Physics Study Guide/Print version/Section One

The SI System of Measurement

Fundamental units

These are basic units upon which most units depends.

Time

Time is defined as the duration between two events. In the international system of measurement (S.I.) the second (s) is the basic unit of time and it is defined as the time it takes a cesium (Cs) atom to perform 9,192,631,770 complete oscillations. The Earth revolves around its own axis in 86400 seconds with respect to the Sun; this is known as the mean solar day, and the 86400th part of one day is known as a second.

Length

In the international system of measurement (S.I.) the metre (m) ('meter' in the US) is the basic unit of length and is defined as the distance travelled by light in a vacuum in 1/299,792,458 second. This definition establishes that the speed of light in a vacuum is precisely 299,792,458 metres per second.

Mass

In the international system of measurement (S.I.) the kilogram (kg) is the basic unit of mass and is defined as the mass of a specific platinum-iridium alloy cylinder kept at the Bureau International des Poids et Mesures in Sèvres, France. A duplicate of the Sèvres cylinder is kept at the National Institute of Standards and Technology (NIST) in Gaithersburg, Maryland. See Wikipedia article.

Current

In the international system of measurement (S.I.) the ampere (A) is the basic measure of electrical current. It is defined as the constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre (m) apart in vacuum, would produce between these conductors a force equal to 2×10^-7 newton (N) per metre of length.

Unit of Thermodynamic Temperature

The kelvin (K), unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.

Unit of Amount of Substance

1. The mole (mol) is the amount of substance of a system that contains as many elementary entities as there are atoms in 0.012 kilograms (12 grams) of carbon 12.

2. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.

Luminous Intensity

The candela (cd) is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x 10¹² hertz and that has a radiant intensity in that direction of 1/683 watt per steradian. (A steradian (sr) is the SI unit of solid angle, equal to the angle at the centre of a sphere subtended by a part of the surface equal in area to the square of the radius.)

Derived Units

These are units obtained by combining two or more fundamental units.

Charge

The SI unit of charge is the coulomb (C). It is equal to ampere times second: $1\ \mathrm {C} =1\ \mathrm {A} \cdot \mathrm {s}$

Velocity

The SI unit for velocity is in m/s or metres per second.

Force

The SI unit of force is the newton ($N$), named after Sir Isaac Newton. It is equal to $1\ \mathrm {kg} \cdot \mathrm {m} /\mathrm {s} ^{2}$.

Energy

The SI unit of energy is the joule (J). The joule has base units of kg·m²/s² = N·m. A joule is defined as the work done or energy required to exert a force of one newton for a distance of one metre. See Wikipedia article.

Pressure

The SI unit of pressure is the pascal (Pa). The pascal has base units of $\mathrm {N} /\mathrm {m} ^{2}$ or $\mathrm {kg} /\mathrm {m} \cdot \mathrm {s} ^{2}$. See Wikipedia article.

Prefixes

Prefix	yotta	zetta	exa	peta	tera	giga	mega	kilo	hecto	deca		deci	centi	milli	micro	nano	pico	femto	atto	zepto	yocto
Symbol	Y	Z	E	P	T	G	M	k	h	da		d	c	m	µ	n	p	f	a	z	y
10n	1024	1021	1018	1015	1012	109	106	103	102	101	100	10-1	10-2	10-3	10-6	10-9	10-12	10-15	10-18	10-21	10-24
1000n	10008	10007	10006	10005	10004	10003	10002	10001						1000-1	1000-2	1000-3	1000-4	1000-5	1000-6	1000-7	1000-8

Astronomical Measurements

The SI units are not always convenient to use, even with the larger (and smaller) prefixes. For astronomy, the following units are prevalent:

Julian Year

The Julian year is defined by the IAU as exactly 365.25 days, a day being exactly 60*60*24 = 86,400 SI seconds. The Julian year is therefore equal to 31,557,600 seconds.

Astronomical Unit

The Astronomical Unit (au or ua), often used for measuring distances in the Solar system, is the average distance from the Earth to the Sun. In 2012 this was defined as exactly 149,597,870,700 metres. Previously it was 149,597,870,691 m, ± 30 m.

Light Year

The light year (ly) is defined as the distance light travels in a homogeneous isotopic non-attenuating medium (a vacuum) in one Julian year. Due to the word "year", the light year is often mistaken for a unit of time in popular culture. It is, however, a unit of length (distance), and is equal to exactly 9,460,730,472,580,800 m.

Parsec

The parsec (pc), or "parallax second", is the distance of an object that appears to move two arc-seconds against the background stars as the Earth moves around the sun, or by definition one arc-second of parallax angle. This angle is measured in reference to a line connecting the object and the Sun, and thus the apparent motion is one arc-second on either side of this "central" position. The parsec is approximately 3.26156 ly.

Kinematics is the description of motion. The motion of a point particle is fully described using three terms - position, velocity, and acceleration. For real objects (which are not mathematical points), translational kinematics describes the motion of an object's center of mass through space, while angular kinematics describes how an object rotates about its centre of mass. In this section, we focus only on translational kinematics. Position, displacement, velocity, and acceleration are defined as follows.

