Mechanics A Level Notes

Physics – Mechanics – notes

1.1.1 Physical quantities and units

Most physical quantities are made up of a numerical magnitude and a unit, which gives the magnitude meaning.

Système Internationale (SI) units are the standard units used when measuring quantity. Some of these are metres (length), amperes (current), kilograms (mass), newtons (weight) and ohms (resistance).

There are different prefixes that go before these units, which indicate sub-multiples or multiples of units. These are:

  • (\text{Pico (p) – }\times 10^{-12})
  • (\text{Nano (n) – }\times 10^{-9})
  • (\text{Micro (}\mu\text{) – }\times 10^{-6})
  • (\text{Milli (m) – }\times 10^{-3})
  • (\text{Centi (c) – }\times 10^{-2})
  • (\text{Kilo (k) – }\times 10^3)
  • (\text{Mega (M) – }\times 10^6)
  • (\text{Giga (G) – }\times 10^9)
  • (\text{Tera (T) – }\times 10^{12})

1.1.2 Scalars and vectors

There are two types of physical quantities, vector and scalar. Scalar quantities have only magnitude, while vector quantities have magnitude and direction. Both types have a unit, which is quantified. An example of a scalar quality is mass, or speed. Vector quantities are weight or velocity, as these also have direction.

You can draw a vector triangle in order to determine the resultant of two forces that are acting in the same plane (co-planar). This can be used to calculate resultant velocity, displacement or force.

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Keeping the direction of the vectors the same, position them head to tail (the end of the first vector meets the start of the second vector) and then join them up, with the third line in the triangle being the resultant vector.

You can calculate the resultant of two perpendicular vectors in this way, or Pythagoras, where the resultant vector is the hypotenuse of a right angled triangle formed by the two original vectors.

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A vector can be resolved into two perpendicular components – these are usually horizontal and vertical, but can also be parallel and perpendicular to a slope. This is done by making a right angled triangle and then using sine and cosine to find a value for the vertical and horizontal components.

1.1.3 Kinematics

Displacement – Displacement is the distance travelled in a particular direction, measured in metres.

Instantaneous Speed – Speed is the distance moved per unit time. Instantaneous speed is the speed at a given moment in time, which can be read off a distance-time graph, as it is the gradient at that particular moment in time.

Average Speed – Average speed is equal to (\frac{\text{total distance travelled}}{\text{time}})

Velocity – Velocity is a vector quantity, defined as the rate of change of displacement, which is measured in m/s. (V=\frac{\text{change in displacement}}{\text{time}})

Acceleration – Acceleration is defined as the rate of change of velocity. (\alpha=\frac{\text{change in velocity}}{\text{time}}) It is important to be able to use the equations (\text{average speed}=\frac{\text{distance}}{\text{time}}) and (\text{acceleration}=\frac{\text{change in velocity}}{\text{time}})

Mechanics_Pic03

You can represent the different relationships between distance/displacement, speed/velocity and time graphically. On a velocity time graph, if the gradient and the magnitude are both positive or negative, the object is accelerating.

On a displacement time graph, the gradient is the velocity at a particular point in time.

From a velocity-time graph, you can determine both displacement and acceleration. (\text{Displacement} = \text{velocity} \times \text{time}) and is therefore the area under the graph. (\alpha=\frac{\text{change in velocity}}{\text{time}}) So it can be found using the gradient of the graph.

1.1.4 Linear Motion

You can derive the constant acceleration equations from a velocity-time graph.

Mechanics_Pic04

(\text{Gradient} = \text{a} = \frac{v-u}{t}) which can be rearranged to (v=u+at) Displacement = area under graph Using trapezium rule (\to s= \frac{1}{2} (u+v)t)

(S = s= \frac{1}{2} (v-u)t+ut) (v-u=at) (s=ut+\frac{1}{2} at^2) (s=\frac{1}{2} (\frac{v-u}{a}(v+u)) (2as=(v-u)(v+u)) (2as=v^2-u^2)

There are 4 equations that are commonly used for linear motion (constant acceleration in a straight line):

  • (v=u+at)
  • (s= \frac{1}{2} (u+v)t)
  • (s=ut+\frac{1}{2} at^2)
  • (v^2=u^2+2as)

When an object is falling in the earth’s atmosphere, assuming no air resistance, it will always fall at a constant rate of 9.81ms(^{-2}). This is regardless of mass or size (when there is no air resistance – with air resistance, the larger the surface area of an object, the smaller its acceleration).

Aristotle believed that objects belonged at the centre of the earth and that they would try to return there and that a moving object would come to a halt once the force that caused it to move ‘ran out.’ This was shown to be incorrect by European scientists such as Galileo, who learned to time things, and use sloped ramps to lessen the effect of gravity, which showed these ideas to be wrong.

