Define weight as the force on an object due to a gravitational field
The weight of an object is the force of gravity acting on it.
Where W is the weight in newtons (N), m is the mass in kilograms (kg) and g can be either:
- The acceleration due to gravity (= 9.8 m/s/s at the Earth’s surface); or
- The gravitational field strength (= 9.8 N/kg at the Earth’s surface).
Define gravitational potential energy as the work done to move an object from a very large distance away to a point in a gravitational field.
Newton’s Law of Universal Gravitation
where G is the universal gravitational constant. The Gravitational Field Surrounding any object with mass is a gravitational field.
As we lift an object from the ground to a height above the ground we do work on it. This work is stored in the object as gravitational potential energy. For an object of mass m at a height h above the Earth’s surface the gravitational potential energy E is given by: E p = mgh
The gravitational potential energy is a measure of the work done in moving an object from infinity to a point in the field. The general expression for the gravitational potential energy of an object of mass m at a distance r from the centre of the Earth (or other planet) is given by:
Where M is the mass of the Earth (or other planet).
Change in Gravitational Potential Energy
The change in potential energy of a mass m1 as it moves from infinity to a distance r from a source of a gravitational field (due to a mass m2) is given by:
Change in Gravitational Potential Energy Near the Earth (when radius increases from A to B)
Describe the trajectory of an object undergoing projectile motion within the Earth’s gravitational field in terms of horizontal and vertical components
Any moving object that moves only under the force of gravity is a projectile. The horizontal motion of a projectile is independent to the vertical motion. The reason for this result is that gravity is the only force acting on the objects and this always acts towards the centre of the Earth.
Projectile motion can be analysed by realising that:
- The horizontal motion is constant velocity.
- The vertical motion of constant acceleration (with acceleration of g).
Equations of Uniformly Accelerated Motion
The Path of a Projectile
The velocity at any point of the path of a projectile is simply the vector sum of the horizontal and vertical velocity components at that point.
- The horizontal component is constant.
- The vertical component changes at g, the acceleration due to gravity.
The path followed by a projectile – its trajectory – is a parabola (or linear)
Describe Galileo’s analysis of projectile motion
Galileo was responsible for deducing the parabolic shape of the trajectory of a projectile. Galileo’s analysis of projectile motion led him to consider reference frames. These are what all
measurements are compared to. The concept of Galilean relativity refers that the laws of mechanics are the same in a frame of reference that is at rest or one that moves with constant velocity.
Explain the concept of escape velocity in terms of the:
– gravitational constant
– mass and radius of the planet
If an object is projected upward with a large enough velocity it can escape the gravitational pull of the Earth (or other planet) and go into space. The necessary velocity to leave the Earth (or other planet) is called the escape velocity. Escape velocity depends on the gravitational constant, the mass and radius of the planet. Suppose an object of mass m is projected vertically upward from the Earth’s surface (mass of M and radius R) with an initial velocity u. The initial mechanical energy, that is, kinetic and potential energy is given by:
Let us assume that the initial speed is just enough so that the object reaches infinity with zero velocity. The value of the initial velocity for which this occurs is the escape velocity ve . When the object is at infinity the mechanical energy is zero (the kinetic energy is zero since the velocity is zero and the potential energy is zero because this is where we selected the zero of potential energy).
Discuss Newton’s analysis of escape velocity
Isaac Newton proposed the idea of artificial satellites of the Earth. He considered how a projectile could be launched horizontally from the top of a high mountain so that it would not fall to Earth. As the launch velocity was increased, the distance that the object would travel before hitting the Earth would increase until such a time that the velocity would be sufficient to put the object into orbit around the Earth. (A higher velocity would lead to the object escaping from the Earth.)
The motion of an object in a circular path with constant speed is called uniform circular motion. Although the speed remains the same in uniform circular motion, it follows that an object travelling in a circular path must be accelerating, since the velocity (that is, the speed in a given direction) is continually changing.
As can be seen, when the change in velocity is placed in the average position between v1 and v2, it is directed towards the centre of the circle. When an object is moving with uniform circular motion, the acceleration (the centripetal acceleration) is directed towards the centre of the circle. For an object moving in a circle of radius r with an orbital velocity of v, the centripetal acceleration a is given by:
A satellite can be put into Earth orbit by lifting it to a sufficient height and then giving it the required horizontal velocity so that it does not fall back to Earth. For the satellite to circle the Earth, the centripetal force required is provided by the gravitational attraction between the satellite and the Earth. Hence the centripetal acceleration is given by:
The human body is relatively unaffected by high speeds. Changes in speed, however, that is, accelerations, can and do affect the human body creating ‘acceleration stress’.