Position

"Position" is a relative term that describes the location of an object RELATIVE to some chosen stationary point that is usually described as the "origin".

A vector is a quantity that has both magnitude and direction, typically written as a column of scalars. That is, a number that has a direction assigned to it.

In physics, a vector often describes the motion of an object. For example, Warty the Woodchuck goes 10 meters towards a hole in the ground.

We can divide vectors into parts called "components", of which the vector is a sum. For example, a two-dimensional vector is divided into x and y components.

One dimensional coordinate system

Two dimensional coordinate system

Three dimensional coordinate system

Displacement

$\Delta {\vec {x}}\equiv {\vec {x}}_{f}-{\vec {x}}_{i}\,$

Displacement answers the question, "Has the object moved?"

Note the $\equiv$ symbol. This symbol is a sort of "super equals" symbol, indicating that not only does ${\vec {x}}_{f}-{\vec {x}}_{i}$ EQUAL the displacement $\Delta {\vec {x}}$, but more importantly displacement is operationally defined by ${\vec {x}}_{f}-{\vec {x}}_{i}$.

We say that ${\vec {x}}_{f}-{\vec {x}}_{i}$ operationally defines displacement, because ${\vec {x}}_{f}-{\vec {x}}_{i}$ gives a step by step procedure for determining displacement.

Namely:

1. Measure where the object is initially.
2. Measure where the object is at some later time.
3. Determine the difference between these two position values.

Be sure to note that displacement is not the same as distance travelled.

For example, imagine travelling one time along the circumference of a circle. If you end where you started, your displacement is zero, even though you have clearly travelled some distance. In fact, displacement is an average distance travelled. On your trip along the circle, your north and south motion averaged out, as did your east and west motion.

Clearly we are losing some important information. The key to regaining this information is to use smaller displacement intervals. For example, instead of calculating your displacement for your trip along the circle in one large step, consider dividing the circle into 16 equal segments. Calculate the distance you travelled along each of these segments, and then add all your results together. Now your total travelled distance is not zero, but something approximating the circumference of the circle. Is your approximation good enough? Ultimately, that depends on the level of accuracy you need in a particular application, but luckily you can always use finer resolution. For example, we could break your trip into 32 equal segments for a better approximation.

Returning to your trip around the circle, you know the true distance is simply the circumference of the circle. The problem is that we often face a practical limitation for determining the true distance travelled. (The travelled path may have too many twists and turns, for example.) Luckily, we can always determine displacement, and by carefully choosing small enough displacement steps, we can use displacement to obtain a pretty good approximation for the true distance travelled. (The mathematics of calculus provides a formal methodology for estimating a "true value" through the use of successively better approximations.) In the rest of this discussion, I will replace $\Delta$ with $\delta$ to indicate that small enough displacement steps have been used to provide a good enough approximation for the true distance travelled.

Velocity

${\vec {v}}_{av}\equiv {\frac {{\vec {x_{f}}}-{\vec {x_{i}}}}{t_{f}-t_{i}}}\equiv {\frac {\Delta {\vec {x}}}{\Delta t}}$

[Δ, delta, upper-case Greek D, is a prefix conventionally used to denote a difference.] Velocity answers the question "Is the object moving now, and if so - how quickly?"

Once again we have an operational definition: we are told what steps to follow to calculate velocity.

Note that this is a definition for average velocity. The displacement Δx is the vector sum of the smaller displacements which it contains, and some of these may subtract out. By contrast, the distance travelled is the scalar sum of the smaller distances, all of which are non-negative (they are the magnitudes of the displacements). Thus the distance travelled can be larger than the magnitude of the displacement, as in the example of travel on a circle, above. Consequently, the average velocity may be small (or zero, or negative) while the speed is positive.

If we are careful to use very small displacement steps, so that they come pretty close to approximating the true distance travelled, then we can write the definition for instantaneous velocity as

${\vec {v}}_{inst}\equiv {\frac {\vec {\delta x}}{\delta t}}$

[δ is the lower-case delta.] Or with the idea of limits from calculus, we have:

${\vec {v}}_{inst}\equiv {\frac {d{\vec {x}}}{dt}}$

[d, like Δ and δ, is merely a prefix; however, its use definitely specifies that this is a sufficiently small difference so that the error--due to stepping (instead of smoothly changing) the quantity--becomes negligible.]

Acceleration

${\vec {a}}_{av}\equiv {\frac {{\vec {v_{f}}}-{\vec {v_{i}}}}{t_{f}-t_{i}}}\equiv {\frac {\Delta {\vec {v}}}{\Delta t}}$

Acceleration answers the question "Is the object's velocity changing, and if so - how quickly?"

Once again we have an operational definition. We are told what steps to follow to calculate acceleration.

