## Module 5: Advanced Mechanics

Contents

## Projectile Motion

A projectile is any object that is moving freely through space without any power source driving it (eg motor or propeller engine). The analysis of the motion of such objects can occur in

- 1 dimension
- 2 dimensions

### Projectile motion in 1 dimension

A projectile moving in 1 dimension means that the projectile is can only move in 2 directions (e.g. up or down, left or right, forwards or backwards). Solving projectile motion questions in 1 dimension requires the following equations of motion: Manipulation of the above equations will allow you to get the quantity which the question wants you to calculate. A good example would be a free-falling projectile.

### Free-falling projectiles

In these cases, the only force acting on a projectile is the weight force (otherwise known as gravity). Therefore, the projectile will experience a constant acceleration downwards.

### Projectile motion in 2-dimensions

Here, the object can move along two different ‘directions’. E.g. a rocket launched at an angle will be travelling forwards as well as upwards. A key property of projectiles moving in 2 dimensions is that it can be split into two different components:

- A horizontal component.
- A vertical component.
- These 2 components are completely independent of each other and therefore our analysis of them can be done separately.

Trigonometry may be used to calculate the vertical and horizontal components of velocity. Quite often, the horizontal component of motion will consist only of the object travelling at constant velocity. The vertical component will often have just gravitational acceleration. The combined effect of the horizontal and vertical motion often results in the projectile travelling in a parabolic trajectory.

## Circular motion

In the HSC, we only look at uniform circular motion. Uniform circular motion means that the object moves in a circle at a constant speed.

- However, the velocity of the object is always changing since the direction of the velocity is always changing and so the object is technically always accelerating.
- There is a force that always points towards the centre of the circle called the centripetal force. The centripetal force is always perpendicular to the velocity vector.

The centripetal force is given by the following formula:

- Where Fc is the centripetal force given in Newtons.
- M is the mass of the moving object given in kilograms.
- V is the velocity of the object given in ms-1.
- R is the radius of the circular path and is given in metres.

Using Newton’s Law we can deduce that the centripetal acceleration is:

### Period, angular velocity and frequency

The time required to travel around the circle is called the period, T, of the motion. The number of rotations each second is called the frequency, f, of the motion. These two quantities can be related through the following equation: The angular velocity is the angle of rotation in a given time. The angular velocity is given by:

### Work and Energy

The most important concept here is the conservation of mechanical energy. Sometimes energy is dissipated or transformed into light, heat and/or sound and thus the total energy of the system is reduced. In a problem question, an external force usually refers to friction or a thrust force from an engine. Therefore, the conservation of mechanical energy would apply given that no other forms of energy are involved (e.g. heat, sound and/or light). The above concept is most often used to calculate the speed of the moving object at certain points in the circular motion (particularly vertical circular motion).

### Work

Work is defined as the transfer of energy from one object to another and/or the transformation of energy from one form to another. A force does work on an object when it acts on a body causing a displacement in the direction of the force. The formula for work is given by: Work and energy are scalar quantities measured in Joules. To find the work done on an object, use the net force. In the context of circular motion

- There is always a net centripetal force acting on the object. Therefore, the object is always being accelerated by the centripetal force.
- However, because the centripetal force is always perpendicular to the direction, therefore the force does no work according to the above formula.

### Torque

This involves an object rotating around a pivot point. E.g. closing a door or turning a steering wheel. In these situations, the force acts to provide a turning effect or torque ( Torque is a vector, so it has a magnitude and a direction. The amount of torque applied on an object is directly proportional to the perpendicular distance between the pivot point and the line of action of the force. This perpendicular distance is called the force arm. It is only the component of the force which is perpendicular to the line of action which generates effective torque. The torque equation is given by: Note that the above 2 equations are equivalent. To convert between the two formulas requires use of trigonometry. An example of this conversion in the context of an opening door can be seen below:

### Examples of uniform circular motion

Banked tracks Often tracks are tilted at an angle to provide additional forces to allow vehicles to negotiate a sharp corner without slipping. The banked track allows the vehicle to take advantage of the normal reaction force from the road to provide the centripetal force. The design speed is the speed at which a vehicle can negotiate a banked track with 0 sideways friction. The situation is illustrated as follows: It is crucial to draw a force diagram to figure out what is going on.

### A mass on a string

The situation here is that a mass is tied to a string and travels in uniform circular motion at a level below the point at which the string is attached to the pole. See diagram below: The tension in the string is the force which provides the centripetal force.

### Cars moving around horizontal circular bends

Cars which move around horizontal bends rely solely on friction to provide the centripetal force. In these cases, we can pretty much ignore gravity and the normal force as they do not affect the horizontal situation. Friction is the only force which provides the centripetal force.

