IB Physics/Mechanics

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Topic 2: Mechanics

Kinematic concepts
2.1.1 Define displacement, velocity, speed and acceleration.


 * Vector quantities always have a direction associated with them.

2.1.2 Define and explain the difference between instantaneous and average values of speed, velocity and acceleration.


 * Average value - over a period of time.


 * Instantaneous value - at one particular time.

2.1.3 Outline the conditions under which the equations for uniformly accelerated motion may be applied.

Acceleration must be constant

Graphical representation of motion
2.1.4 Draw and analyse distance–time graphs, displacement–time graphs, velocity–time graphs and acceleration–time graphs.

2.1.5 ''Analyse and calculate the slopes of displacement–time graphs and velocity – time graphs, and the areas under velocity–time graphs and acceleration–time graphs. Relate these to the relevant kinematic quantity.''

Uniformly accelerated motion
Determine the velocity and acceleration from simple timing situations

$$ \begin{align} &\left[\vec{a}=\dfrac{\vec{v}-\vec{u}}{t}\right] && \left[\vec{s}=\dfrac{(\vec{u}+\vec{v})t}{2}\right] &&& \left[\vec{s}=\vec{u}t+\dfrac{1}{2}\vec{a}t^2\right] &&&& \left[\vec{v}^2=\vec{u}^2+2\vec{a}\vec{s}\right]. \\ \end{align} $$

Derive the equations for uniformly accelerated motion.

There are three equations for uniformly accelerated motion. The first one is derived form the definition of acceleration as the rate of change of velocity,

i.e

$$ \text{acceleration}=\vec{a}=\dfrac{\partial}{\partial t}\,\vec{v}=\dfrac{\text{change in velocity}}{\text{change in time}}. $$

Let the initial and final velocities be designated by $$u$$ and $$v$$ respectively, acceleration by $$a$$ and time by $$t$$, then

$$ \vec{a}=\dfrac{\vec{v}-\vec{u}}{t}. $$

Making v the subject we get,

$$ \vec{v}=\vec{u}+\vec{a}t, $$

which is the first equation of motion.

Describe the vertical motion of an object in a uniform gravitational field.

Describe the effects of air resistance on falling objects.

Solve problems involving uniformly accelerated motion.

Forces and free-body diagrams
Identify the forces acting on an object and draw free-body diagrams representing the forces acting. Each force should be labelled by name or given a commonly accepted symbol. Vectors should have lengths approximately proportional to their magnitudes.

Newton’s first law
Newton's First Law of Motion states that a body will remain at rest or moving with a constant velocity unless acted upon by an unbalanced force.

Equilibrium
Equilibrium is the condition of a system in which competing influences (such as forces) are balanced.

Newton’s second law
$$ \sum \vec{F}=m\vec{a}. $$

Alternately:

$$ \sum \vec{F}=\dfrac{\partial}{\partial t}\,\vec{p}=\dfrac{\text{change in momentum}}{\text{change in time}}. $$

In words, the resultant force is all that matters in the second law. The direction of motion depends on the direction of the resultant force.

Newton’s third law
If body A exerts a force on body B, then body B exerts an equal and opposite force on body A along the line of centers.

Every action force has an equal and opposite reaction force. This is why you are able to place your coffee mug on the table without it falling into the table. The table provides a normal reaction force (up) to counter the gravitational force (down) hence F(net) = 0 and your coffee mug stays on the table. Important to note is that the opposite reaction must happen along the lines of centers, i.e, the vector arrow of the opposite force must be aimed at the first force while still being opposite.

Inertial Mass, Gravitational Mass and Weight (2.3)
An object's inertial mass is defined as the ratio of the applied force $$\vec{F}$$, to its acceleration, $$\vec{a}$$, where $$\vec{F}$$ and $$\vec{a}$$ are no higher than one dimensional vectors, given that vector division is not well defined beyond one dimension.

$$m_{inertial}=\dfrac{\vec{F}}{\vec{a}}$$

State Newton's first law of motion (2.2.4)
In ancient times, Aristotle had maintained that a force is what is required to keep a body in motion. The higher the speed, the larger the force needed. Aristotle's idea of force is not unreasonable and is in fact in accordance with experience from everyday life: It does require a force to push a piece of furniture from one corner of a room to another. What Aristotle failed to appreciate is that everyday life is plagued by friction. An object in motion comes to rest because of friction and thus a force is required if it is to keep moving. This force is needed in order to cancel the force of friction that opposes the motion. In an idealized world with no friction, a body that is set into motion does not require a force to keep it moving. Galileo, 2000 years after Aristotle, was the first to realize that the state of no motion and the state of motion with constant speed in a straight line are indistinguishable from each other. Since no force is present in the case of no motion, no forces are required in the case of motion in a straight line with constant speed either. Force is related to changes in velocity (i.e. acceleration)

Newton's first law (generalizing Galileo's statements) states the following:

When no forces act on a body, that body will either remain at rest or continue to move along a straight line at constant speed.

A body that moves with acceleration (i.e. changing speed or changing direction of motion) must have a force acting on it. An ice hockey puck slides on ice with practically no friction and will thus move with constant speed in a straight line. A spacecraft leaving the solar system with its engines off has no force acting on it and will continue to move in a straight line at constant speed (until it encounters another body that will attract or hit it). Using the first law, it is easy to see if a force is acting on a body. For example, the earth rotates around the sun and thus we know at once that a force must be acting on the Earth.

Newton's first law is also called the law of Inertia

Inertia is the reluctance of a body to change its state of motion. Inertia keeps the body in the same state of motion when no forces act on the body. When a car accelerates forward, the passengers are thrown back into their seats. If a car brakes abruptly, the passengers are thrown forward. This implies that a mass tends to stay in the state of motion it was in before the force acted on it. The reaction of a body to a change in its state of motion is inertia.