Again, also note that technically we have a definition for average acceleration. As for displacement, if we are careful to use a series of small velocity changes, then we can write the definition for instantaneous acceleration as:

${\vec {a}}_{inst}\equiv {\frac {\delta {\vec {v}}}{\delta t}}$

Or with the help of calculus, we have:

${\vec {a}}_{inst}\equiv {\frac {d{\vec {v}}}{dt}}={\frac {d^{2}{\vec {x}}}{dt^{2}}}$

Vectors

Notice that the definitions given above for displacement, velocity and acceleration included little arrows over many of the terms. The little arrow reminds us that direction is an important part of displacement, velocity, and acceleration. These quantities are vectors. By convention, the little arrow always points right when placed over a letter. So for example, ${\vec {v}}$ just reminds us that velocity is a vector, and does not imply that this particular velocity is rightward.

Why do we need vectors? As a simple example, consider velocity. It is not enough to know how fast one is moving. We also need to know which direction we are moving. Less trivially, consider how many different ways an object could be experiencing an acceleration (a change in its velocity). Ultimately, there are three distinct ways an object could accelerate:

1. The object could be speeding up.
2. The object could be slowing down.
3. The object could be traveling at constant speed, while changing its direction of motion.

More general accelerations are simply combinations of 1 and 3 or 2 and 3.

Importantly, a change in the direction of motion is just as much an acceleration as is speeding up or slowing down.

In classical mechanics, no direction is associated with time (you cannot point to next Tuesday). So the definition of ${\vec {a}}_{av}$ tells us that acceleration will point wherever the change in velocity $\Delta {\vec {v}}$ points.

Understanding that the direction of $\Delta {\vec {v}}$ determines the direction of ${\vec {a}}$ leads to three non-mathematical but very powerful rules of thumb:

1. If the velocity and acceleration of an object point in the same direction, the object's speed is increasing.
2. If the velocity and acceleration of an object point in opposite directions, the object's speed is decreasing.
3. If the velocity and acceleration of an object are perpendicular to each other, the object's initial speed stays constant (in that initial direction), while the speed of the object in the direction of the acceleration increases. Think of a bullet fired horizontally in a vertical gravitational field. Since velocity in the one direction remains constant, and the velocity in the other direction increases, the overall velocity (absolute velocity) also increases.

Again, more general motion is simply a combination of 1 and 3 or 2 and 3.

Using these three simple rules will dramatically help your intuition of what is happening in a particular problem. In fact, much of the first semester of college physics is simply the application of these three rules in different formats.

Equations of motion (constant acceleration)

A particle is said to move with constant acceleration if its velocity changes by equal amounts in equal intervals of time, no matter how small the intervals may be

${\frac {d{\vec {a}}}{dt}}=0\ \mathrm {\frac {m}{s^{3}}}$

Since acceleration is a vector, constant acceleration means that both direction and magnitude of this vector don't change during the motion. This means that average and instantaneous acceleration are equal. We can use that to derive an equation for velocity as a function of time by integrating the constant acceleration.

${\boldsymbol {v}}(t)={\boldsymbol {v}}(0)+\int \limits _{0}^{t}{\boldsymbol {a}}\ dt$

Giving the following equation for velocity as a function of time.

${\boldsymbol {v}}(t)={\boldsymbol {v}}_{0}+{\boldsymbol {a}}t$

To derive the equation for position we simply integrate the equation for velocity.

${\boldsymbol {x}}(t)={\boldsymbol {x}}(0)+\int \limits _{0}^{t}{\boldsymbol {v}}(t)\ dt$

Integrating again gives the equation for position.

${\boldsymbol {x}}(t)={\boldsymbol {x}}_{0}+{\boldsymbol {v}}_{0}t+{\frac {1}{2}}{\boldsymbol {a}}t^{2}$

The following are the equations of motion:

Equations of Motion

Equation	Description
${\vec {x}}={\vec {x}}_{0}+{\vec {v}}_{0}t+{\frac {{\vec {a}}t^{2}}{2}}\$	Position as a function of time
${\vec {v}}={\vec {v}}_{0}+{\vec {a}}t\$	Velocity as a function of time

The following equations can be derived from the two equations above by combining them and eliminating variables.

$v^{2}=v_{0}^{2}+2{\vec {a}}\cdot ({\vec {x}}-{\vec {x}}_{0})\$	Eliminating time (very useful, see the section on Energy)
${\vec {x}}={\vec {x}}_{0}+{\frac {{\vec {v}}_{0}t+{\vec {v}}t}{2}}$	Eliminating acceleration

Symbols

Symbol	Description
${\vec {v}}$	velocity (at time t)
${\vec {v_{0}}}$	initial velocity
${\vec {a}}$	(constant) acceleration
$t\$	time (taken during the motion)
${\vec {x}}$	position (at time t)
${\vec {x_{0}}}$	initial position

Acceleration in Two Dimensions

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Acceleration in Three Dimensions

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What does force in motion mean?

Force means strength and power. Motion means movement. That’s why we need forces and motions in our life. We need calculation when we want to know how fast things go, travel and other things which have force and motion. ...

