. He used a horizontal torsion beam with lead balls whose inertia (in relation to the torsion constant) he could tell by timing the beam's oscillation. Their faint attraction to other balls placed alongside the beam was detectable by the deflection it caused. However, it is worth mentioning that the aim of Cavendish was not to measure the gravitational constant but rather to measure the mass and density relative to water of the Earth through the precise knowledge of the gravitational interaction. The value that he calculated, in SI units, was 6.754x10−11 m3 Kg−1 s−2.
The accuracy of the measured value of G has increased only modestly since the original experiment of Cavendish. G is quite difficult to measure, as gravity is much weaker than other fundamental forces, and an experimental apparatus cannot be separated from the gravitational influence of other bodies. Furthermore, gravity has no established relation to other fundamental forces, so it does not appear possible to measure it indirectly. Published values of G have varied rather broadly, and some recent measurements of high precision are, in fact, mutually exclusive.
Source: www.wikipedia.org