Use the term ‘ g forces ’ to explain the forces acting on an astronaut during launch
g-forces on Astronauts Humans can withstand 4g without undue concern. Accelerations up to ~10g are tolerable for short times when the acceleration is directed parallel to a line drawn between the person’s front and back.
Acceleration forces – g-forces – are measured in units of gravitational acceleration g. For example, a force of 5g is equivalent to acceleration five times the acceleration due to gravity.
If the accelerations are along the body’s long axis then two distinct effects are possible:
- If the acceleration is in the direction of the person’s head they may experience a ‘black out’ as the blood rushes to their feet; or
- If the acceleration is towards their feet, they may experience a ‘red out’ where the blood rushes to their head and retina.
Compare the forces acting on an astronaut during launch with what happens during a roller coaster ride
As you ‘fall’ from a height, you experience negative g-forces (you feel lighter). When you ‘pull out’ of a dip after a hill or follow an ‘inside loop’, you experience positive g-forces (you feel heavier). The positive g-forces are like those astronauts experience at lift-off.
Consider a rider in a car at the bottom of an inside loop. The rider has two forces acting on them:
- Their normal weight (mg) acting down; and
- The ‘normal reaction force’ (N) acting up. This is the push of the seat upwards on
Assume that the loop is part of a circle of radius R. A centripetal force is required for the rider to travel in a circle. This is the difference between the normal force and the weight force, that is:
The g-forces are found from the ‘normal force’ divided by the weight. That is:
Discuss the impact of the Earth’s orbital motion and its rotational motion on the launch of a rocket
A moving platform offers a boost to the velocity of a projectile launched from it, if launched in the direction of motion of the platform. This principle is used in the launch of a rocket by considering that the Earth revolves around the Sun at 107,000km/h relative to the Sun and rotates
once on its axis per day so that a point on the Equator has a rotational velocity of approximately 1,700km/h relative to the Sun. Hence, the Earth is itself a moving platform with two different motions which can be exploited in a rocket launch to gain a boost in velocity.
A rocket heading into orbit is launched to the east to receive a velocity boost from the Earth’s rotational motion.
An Interplanetary Trip
The flight of a rocket heading into space is timed so that it can head out in the direction of the Earth’s motion and thereby receive an extra boost.
Analyse the changing acceleration of a rocket during launch in terms of the:
Law of Conservation of Momentum
Forces experienced by astronaut
Law of Conservation of Momentum
Rocket engines generate thrust by burning fuel and expelling the resulting gases. Conservation of momentum means that as the gases move one way, the rocket moves the other. (Momentum
before the burning is zero; hence the momentum after is also zero. The gases carry momentum in one direction down, and so the rocket carries an equal momentum in the opposite direction up.) As fuel is consumed and the gases expelled, the mass of the system decreases. Since acceleration is proportional to the thrust and inversely proportional to the mass, as the mass decreases, the acceleration increases. Hence the forces on the astronauts increase.
Forces Experienced by Astronauts
g forces varied during the launch of Saturn V, a large three-stage rocket used to launch the Apollo spacecraft. This is attributed to the sequential shutdown of the multiple rocket engines of each stage
– a technique designed specifically to avoid extreme g forces.
Analyse the forces involved in uniform circular motion for a range of objects, including satellites orbiting the Earth
Compare qualitatively and quantitatively low Earth and geostationary orbits
Low Earth Orbit
A low Earth orbit is generally an orbit higher than approximately 250 km, in order to avoid atmospheric drag, and lower than approximately 1000 km, which is the altitude at which the Van Allen radiation belts start to appear. The space shuttle utilises a low Earth orbit somewhere between
250 km and 400 km depending upon the mission. At 250 km, an orbiting spacecraft has a velocity of 27,900km/h and takes just 90 minutes to complete an orbit of the Earth.
A geostationary orbit is at an altitude at which the period of the orbit precisely matches that of the Earth. If over the Equator, such an orbit would allow a satellite to remain ‘parked’ over a fixed point on the surface of the Earth throughout the day and night. From the Earth such a satellite appears to be stationary in the sky, always located in the same direction regardless of the time of day. This is particularly useful for communications satellites because a receiving dish need only point to a fixed spot In the sky in order to remain in contact with the satellite.