How do we calculate the speed?

If you want to calculate the average speed, distance travelled or time taken you need to use this formula and remember it:

${speed}={\frac {distance}{time\ taken}}$

This is an easy formula to use, you can find the distance travelled, time taken or average speed, you need at least 2 values to find the whole answer.

Is velocity the same thing as speed?

Velocity is a vector quantity that refers to "the rate at which an object changes its position", whereas speed is a scalar quantity, which cannot be negative. Imagine a kid moving rapidly, one step forward and one step back, always returning to the original starting position. While this might result in a frenzy activity, it would result in a zero velocity, because the kid always returns to the original position, the motion would never result in a change in position, in other words $\Delta {\vec {x}}$ would be zero.

Speed is measured in the same physical units of measurement as velocity, but does not contain an element of direction. Speed is thus the magnitude component of velocity. Velocity contains both the magnitude and direction components. You can think of velocity as the displacement/duration, whereas speed can be though as distance/duration.

Acceleration

When a car is speeding up we say that it is accelerating, when it slows down we say it is decelerating.

How do we calculate it?

When we want to calculate it, the method goes like that: A lorry driver brakes hard, and slows from 25 m/s to 5 m/s in 5 seconds. What was the vehicle's acceleration?

${acceleration}={\frac {change\ in\ velocity}{time\ taken}}={\frac {5-25}{5}}={\frac {-20}{5}}=-4\ ms^{-2}$

What is initial velocity and final velocity? Initial velocity is the beginning before motion starts or in the middle of the motion, final velocity is when the motion stops.

There is another way to calculate it and it is like that This equations which are written is the primary ones, which means that when you don’t have lets say final velocity, how will you calculate the equation?

This is the way you are going to calculate.

Observing motion

When you want to know how fast an athletic person is running, what you need is a stopwatch in your hand, then when the person starts to run, you start the stopwatch and when the person who is sprinting stops at the end point, you stop the watch and see how fast he ran, and if you want to see if the athlete is wasting his energy, while he is running look at his movement, and you will know by that if he is wasting his energy or not.

This athletic person is running, and while he is running the scientist could know if he was wasting his energy if they want by the stop watch and looking at his momentum.

Measuring acceleration

Take a slope, a trolley, some tapes and a stop watch, then put the tapes on the slope and take the trolley on the slope, and the stopwatch in your hand, as soon as you release the trolley, start timing the trolley at how fast it will move, when the trolley stops at the end then stop the timing. After wards, after seeing the timing , record it, then you let the slope a little bit high, and you will see, how little by little it will decelerate.

Newton

Isaac Newton was an English physicist, mathematician (described in his own day as a "natural philosopher") , astronomer and alchemist. Newton is one of the most influential scientists of all time, and he is known, among other things, for contributing to development of classical mechanics and for inventing, independently from Gottfried Leibniz, calculus.

Newton's laws of motion

Newton is also known by his three laws of motion, which describe the relationship between a body and the forces acting upon it, and its motion in response to said forces.

1. First Law (also known as the law of inertia) states that every body continues in its state of rest or state of uniform motion unless compelled to change that state by being subject to an external force. The moment of inertia is defined as the tendency of matter to resist any change in its state of motion or state of rest.
2. Second Law The vector sum of the external forces $F$ on an object is equal to the mass $m$ of that object multiplied by the acceleration vector $a$ of the object, or algebraically $F=ma$.
3. Third Law states that when one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.

Symbols

Some useful symbols seen and that we will see:

Name	Symbol
Distance travelled	$d$ or $s$
Force	$F$
Velocity	${\vec {v}}$
Initial velocity	${\vec {v}}_{0}$ or ${\vec {v}}_{i}$
Final velocity	${\vec {v}}_{f}$
Change in velocity	$\Delta {\vec {v}}$
Acceleration	${\vec {a}}$
Mass	$m$
Newton	$N$
Gravity	$G$
Weight	$W$

Force

A force is any interaction that tends to change the motion of an object. In other words, a force can cause an object with mass to change its velocity. Force can also be described by intuitive concepts such as a push or a pull. A force has both magnitude and direction, making it a vector quantity. It is measured in the SI unit of newtons and represented by the symbol $F$.

How to calculate the force?

When we want to calculate the force, and we have the mass and acceleration, we can simply use the simple formula stated in the Newton's second law above, that is $F=ma$, where $m$ is the mass (or the amount of matter in a body), and $a$ is the acceleration. Note that the Newton’s second law is defined as a numerical measure of inertia.

What is inertia?

Inertia is the tendency of a body to maintain its state of rest or uniform motion, unless acted upon by an external force.

Robert Hooke

Robert Hooke was an English polymath who played an important role in the scientific revolution, through both experimental and theoretical work.

Hooke's law

Hooke's law is a principle of physics that states that the force $F$ needed to extend or compress a spring by some distance $X$ is proportional to that distance, or algebraically $F=-kX$, where $k$ is a constant factor characteristic of the spring, its stiffness.