Aristotle also believed that objects with a greater mass fall with a greater acceleration, a theory which was generally believed until Galileo showed that acceleration due to gravity is constant for all masses (f=ma and f is directly proportional to m). He showed this by dropping two balls from the top of the leaning tower of Pisa, with different masses, and they landed on the ground at the same time.

For projectiles, with no air resistance, horizontal velocity is constant, and vertical acceleration is constant, so the path of the projectile is independent of the mass.

You can explain the parabolic shape of a projectile path using the constant acceleration equations. In the horizontal direction, the projectile will continue to move at constant velocity until it hits the ground, as (assuming no air resistance) there is no further force in this direction, as g acts perpendicularly to the horizontal. In the vertical direction, the acceleration is constant, which means it falls slowly at first, and gets faster as it gets closer to the ground.

1.2.1 Forces

The equation F=ma can be used to relate force, mass and acceleration. Net force and acceleration are always in the same direction.

The Newton is defined as the force that will propel a mass of 1kg at 1ms(^{-2}).

You can use the equations of constant acceleration in conjunction with f=ma to analyse the effect of force on the motion of objects. If you have a final and initial speed and a time, you can calculate the acceleration of an object, and provided you know its mass, you can calculate the resultant force on that object.

This equation becomes less useful at speeds approaching 3x10(^8) m/s (speed of light), as this equation is only valid when the mass of an object is constant, and at these speeds, mass changes.

1.2.2 Non Linear Motion

Non-linear motions refers to a motion where acceleration is not constant. This means that the equations of linear motion cannot be used, as they require a to be a constant.

Non-linear motion can arise from a changing mass, or a changing force. When travelling in a medium, an object will experience a resistive, or frictional force, known as drag. In air, this is known as air resistance.

The magnitude of the drag force is affected by velocity, cross-sectional area, roughness of surface, and whether or not the object is streamlined. Vehicles and animals such as fish are designed to be as streamlined as possible, whereas parachutes have a large surface area in order to create as much drag as possible.

You can calculate the acceleration of an object with drag using F=ma, by calculating the resultant force without drag using W=mg, and then taking down as the positive direction, subtract the force of the drag. You can then use f=ma with the total of the force and the mass of the object.

The weight of an object is defined as the gravitational force that is acting on an object.

Weight = mass (\times) acceleration of free fall (W=mg)

In a lift

  1. On the ground floor – W=R before the lift moves. The ‘experience’ of weight is due to reaction from the floor.
  2. As the lift starts to move up – R > W so she feels heavy. The lift accelerates upwards.
  3. Between floors – R=W, at constant velocity, she feels normal
  4. Coming to a stop – R < W – she feels light. Resultant force is downwards, opposite to motion, so deceleration occurs.

When a body begins to fall in a uniform gravitational field with drag, it will initially accelerate, due to its weight. As it continues to accelerate, its drag will continually increase as velocity increases. Eventually the drag (upwards) becomes equal to the weight (downwards) and so there is zero resultant force on the object and so it will not accelerate further, but move at constant velocity. This velocity is known as its terminal velocity. If an object has a bigger mass, it will reach a higher terminal velocity, as W is larger, but it will take a greater time.

When t=0, A=0, where A is air resistance (drag). (A=kv^2) (\alpha= \frac{\sum F}{m}=\frac{w-A}{m}=\frac{w}{m}) This is the greatest value of a in the journey, which is equal to g on earth.

Later time t, reached speed v

(a= \frac{\sum f}{m}=\frac{w-A}{m}) So acceleration increases, even as speed increases.

Eventually terminal velocity is reached where A=W and therefore resultant force = 0

1.2.3 Equilibrium

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You can represent forces in equilibrium on a point using a diagram. These can then be drawn in a closed triangle, which shows that the forces are in equilibrium. The net force and the net moment on an object in equilibrium is always zero.

Mechanics_Pic06

The centre of gravity of an object is the point through which the entire weight of the object appears to act. This can be determined in a flat object by doing a simple experiment:

  1. Hook the object over a pivot so that it is hanging freely
  2. Hang a plumb line on the same pivot and let it rest
  3. Draw a vertical line on the object along the plumb line
  4. Rotate the object around so it has a different orientation – then repeat steps 2 and 3
  5. Where the two lines cross in the centre of gravity

A couple is a pair of forces that are equal and opposite each other, which cancel each other out linearly and that produce rotation only.

The torque of a couple describes the turning effect of a couple. A torque is calculated is one of the forces multiplied by the perpendicular distance between them. A torque is measured in Nm.