A well-known example of inertia is that of a magician who very suddenly pulls the tablecloth off a table leaving all the plates, glasses, etc., behind on the table. The inertia of these objects make them 'want' to stay on the table where they are. Similarly, if you pull very suddenly on a roll of toilet papers you will tear off a sheet. But if you pull gently you will only succeed in making the paper roll rotate.

Work
Work refers to an activity involving a force and movement along the direction of the force. It is a scalar quantity that is measured in Joules (Newton meters in SI units) which can be defined as:

$$ W=\vec{F}\cdot \vec{s}\,\text{cos}(\theta) $$

Where $$\vec{F}$$ is the force applied to the object, $$\vec{s}$$ is the displacement of the object and $$\text{cos}(\theta)$$ is the cosine of the angle between the force and the displacement, and where $$\cdot$$ is the dot product.

In a linear example (with the force being exerted in the same direction as the displacement), the $$\text{cos}(\theta)$$ is equal to $$1$$ and the equation simplifies to $$W=\vec{F}\cdot \vec{s}$$.

Example calculation:

If a force of $$20$$ newtons pushes an object $$5$$ meters in the same direction as the force what is the work done?

$$ \begin{align} &F=20\,\text{N}\\ &s=5\,\text{m} \\ &W=\vec{F}\cdot \vec{s}=20\times 5=100\,\text{J} \\ \end{align} $$

Answer: $$100$$ Joules of work is done.

Examples (when is work done?): Force making an object move faster (accelerating) Lifting an object up (moving it to a higher position in the gravitational field) Compressing a spring

When is work not done?: When there is no force Object moving at a constant speed Object not moving

Some useful equations;

If an object is being lifted vertically the work done to it can be calculated using the equation

$$ W=m\,g\,h $$

Where $$m$$ is the mass in kilograms, $$g$$ is the earth's gravitational field strength ($$10\;N\,kg^{-1}$$), and $$h$$ is the height in meters.

Work done in compressing or extending a spring:

$$ W=\dfrac{1}{2}\,k\,\vec{s}^2 $$

Where $$k$$ is Hooke's constant and $$\vec{s}$$ is the displacement.

Energy and Power
Energy is the capacity for doing work. The amount of energy you transfer is equal to the work done. Energy is a measure of the amount of work done, this means that the units for energy and work must be the same- joules. Energy is like the "currency" for performing work. To do $$100$$ joules of work, you must expend $$100$$ joules of energy.

Conservation of energy

In any situation the change in energy must be accounted for. If it is 'lost' by one object it must be gained by another. This is the principle of conservation of energy which can be stated in several ways:


 * The total overall energy of a closed system must be constant
 * Energy is neither created or destroyed, it just changes form.
 * there is no change in the total energy of the universe

Energy can be in many different types these include:


 * Kinetic energy, Gravitational potential energy, Elastic potential energy, Electrostatic potential energy, Thermal energy, Electrical energy, Chemical energy, Nuclear energy, Internal energy, Radiant energy, Solar energy, and light energy.

You will need equations for the first three
 * $$\text{Kinetic energy}=E_{K}=\dfrac{1}{2}\,m\vec{v}^2$$ where $$m$$ is the mass in $$\text{kg}$$, $$v$$ is the velocity (in $$m\,s^{-1}$$)
 * $$\text{Gravitational potential energy}=E_{p}=mg\Delta h$$ where $$m$$ is the mass in $$\text{kg}$$, $$g$$ is the gravitational field strength, and $$\Delta h$$ is the change in height.
 * $$\text{Elastic potential energy}=E_{p}=\dfrac{1}{2}\,k\,\left(\Delta x\right)^2$$ where k is the spring constant and $$\Delta x$$ is the extension.

Power - measured in Watts ($$\text{W}$$) or Joules per second ($$\text{J}\,\text{s}^{-1}$$)- is the rate of doing work or the rate at which energy is transferred.

$$ \text{Power}=P=\dfrac{\text{energy transferred}}{\text{time taken}}=\dfrac{E}{t}. $$

If something is moving at a constant velocity $$v$$ against a constant frictional force $$\vec{f}$$, the power $$P$$ needed is $$P=\vec{f}\cdot \vec{v}$$.

If you do $$100$$ joules of work in one second (using $$100$$ joules of energy), the power is $$100$$ watts.

Efficiency is the ratio of useful energy to the total energy transferred.

Work-Energy Principle
The change in the kinetic energy of an object is equal to the net work done on the object.

$$ W_{net}=\dfrac{1}{2}mv_{final}^2-\dfrac{1}{2}mv_{initial}^2 $$

This fact is referred to as the Work-Energy Principle and is often a very useful tool in mechanics problem solving. It is derivable from conservation of energy and the application of the relationships for work and energy, so it is not independent of the conservation laws. It is in fact a specific application of conservation of energy. However, there are so many mechanical problems which are solved efficiently by applying this principle that it merits separate attention as a working principle.

For a straight-line collision, the net work done is equal to the average force of impact times the distance traveled during the impact:

$$ W_{net}=\vec{F}_{avg}\cdot \vec{s}. $$

$$ \text{Average impact force}\cdot \text{distance traveled}=\vec{F}_{avg}\,d=\dfrac{1}{2}\,m\vec{v}^2=\text{change in kinetic energy} $$

If a moving object is stopped by a collision, extending the stopping distance will reduce the average impact force.

Uniform Circular Motion (2.6)
The centripetal force with constant speed  $$v$$, at a distance $$R$$ from the center is defined as:

$$F = m\dfrac{v^2}{R}$$