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Force

A net force on a body causes a body to accelerate. The amount of that acceleration depends on the body's inertia (or its tendency to resist changes in motion), which is measured as its mass. When Isaac Newton formulated Newtonian mechanics, he discovered three fundamental laws of motion.

Later, Albert Einstein proved that these laws are just a convenient approximation. These laws, however, greatly simplify calculations and are used when studying objects at velocities that are small compared with the speed of light.

Friction

It is the force that opposes relative motion or tendency of relative motion between two surfaces in contact represented by f. When two surfaces move relative to each other or they have a tendency to move relative to each other, at the point (or surface) of contact, there appears a force which opposes this relative motion or tendency of relative motion between two surfaces in contact. It acts on both the surfaces in contact with equal magnitude and opposite directions (Newton's 3rd law). Friction force tries to stop relative motion between two surfaces in contact, if it is there, and when two surfaces in contact are at rest relative to each other, the friction force tries to maintain this relative rest. Friction force can assume the magnitude (below a certain maximum magnitude called limiting static friction) required to maintain relative rest between two surfaces in contact. Because of this friction force is called a self adjusting force.

Earlier, it was believed that friction was caused due to the roughness of the two surfaces in contact with each other. However, modern theory stipulates that the cause of friction is the Coulombic force between the atoms present in the surface of the regions in contact with each other.

Formula: Limiting Friction = (Friction Coefficient)(Normal reaction)

Static Friction = the friction force that keeps an object at relative rest.

Kinetic Friction = sliding friction

Newton's First Law of Motion

(The Law of Inertia)

A static object with no net force acting on it remains at rest or if in movement it will maintain a constant velocity

This means, essentially, that acceleration does not occur without the presence of a force. The object tends to maintain its state of motion. If it is at rest, it remains at rest and if it is moving with a velocity then it keeps moving with the same velocity. This tendency of the object to maintain its state of motion is greater for larger mass. The "mass" is, therefore, a measure of the inertia of the object.

In a state of equilibrium, where the object is at rest or proceeding at a constant velocity, the net force in every direction must be equal to 0.

At a constant velocity (including zero velocity), the sum of forces is 0. If the sum of forces does not equal zero, the object will accelerate (change velocity over time).

It is important to note, that this law is applicable only in non-accelerated coordinate systems. It is so, because the perception of force in accelerated systems are different. A body under balanced force system in one frame of reference, for example a person standing in an accelerating lift, is acted upon by a net force in the earth's frame of reference.

Inertia is the tendency of an object to maintain its velocity i.e. to resist acceleration.

• Inertia is not a force.
• Inertia varies directly with mass.

Newton's Second Law of Motion

• The time rate of change in momentum is proportional to the applied force and takes place in the direction of the force.

• 'The acceleration of an object is proportional to the force acting upon it.

These two statements mean the same thing, and is represented in the following basic form (the system of measurement is chosen such that constant of proportionality is 1) :

${\vec {F}}={\frac {d}{dt}}(m{\vec {v}})$

The product of mass and velocity i.e. mv is called the momentum. The net force on a particle is thus equal to rate change of momentum of the particle with time. Generally mass of the object under consideration is constant and thus can be taken out of the derivative.

${\vec {F}}=m{\frac {d{\vec {v}}}{dt}}=m{\vec {a}}$

Force is equal to mass times acceleration. This version of Newton's Second Law of Motion assumes that the mass of the body does not change with time, and as such, does not represent a general mathematical form of the Law. Consequently, this equation cannot, for example, be applied to the motion of a rocket, which loses its mass (the lost mass is ejected at the rear of the rocket) with the passage of time.

An example: If we want to find out the downward force of gravity on an object on Earth, we can use the following formula:

$\|{\vec {F}}\|=m\|{\vec {g}}\|$

Hence, if we replace m with whatever mass is appropriate, and multiply it by 9.806 65 m/s², it will give the force in newtons that the earth's gravity has on the object in question (in other words, the body's weight).

Newton's Third Law of Motion

Forces occur in pairs equal in magnitude and opposite in direction

This means that for every force applied on a body A by a body B, body B receives an equal force in the exact opposite direction. This is because forces can only be applied by a body on another body. It is important to note here that the pair of forces act on two different bodies, affecting their state of motion. This is to emphasize that pair of equal forces do not cancel out.

There are no spontaneous forces.

It is very important to note that the forces in a "Newton 3 pair", described above, can never act on the same body. One acts on A, the other on B. A common error is to imagine that the force of gravity on a stationary object and the "contact force" upwards of the table supporting the object are equal by Newton's third law. This is not true. They may be equal - but because of the second law (their sum must be zero because the object is not accelerating), not because of the third.

The "Newton 3 pair" of the force of gravity (= earth's pull) on the object is the force of the object attracting the earth, pulling it upwards. The "Newton 3 pair" of the table pushing it up is that it, in its turn, pushes the table down.