The moment of a force is defined as the turning effect of a force around a pivot.

Moment of a force = force (\times) perpendicular distance from the point measured from the line of action of the force

The Principle of moments: If an object is in equilibrium, the sum of the clockwise moments about any axis is equal to the sum of the anticlockwise moments.

You can use the principle of moments to solve problems, by making a chosen point the chosen axis. This can also be used to calculate how much force is exerted by the biceps when a weight is held, using the elbow as a pivot.

Density is defined as mass per unit volume. (Density=\frac{\text{mass}}{volume}) (\rho=\frac{m}{v}) It is measured in kg/m(^3). The density of water is 1000 kg/m(^3)

Pressure is defined as force per unit area. (Pressure=\frac{\text{force}}{\text{area}}) (P=\frac{F}{A}) It is measured in Nm(^{-2}) or Pascals.

1.2.4 Car Safety

Thinking Distance – The distance travelled in the time between the driver seeing the obstruction and pressing the brake. It can be calculated by using:

Thinking distance = speed of vehicle (\times) driver’s reaction time ((vt))

Braking Distance – The distance travelled between the moment the brake is applied and the car coming to a complete stop. It can be calculated using (v^2 = u^2 + 2as), assuming deceleration is constant. This can be rearranged to give (s=\frac{v^2}{2a})

Stopping distance = thinking distance + braking distance

If the speed of the car is doubled, then it will have 4 times the kinetic energy and so its braking distance will be four times as far. The thinking distance will double at double speed, as in the same time, the car will travel twice as far.

A number of factors can affect stopping distances, either by effecting braking distance or thinking distance – speed affects both thinking and braking distance:

  • Mass of car – kinetic energy = (\frac{1}{2}mv^2) – so an increased mass increases the braking distance of the car.
  • Condition of the cars brakes – this can lessen the car’s braking force, which can increase braking distance.
  • Road conditions (ice or rain) – friction is reduced and so braking distance increases.
  • Drugs/alcohol/tiredness – the thinking time is increased.

There are many features in cars that make them safer for consumers, including air bags, seatbelts and crumple zones. All of these work on the same principle, which is to reduce the forces involved in a collision. (F = ma)

For a given change in velocity, mass is constant and so (f \propto \frac{1}{t}) where (t) is the time taken for the collision.

Seat Belts – without a seatbelt, the driver would obey N(I) and continue with constant velocity into the windscreen. A seatbelt is made of stiff material that extends slightly during the crash, which increases the time of the force on the driver.

Crumple Zone – The region of the car between the screen and the front of the bonnet, designed to squash during the crash. This increases the time of collision and so reduces force on the car.

Air Bags – Need to be inflated at the start of the collision, which is achieved by an accelerometer detecting rapid deceleration. This sends an electrical signal to the firing mechanism which sends gas into the airbag, which then deflates during the collision, reducing the force on the driver.

GPS works by a satellite sending a microwave signal to a receiver, which contains an atomic clock to indicate the exact time the signal was sent. The distance is then calculated using the speed of light (\times) the time taken for the wave to reach the receiver. This locates the sat-nav on the surface of a sphere, which projects to a circle on earth. This is carried out with three satellites simultaneously, which is known as trilateration.

Mechanics_Pic07

1.3.1 Work and conservation of energy

Work done is the process of energy conversion when a force moves, defined as:

Work = force (\times) distance moved in the direction of the force.

(W=f \times \cos \theta), Where (\theta) = the angle between the force and the displacement If (\theta)=90, (\cos \theta) = 0 (no work is done) – this occurs in the orbit of the moon, moments and any circular motion.

If (\theta)=0 then (\cos \theta) = 1 and so the equation becomes (W=fx) if the movement of the force is in the same direction as the force.

The joule is defined as the energy transferred when a force of one newton is displaced by a distance of 1m.

The Principle of Conservation of Energy: In any closed system, energy may be only converted from one form to another, never created or destroyed. There are many different forms of energy, some of them are:

  • Chemical Energy – this is the energy that can be released when the arrangement of atoms is altered.
  • Electrical Potential Energy – such as when a positive charge is placed next to a positive charge, and the two repel.
  • Electromagnetic energy – Waves (EM spectrum) that hold their energy in electrical and magnetic fields.
  • Gravitational potential energy – the energy an object has due to its position in a gravitational field, at a height.
  • Heat Energy – The molecules in all objects have potential energy when they are close to each other. This is also called internal energy.
  • Kinetic energy – when an object is moving, it has KE.
  • Nuclear Energy – potential energy that can be released by reorganising an atoms nucleus, this is also known as atomic energy.
  • Sound energy – caused by the movement of atoms. Work done is equal to total energy transferred – we can use this to do calculations.