The altitude of such an orbit is approximately 38,800 km. If a satellite at this height is not positioned over the Equator but at some other latitude, it will not remain fixed at one point in the sky. Instead, from the Earth the satellite will appear to trace out a ‘figure of eight’ path each 24 hours. It still has a period equal to the Earth’s, however, and so this orbit is referred to as geosynchronous.
Discuss the important of Newton’s Law of Universal Gravitation in understanding and calculating the motion of satellites
Using Newton’s Law of Universal Gravitation combined with the expression for centripetal force, we can see that the orbital velocity required for a particular orbit depends only on the mass of the Earth, the radius of the Earth and the altitude of the orbit (distance from the surface of the Earth) . Given that the mass and radius of the Earth have fixed values, this means that altitude is the only variable that determines the specific velocity required. In addition, the greater the radius of the orbit, the lower the orbital velocity required.
Once a launched rocket has achieved a sufficient altitude above the surface of the Earth, it can be accelerated into an orbit. It must attain a specific speed that is dependent only upon the mass and radius of the Earth and the altitude above it. If that speed is not reached, the spacecraft will spiral
back in until it re-enters the atmosphere; if the speed is exceeded, it will spiral out. This can be considered by appreciating that the simplest orbital motion is a uniform speed along a circular path around the Earth.
Uniform circular motion, as already mentioned, is a circular motion with a uniform orbital velocity. According to Newton’s First Law of Motion, a spacecraft in orbit around the Earth, or any object in circular motion, requires some force to keep it there, otherwise it would fly off at a tangent to the circle. This force is directed back towards the centre of the circle. In the case of spacecraft, it is the gravitational attraction between the Earth and the spacecraft that acts to maintain the circular motion that is the orbit. The force required to maintain circular motion, known as centripetal force, can be determined using the following equation:
The application of Newton’s Law of Universal Gravitation to the orbital motion of a satellite will produce an expression for the critical orbital velocity mentioned earlier. Recall that this law states that the gravitational attraction between a satellite and the Earth would be given by the following expression:
Further, we can use the expression for orbital velocity to prove Kepler’s Third Law – the Law of Periods. The period or the time taken to complete one full orbit can be found by dividing the length of the orbit (the circumference of the circle) by the orbital velocity, v.
- Describe how a slingshot effect is provided by planets for space probes
Many of today’s space probes to distant planets such as Jupiter use a gravitational ‘slingshot’ effect (also known as a gravity-assist trajectory) that brings the probe close to other planets to increase the probe’s velocity. In 1974, Mariner 10 was directed past Venus on its way to Mercury. The
Pioneer and Voyager probes also used this method.
Consider a trip to Jupiter such as the Galileo probe that involved a single fly-by of Venus and two of the Earth. As the probe approaches Venus, it is accelerated by Venus’ gravitational attraction, causing it to speed up relative to Venus. (By Newton’s Third Law, Venus will also experience a force slowing it down. It’s mass, however, is so much greater than that of the probe that the velocity decrease is imperceptible.)
As the probe passes Venus, its speed is reduced (relative to Venus). Relative to the Sun, however, its speed has increased. The probe picks up angular momentum from the planet (which loses an equal amount of an angular momentum). Gravity allows the ‘coupling’ between the probe and planet to facilitate the transfer. For this reason, gravity-assist trajectories should more correctly be called angular momentum-assist trajectories
Account for the orbital decay of satellites in low Earth orbit
All satellites in low Earth orbit are subject to some degree of atmospheric drag that will eventually decay their orbit and limit their lifetimes. As a satellite slows, it loses altitude and begins a slow spiral downwards. As it descends, it encounters higher density air and higher drag, speeding up
the process. By the time the satellite is below an altitude of 200 km it has only a few hours left before colliding with the Earth. The re-entry process generates much heat and most satellites burn up (vaporise) before impacting.
Discuss issues associated with safe re-entry into the Earth’s atmosphere and landing on the Earth’s surface
There are significant technical difficulties involved in safe re-entry, the most important being:
- The heat generated as the spacecraft contacts the Earth’s atmosphere; and
- Keeping the retarding-forces (g-forces) within safe limits for humans.
The Earth’s atmosphere provides aerodynamic drag on the spacecraft and as a result high temperatures are generated by friction with air molecules.
Identify that there is an optimum angle for re-entry into the Earth’s atmosphere and the consequences of failing to achieve this angle.
The angle of re-entry is critical: too shallow and the spacecraft will bounce off the atmosphere back into space; too steep and the g– forces will be too great for the crew to survive (and the temperatures generated with the atmosphere will be too high even for the refracting materials used). The ‘allowed’ angle of re-entry is –6.2° ± 1° relative to the Earth’s horizon.