Equations

To find Displacement

$\Delta x=t{\vec {v}}_{i}+{\frac {t^{2}{\vec {a}}}{2}}$

To find Final Velocity

${\vec {v}}_{f}={\vec {v}}_{i}+t{\vec {a}}$

To find Final Velocity

$(v_{f})^{2}=(v_{i})^{2}+2a\Delta x$

To find Force when mass is changing

${\vec {F}}={\frac {d}{dt}}(m{\vec {v}})$

To find Force when mass is a constant

${\vec {F}}=m{\frac {d{\vec {v}}}{dt}}=m{\vec {a}}$

Variables
${\vec {F}}$ Force (N) $m\$ Mass (kg) ${\vec {a}}$ Acceleration (m/s2) ${\vec {p}}$ Momentum (kg m/s) $t\$ time (s) ${\vec {T}}$ Tension (N) ${\vec {g}}$ Acceleration due to gravity near the earth's surface ($\|{\vec {g}}\|=9.806\ 65\mathrm {m/s^{2}}$ see Physics Constants)

Definitions

Force (F): Force is equal to rate change of momentum with time. (Newton’s second law). A vector. Units: newtons (N) The newton (N): defined as the force it takes to accelerate one kilogram one metre per second squared (U.S. meter per second squared), that is, the push it takes to speed up one kilogram from rest to a velocity of 1 m/s in 1 second $1\mathrm {N} =1\mathrm {kg} \cdot \mathrm {m} /\mathrm {s} ^{2}$ Mass (m) : Also called inertia. The tendency of an object to resist change in its motion, and its response to a gravitational field. A scalar. Units: kilograms (kg) Acceleration (a): Change in velocity (Δv) divided by time (t). A vector. Units: meters per second squared (U.S. meters per seconds squared) (m/s2) Momentum (p): Mass times velocity. Expresses the motion of a body and its resistance to changing that motion. A vector. Units: kg m/s

Momentum

Linear momentum

Momentum is equal to mass times velocity.

${\vec {p}}=m{\vec {v}}$

Angular momentum

Angular momentum of an object revolving around an external axis $O$ is equal to the cross-product of the position vector with respect to $O$ and its linear momentum.

${\vec {L}}={\vec {r}}\times {\vec {p}}={\vec {r}}\times m{\vec {v}}$

Angular momentum of a rotating object is equal to the moment of inertia times angular velocity.

${\vec {L}}=I{\vec {\omega }}$

Force and linear momentum, torque and angular momentum

Net force is equal to the change in linear momentum over the change in time.

${\vec {F}}={\frac {\Delta {\vec {p}}}{\Delta t}}$

Net torque is equal to the change in angular momentum over the change in time.

${\vec {\tau }}={\frac {\Delta {\vec {L}}}{\Delta t}}$

Conservation of momentum

${\begin{matrix}{\vec {p}}_{i}={\vec {p}}_{f}\\{\vec {L}}_{i}={\vec {L}}_{f}\end{matrix}}$

Let us prove this law.

We'll take two particles $a,b$ . Their momentums are ${\vec {p}}_{a},{\vec {p}}_{b}$ . They are moving opposite to each other along the $x$-axis and they collide. Now force is given by:

${\vec {F}}={\frac {d{\vec {p}}}{\mathrm {d} t}}$

According to Newton's third law, the forces on each particle are equal and opposite.So,

${\frac {d{\vec {p}}_{a}}{dt}}=-{\frac {d{\vec {p}}_{b}}{dt}}$

Rearranging,

${\frac {d{\vec {p}}}{dt}}+{\frac {d{\vec {p}}_{b}}{dt}}=0$

This means that the sum of the momentums does not change with time. Therefore, the law is proved.

Variables

p: momentum, (kg·m/s) m: mass, (kg) v: velocity (m/s) L: angular momentum, (kg·m2/s) I: moment of inertia, (kg·m2) ω: angular velocity (rad/s) α: angular acceleration (rad/s2) F: force (N) t: time (s) r: position vector (m) Bold denotes a vector quantity. Italics denotes a scalar quantity.

Definition of terms

Momentum (p): Mass times velocity. (kg·m/s) Mass (m) : A quantity that describes how much material exists, or how the material responds in a gravitational field. Mass is a measure of inertia. (kg) Velocity (v): Displacement divided by time (m/s) Angular momentum (L): A vector quantity that represents the tendency of an object in circular or rotational motion to remain in this motion. (kg·m2/s) Moment of inertia (I): A scalar property of a rotating object. This quantity depends on the mass of the object and how it is distributed. The equation that defines this is different for differently shaped objects. (kg·m2) Angular speed (ω): A scalar measure of the rotation of an object. Instantaneous velocity divided by radius of motion (rad/s) Angular velocity (ω): A vector measure of the rotation of an object. Instantaneous velocity divided by radius of motion, in the direction of the axis of rotation. (rad/s) Force (F): mass times acceleration, a vector. Units: newtons (N) Time (t) : (s) Isolated system: A system in which there are no external forces acting on the system. Position vector (r): a vector from a specific origin with a magnitude of the distance from the origin to the position being measured in the direction of that position. (m)

Calculus-based Momentum

Force is equal to the derivative of linear momentum with respect to time.