## Motion in Gravitational Fields

### Newton’s Law of Universal Gravitation

- Where r is the distance of mass m1 from mass m2 in metres.
- G is Universal Gravitational Constant: 6.67 x 10-11 N m2 kg-2.
- M1 and m2 are masses in kg.

This measures the gravitational force of attraction of between two objects. The direction is usually towards the object with larger central mass. G can also be found on other planets by:

- Where G is the gravitational constant.
- R is the radius of the planet.
- Mp is the mass of the planet.

This equation assumes that the mass of Earth is uniformly distributed. This value of g is also the value of free fall acceleration at the point.

## Weight force on various locations on Earth

The weight force which an object experiences at different locations on Earth vary according to different locations. This is because at different locations, the strength of the gravitational field of the Earth varies as the Earth is not a perfect sphere.

### Orbital velocity and orbital period

For satellites and planets that undergo roughly centripetal motion around the Earth or Sun, the gravitational force provides the centripetal force. Therefore, we can equate the centripetal force formula with the Newton’s formula for gravitational force to calculate orbital velocity. Knowing the length of the orbital path and the orbital velocity we can then calculate the orbital period.

### Satellites

Satellites orbit the Earth at high, medium and light orbit.

Type of orbit | Altitude (km) | Use |

Low | 180 – 2000 | Often have a period of 45 min to 1 hr. They encounter orbital decay due to the low altitude of the satellites. Low orbital period so rapid coverage of the Earth. Reaching low orbits also require less fuel and less power required to transmit to the satellites. Uses include spying, weather surveying, geotopographic studies.; |

Medium | 2000 – 36000 | These are often geostationary satellites. Therefore, they normally have a period of 24 hr. Negligible atmospheric friction. Geostationary means that the satellite’s location is always the same in the sky. Useful for GPS, television, global radio communication. |

High | > 36000 | Used for deep space weather pictures and communication satellites |

### Kepler’s laws

- Kepler’s laws are as follows
- The planets move in elliptical orbits with the Sun at one focus.
- The line connecting a planet to the Sun sweeps out equal areas in equal intervals of time.

For every planet, the ratio of the cube of the average orbital radius, r, to the square of the period, T, of revolution is the same.

### Escape velocity and energy

Escape velocity is the minimum velocity at which an object is able to escape the influence of gravity. As the rocket gets closer to the edge of the atmosphere, it can be assumed that no energy is being dissipated as heat because there is no air resistance. No work is being done by an external force so mechanical energy is conserved. Note: Gravitational force still does negative work. Thus kinetic energy is converted into potential energy. Therefore loss in kinetic energy = Gain in potential energy. Note, this can be generalised as:

- Where G is the gravitational constant.
- Rp is the distance from centre of mass to launch point. (m)
- Mp is the mass of the planet. (kg)
- Ve is escape velocity (ms-1).

It can be seen that the mass of the object does not come into play when calculating its escape velocity. Thus escape velocity is uniform regardless of object launched from a planet. The launch velocity of a rocket does not have to be vertical. Even if it is launched parallel to the surface of the Earth, it will have enough kinetic energy to achieve the gain in potential energy to escape the Earth’s gravitational field as long as its initial speed > Ve.

### Gravitational Potential Energy

- G is gravitational constant.
- M1 and m2 are the masses of the two objects. (kg).
- R is the distance between the two objects.
- Ep is in Joules.
- GPE is stored energy, if moving away from the central body it gains GPE.

The zero reference point for G.P.E calculation in space is set at a point which is outside the influence of the gravitational field of the object. i.e. an infinite distance from the object. Thus Ep is the amount of energy required to move an object from a point in the gravitational field to the zero reference point. When Ep = 0, the object is outside the gravitational field. At a position very far from Earth, gravitational attraction is negligible. Thus 0 Ep. If it moves closer to Earth, it is losing GPE hence it is negative. Likewise, to reach the infinite point, positive work needs to be done to move object. If object ends up with negative Ep it follows that object has negative Ep closer to Earth. When performing calculations on satellites, mass of Earth and mass of satellite is used in the Ep equation. Thus to calculate the change in Ep between two points for a satellite, use:

- Where r1 is the initial distance from Earth.
- R2 is the final distance from Earth.
- Both r’s are taken as distance to Earth’s Core. Not surface. (m)
- Everything else is the same.

### Total energy in a non-constant gravitational field

A satellite has two important forms of energy: gravitational potential energy (U) and kinetic energy (K). The sum of these two energies is known as the total mechanical energy Using the centripetal force of a satellite and the gravitational force formula we will get: The principle of conservation of mechanical energy is again very useful for calculating the orbital speed of satellites.