Discuss the limitation of current maximum velocities being too slow for extended space travel to be viable
Scientists have not yet been able to produce speeds of spacecraft more than a few tens of thousands of kilometres per hour. When travelling to distant planetary objects, the engines of spacecraft are not on as spacecraft rely on inertia to move along. To increase the speed significantly would require the engines to be operating, which would require more fuel. More fuel would require more thrust putting the spacecraft into orbit, which would require more fuel and so on. To increase the speed of spacecraft to values that would make interplanetary travel feasible requires a whole new technology (one not based on the emission of gases produced by combustion). Clearly, while current maximum velocities are just adequate for interplanetary travel, they are entirely inadequate for interstellar travel.
Describe difficulties associated with effective and reliable communications between satellites and earth caused by:
van Allen radiation belts
Microwaves and radio waves, like all EM waves, travel through space at the speed of light. This is between satellites and earth of space communications. The immense distance involved in space communications creates a caused by: distance-related time lag. Also, as EM radiation obeys an inverse square law, there is a loss of signal strength as distance increases. This is referred to as space loss.
Van Allen radiation belts
There are two belts of energetic charged particles, mainly electrons and protons, lying at right angles to the equator of the Earth. Some of the solar wind particles become trapped in the Van Allen
radiation belts. Intense solar activity can disrupt the Van Allen Belts. This in turn is associated with auroras and magnetic storms. The charged particles drifting around the Earth in the outer belt corresponds to an electric current and hence has an associated magnetic field. Once or twice a month this current increases and as a result its magnetic field increases. This can lead to interference of short wave radio communication, errors in communication satellites and even failure of electrical transmission lines.
Sunspots are associated with the solar wind (consisting of a stream of charged particles). The solar wind affects the Earth’s magnetic field and this in turn affects radio communication
Outline the features of the ether model for the transmission of light
It was believed that light waves require a medium to propagate. Although nobody could find such a medium, belief in its existence was so strong that it was given a name – the ether. The ether:
- Filled all of space, had low density and was perfectly transparent
- Permeated all matter and yet was completely permeable to material objects
- Had great elasticity to support and propagate the light waves
Describe and evaluate the Michelson-Morley attempt to measure the relative velocity of the Earth through the ether
The Ether Wind Because the Earth was moving around the Sun, it was reasoned that an ether wind should be blowing past the Earth. However, if a wind blows, the speed of sound relative to the stationary observer would vary. Thus it was believed that the speed of light should vary due to the presence of the “ether wind”. It was in an attempt to detect this difference that Michelson and Morley did their famous experiment.
The Michelson-Morley Experiment Light sent from S is split into two perpendicular beams by the half-silvered mirror at A. These two beams are then reflected back by the mirrors M1 and M2 and are recombined in the observer’s eye. An interference pattern results from these two beams.
The beam AM1 travelled across the ether, whilst AM2 travelled with and against the ether. The times to do this can be shown to be different and so introduce a phase difference between the beams. When the entire apparatus was rotated through 90°, a change in the interference pattern was expected. None was observed.
The result of the Michelson-Morley experiment was that no motion of the Earth relative to the ether was detectable
Discuss the role of critical experiments in science, such as Michelson-Morley’s, in making determinations about competing theories
From a hypothesis, predictions are made of what should happen if a particular experiment is performed. If the results are not in agreement with the prediction, the hypothesis is incorrect. As we have seen, the fact that a null result was found from this experiment showed the ether hypothesis to be invalid. This opened up a completely revolutionary view of space and time with the work of Einstein.
Outline the nature of inertial frames of reference
Frames of Reference Frames of reference are objects or coordinate systems with respect to which we take measurements.
Position In maths, the Cartesian coordinate system is used and position is referred to the axes x, y and z. In experiments in class, the laboratory is the frame of reference.
Velocity An object P travels with velocity v with respect to a reference frame S. Another frame S’ moves with velocity u relative to S. The velocity of P relative to S’ is v’ = v – u. Velocity thus depends upon the reference frame.
Inertial Frames of Reference
An inertial frame of reference is one that is moving with constant velocity or is at rest (the two conditions being indistinguishable). In such reference frames, Newton’s Law of Inertia holds.
A non-inertial frame of reference is one that is accelerating.
Discuss the principle of relativity
Three hundred years before Einstein, Galileo posed a simple idea, now called the principle of relativity, which states that all steady motion is relative and cannot be detected without reference to an outside point. This idea can be found built into Newton’s First Law of Motion as well.