${\vec {F}}={\frac {d{\vec {p}}}{dt}}$

Torque is equal to the derivative of angular momentum with respect to time.

${\vec {\tau }}={\frac {d{\vec {L}}}{dt}}$

The Normal Force

Why is it that we stay steady in our chairs when we sit down? According to the first law of motion, if an object is translationally in equilibrium (velocity is constant), the sum of all the forces acting on the object must be equal to zero. For a person sitting on a chair, it can thus be postulated that a normal force is present balancing the gravitational force that pulls the sitting person down. However, it should be noted that only some of the normal force can cancel the other forces to zero like in the case of a sitting person. In Physics, the term normal as a modifier of the force implies that this force is acting perpendicular to the surface at the point of contact of the two objects in question. Imagine a person leaning on a vertical wall. Since the person does not stumble or fall, he/she must be in equilibrium. Thus, the component of his/her weight along the horizontal is balanced or countered (opposite direction) by an equal amount of force -- this force is the normal force on the wall. So, on a slope, the normal force would not point upwards as on a horizontal surface but rather perpendicular to the slope surface.

The normal force can be provided by any one of the four fundamental forces, but is typically provided by electromagnetism since microscopically, it is the repulsion of electrons that enables interaction between surfaces of matter. There is no easy way to calculate the normal force, other than by assuming first that there is a normal force acting on a body in contact with a surface (direction perpendicular to the surface). If the object is not accelerating (for the case of uniform circular motion, the object is accelerating) then somehow, the magnitude of the normal force can be solved. In most cases, the magnitude of the normal force can be solved together with other unknowns in a given problem.

Sometimes, the problem does not warrant the knowledge of the normal force(s). It is in this regard that other formalisms (e.g. Lagrange method of undertermined coefficients) can be used to eventually solve the physical problem.

Friction

When there is relative motion between two surfaces, there is a resistance to the motion. This force is called friction. Friction is the reason why people could not accept Newton's first law of Motion, that an object tends to keep its state of motion. Friction acts opposite to the direction of the original force. The frictional force is equal to the frictional coefficient times the normal force.

Friction is caused due to attractive forces between the molecules near the surfaces of the objects. If two steel plates are made really flat and polished and cleaned and made to touch in a vacuum, it bonds together. It would look as if the steel was just one piece. The bonds are formed as in a normal steel piece. This is called cold welding. And this is the main cause of friction.

The above equation is an empirical one--in general, the frictional coefficient is not constant. However, for a large variety of contact surfaces, there is a well characterized value. This kind of friction is called Coulomb friction. There is a separate coefficient for both static and kinetic friction. This is because once an object is pushed on, it will suddenly jerk once you apply enough force and it begins to move.

Also, the frictional coefficient varies greatly depending on what two substances are in contact, and the temperature and smoothness of the two substances. For example, the frictional coefficients of glass on glass are very high. When you have similar materials, in most cases you don't have Coulomb friction.

For static friction, the force of friction actually increases proportionally to the force applied, keeping the body immobile. Once, however, the force exceeds the maximum frictional force, the body will begin to move. The maximum frictional force is calculated as follows:

$\left|{\vec {F}}_{f}\right|\leq \mu _{s}\left|{\vec {N}}\right|$

The static frictional force is less than or equal to the coefficient of static friction times the normal force. Once the frictional force equals the coefficient of static friction times the normal force, the object will break away and begin to move.

Once it is moving, the frictional force then obeys:

$\left|{\vec {F}}_{f}\right|=\mu _{k}\left|{\vec {N}}\right|$

The kinetic frictional force is equal to the coefficient of kinetic friction times the normal force. As stated before, this always opposes the direction of motion.

Variables

Symbol	Units	Definition
${\vec {F}}_{f}$	$\mathrm {N} \$	Force of friction
$\mu \$	none	Coefficient of friction

Definition of Terms

Normal force (N): The force on an object perpendicular to the surface it rests on utilized in order to account for the body's lack of movement. Units: newtons (N) Force of friction (Ff): The force placed on a moving object opposite its direction of motion due to the inherent roughness of all surfaces. Units: newtons (N) Coefficient of friction (μ): The coefficient that determines the amount of friction. This varies tremendously based on the surfaces in contact. There are no units for the coefficient of either static or kinetic friction

It's important to note, that in real life we often have to deal with viscose and turbulent friction - they appear when you move the body through the matter.

Viscose friction is proportional to velocity and takes place at approximately low speeds. Turbulent friction is proportional to $V^{2}$ and takes place at higher velocities.

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Work

Work is equal to the scalar product of force and displacement.

$W={\vec {F}}\cdot {\vec {d}}$

The scalar product of two vectors is defined as the product of their lengths with the cosine of the angle between them. Work is equal to force times displacement times the cosine of the angle between the directions of force and displacement.