1.3.2 Kinetic and Potential Energies

A moving object has kinetic energy, which is calculated by (E_k=\frac{1}{2} mv^2)

This has units of kgm(^2)s(^{-2}), which is equal to the joule.

Gravitational potential energy is the energy stored in an object (the work an object can do) due to its position in a gravitational field.

(W=mg), and work done = force (\times) distance, so GPE lost when an object falls through a height, (h), is equal to (Wh). Therefore GPE = (mgh) (E_p=mgh) When an object falls through the air, its GPE is converted into KE. Realistically, some is lost as heat and sound due to frictional forces, but with no air resistance, the moment before an object hits the ground, all its GPE has been converted into KE. Using the principle of conservation of energy, you can calculate the speed at which an object falls, given its mass and its initial GPE. This is converted into KE, where the equation can then be rearranged to give: (v= \sqrt2gh)

1.3.3 Power

Power is defined as the rate of work done. (\text{Power} = \frac{\text{work done}}{\text{time taken}})

It is measured in joules per second, or watts. It does not only refer to electrical power, the human body uses power. By calculating the energy transfer of an action, if the time taken is given, you can calculate the power used.

(\text{Efficiency} = \frac{\text{total output energy}}{\text{total input energy}} \times 100)

The efficiency of a device is almost always less than 100% - this is because some energy is transferred to heat. A radiator, where heat energy is the useful energy, can have 100% efficiency. We can show the useful and non-useful energy output using a Sankey diagram.

Mechanics_Pic08

This diagram shows the energy outputs from a power station. 3000MW of energy is put into the system, and 1000 comes out as electrical power, which is the useful energy. The width of the arrows is proportional to the amount of energy used in that process.

1.3.4 Behaviour of strings and materials

Deformation of a material is caused by a force, in one dimension. This force can either be tensile, if the material is being stretched by equal forces on either end, or compressive, if the material is being squashed. The forces on either end of the spring must be equal, otherwise the spring accelerates in one particular direction.

Hooke’s Law: The extension of a spring is directly proportional to the force applied, up to the elastic limit.

In a material which obeys Hooke’s law, the extension is proportional to the applied force, up to the elastic limit. This means that (f=kx), where (k) is the force constant, which is a measure of stiffness, constant for a specific material. This can be found from the gradient of a graph of force against extension.

Once the elastic limit is passed, the material will stretch with plastic flow, which is shown on a graph by a straight line curving.

The work done to stretch a wire in the elastic region can be calculated by the area under the graph, which is equal to (\frac{1}{2}Fx), where (F=kx), so the work done = (\frac{1}{2}kx^2). In the elastic region, this is the same as the elastic potential energy so:

(E_E=\frac{1}{2} kx^2)

If two springs are in parallel, then their effective (k=2k), but if two springs are in series, then their effective (k = 0.5k).

You can do an experiment, stretching a wire, to find (k) in different materials.

Mechanics_Pic09

Using the apparatus shown, record the force applied to the wire, and the extension of the wire. For the elastic region, (k = \frac{f}{x})

However to assign a number to the metal itself, the C.S.A. and the length need to be accounted for. The c.s.a can be measured using a micrometre.

Stress is defined as force per unit cross sectional area. Stress= (\frac{\text{force applied}}{\text{c.s.a.}}) This is measured in Nm(^{-2})

Strain is defined as extension per unit length. Strain= (\frac{\text{extension}}{\text{original length}}) This has no units, as it is m/m.

Stress on a material causes strain, the magnitude of which is dependent on the material. Youngs modulus of elasticity is the ratio between stress and strain, it is represented by the letter E, and is a constant.

Ductile materials (such as copper) can be drawn out into a wire. They have a large plastic region to enable their shape to be altered easily. The maximum stress that can applied to a material before it breaks is known as its maximum tensile stress. Most metals are ductile. Tungsten is not a ductile metal – a wire made of tungsten is formed by forcing finely powdered metal into wire form. Mechanics_Pic10

Brittle materials distort very little, and do not have plastic flow – they will snap if subjected to enough stress. Very little elastic potential energy is stored in brittle materials. Biscuits and concrete are both brittle materials but have a very different maximum tensile stress. Mechanics_Pic11

Polymeric materials are made up of long polymer chains that can be stretched out. Polymeric materials may reach a strain of up to 300%. At first, not a lot of stress is required to pull the molecules apart and then the chain unravels. Once the chain is no longer twisted, it becomes very difficult to stretch the material, shown by the shape of the graph. It will break soon after this point. The strain of a polymeric material can be 10x that of a brittle or ductile material. Mechanics_Pic12

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