Two points to be reinforced:
- The principle of relativity applies only for non-accelerated steady motion
- This principle states that within an inertial frame of reference you cannot perform any mechanical experiment or observation that would reveal to you whether you were moving with uniform velocity or standing still.
Identify the significance of Einstein’s assumption of the constancy of the speed of light
In 1905, Albert Einstein proposed that the speed of light is constant and is independent of the speed of the source or the observer. This premise explained the ‘negative’ result of the Michelson-Morley experiment and showed that the ether concept was not needed. As a consequence of this ‘law of light’ it can be shown that there is no such thing as an absolute frame of reference. All inertial reference frames are equivalent. That is, all motion is relative. The laws of physics are the same in all frames of reference; that is, the principle of relativity always holds.
Recognise that if c is constant then space and time become relative
In Newtonian physics, distance and velocity can be relative terms, but time is an absolute and fundamental quantity. Einstein radically altered the assumptions of Newtonian physics so that now the speed of light is absolute, and space and time are both relative quantities that depend upon the motion of the observer. (Our reality is what we measure it to be. Reality and observation cannot be separated. Remember this as we proceed).
Discuss the concept that length standards are defined in terms of time with reference to the original meter
…In other words, the measured length of an object and the time taken by an event depend entirely upon the velocity of the observer. (This is why our current standard of length is defined in terms of time – the metre is the distance travelled by light in a vacuum in the fraction 1/299792458 of a second).
Identify the usefulness of discussing space/time, rather than simple space
…Further to this, since neither space nor time is absolute, the theory of relativity has replaced them with the concept of a space-time continuum. (Space and time, not just space, are relative quantities).
Account for the need, when considering space/time, to define events using four dimensions
…Any event then has four dimensions (three space coordinates plus a time coordinate) that fully define its position within its frame of reference.
Explain qualitatively and quantitatively the consequence of special relativity in relation to:
The relativity of simultaneity
The equivalence between mass and energy
The Relativity of Simultaneity (simultaneity and the velocity of light) quantitatively the Observers in relative motion will disagree on the simultaneity of events separated in space.
The Equivalence: Between Mass and Energy The mass of a ‘moving’ object is greater than when it is ‘stationary’ – it experiences mass dilation (covered later)
Since c is the maximum speed in the universe it follows that a steady force applied to an object cannot continue to accelerate. It follows that the inertia, that is the resistance to acceleration,must increase. But inertia is a measure of mass and so the mass has increased.
It is this increase in mass that prevents any object from exceeding the speed of light, because as it accelerates to higher velocities its mass increases, which means that further accelerations will require even greater force. This is further complicated by time dilation because, as speeds increase to near light speed, any applied force has less and less time in which to act. The combined effect is that as mass becomes infinite and time dilates, an infinite force would be required to achieve any acceleration at all. Sufficient force can never be supplied to accelerate beyond the speed of light. If force is applied to an object, then work is done on it – energy is given to the object. This energy would take the form of increased kinetic energy as the object speeds up. But at near light speed the object does not speed up. The applied force is giving energy to the object and the object does not acquire the kinetic energy we would expect. Instead, it acquires extra mass. Einstein made an inference here and stated that the mass (or inertia) of the object contained the extra energy. Relativity results in a new definition of energy as follows:
To measure speed we need to measure distance and time. If c remains constant, then it follows that distance (length) and time must change. Space and time are relative concepts.
Length Contraction (the Lorentz-FitzGerald Contraction) The length of a ‘moving’ rod appears to contract in the direction of motion relative to a ‘stationary’ observer.
where l is the moving length, l0 is the ‘rest’ length (that is, the length as measured by an observer at rest with respect to the rod) and v is the speed of the rod.
Time Dilation Time in a ‘moving’ frame appears to go slower relative to a ‘stationary’ observer
where t is the observed time for a ‘stationary’ observer and t 0 is the time for an observer travelling in the frame. t 0 is called the proper time (this is the time measured by an observer present at the same location as the events that indicate the start and end of an event).
Mass Dilation The mass of a ‘moving’ object is greater than when it is ‘stationary’.
where m is the mass for a ‘moving’ object and m0 is the mass for that object when it is ‘stationary.’
Discuss the implications of time dilation and length contraction for space travel
The relativity of time allows for space travel into the future but not into the past. When travelling at relativistic speeds (0.1c or faster), relativity influences the time that passes on the spacecraft. Astronauts on a relativistic interstellar journey would find their trip has taken fewer years than observed on Earth.