$W=\|{\vec {F}}\|\ \|{\vec {d}}\|\cos \theta$

Work is equal to change in kinetic energy plus change in potential energy for example the potential energy due to gravity.

$W=\Delta \mathrm {KE} +\Delta \mathrm {PE} _{g}\$

Work is equal to average power times time.

$W=Pt\$

The Work done by a force taking something from point 1 to point 2 is

$W_{1,2}=\int _{{\vec {x}}_{1}}^{{\vec {x}}_{2}}{\vec {F}}\cdot d{\vec {l}}$

Work is in fact just a transfer of energy. When we 'do work' on an object, we transfer some of our energy to it. This means that the work done on an object is its increase in energy. Actually, the kinetic energy and potential energy is measured by calculating the amount of work done on an object. The gravitational potential energy (there are many types of potential energies) is measured as 'mgh'. mg is the weight/force and h is the distance. The product is nothing but the work done. Even kinetic energy is a simple deduction from the laws of linear motion. Try substituting for v^2 in the formula for kinetic energy.

Variables

W: Work (J) F: Force (N) d: Displacement (m)

Definition of terms

Work (W): Force times distance. Units: joules (J) Force (F): mass times acceleration (Newton’s classic definition). A vector. Units: newtons (N)

When work is applied to an object or a system it adds or removes kinetic energy to or from that object or system. More precisely, a net force in one direction, when applied to an object moving opposite or in the same direction as the force, kinetic energy will be added or removed to or from that object. Note that work and energy are measured in the same unit, the joule (J).

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Advanced work topics

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Energy

Kinetic energy is simply the capacity to do work by virtue of motion.

(Translational) kinetic energy is equal to one-half of mass times the square of velocity.

$\mathrm {KE} _{T}={\frac {1}{2}}m\|{\vec {v}}\|^{2}$

(Rotational) kinetic energy is equal to one-half of moment of inertia times the square of angular velocity.

$\mathrm {KE} _{R}={\frac {1}{2}}I\omega ^{2}$

Total kinetic energy is simply the sum of the translational and rotational kinetic energies. In most cases, these energies are separately dealt with. It is easy to remember the rotational kinetic energy if you think of the moment of inertia I as the rotational mass. However, you should note that this substitution is not universal but rather a rule of thumb.

Potential energy is simply the capacity to do work by virtue of position (or arrangement) relative to some zero-energy reference position (or arrangement).

Potential energy due to gravity is equal to the product of mass, acceleration due to gravity, and height (elevation) of the object.

$\mathrm {PE} _{g}=-m{\vec {g}}\cdot {\vec {x}}=mgy$

Note that this is simply the vertical displacement multiplied by the weight of the object. The reference position is usually the level ground but the initial position like the rooftop or treetop can also be used. Potential energy due to spring deformation is equal to one-half the product of the spring constant times the square of the change in length of the spring.

$\mathrm {PE} _{e}={\frac {1}{2}}k\|{\vec {x}}-{\vec {x}}_{e}\|^{2}={\frac {1}{2}}k\|\Delta {\vec {x}}\|^{2}$

The reference point of spring deformation is normally when the spring is "relaxed," i.e. the net force exerted by the spring is zero. It will be easy to remember that the one-half factor is inserted to compensate for finite '"change in length" since one would want to think of the product of force and change in length $(k\Delta {\vec {x}})\cdot \Delta {\vec {x}}$ directly. Since the force actually varies with $\Delta {\vec {x}}$, it is instructive to need a "correction factor" during integration.

Variables

K: Kinetic energy (J) m: mass (kg) v: velocity (m/s) I: moment of inertia, (kg·m2) ω: ("omega") angular momentum (rad/s) Ug: Potential energy (J) g: local acceleration due to gravity (on the earth’s surface, 9.8 m/s2) h: height of elevation (m) Ue: Potential energy (J) k: spring constant (N/m) Δx: change in length of spring (m)

Definition of terms

Energy: a theoretically indefinable quantity that describes potential to do work. SI unit for energy is the joule (J). Also common is the calorie (cal). The joule: defined as the energy needed to push with the force of one newton over the distance of one meter. Equivalent to one newton-meter (N·m) or one watt-second (W·s). 1 joule = 1 J = 1 newton • 1 meter = 1 watt • 1 second Energy comes in many varieties, including Kinetic energy, Potential energy, and Heat energy. Kinetic energy (K): The energy that an object has due to its motion. Half of velocity squared times mass. Units: joules (J) Potential energy due to gravity (UG): The energy that an object has stored in it by elevation from a mass, such as raised above the surface of the earth. This energy is released when the object becomes free to move. Mass times height time acceleration due to gravity. Units: joules (J) Potential energy due to spring compression (UE): Energy stored in spring when it is compressed. Units: joules (J) Heat energy (Q): Units: joules (J) Spring compression (Dx): The difference in length between the spring at rest and the spring when stretched or compressed. Units: meters (m) Spring constant (k): a constant specific to each spring, which describes its “springiness”, or how much work is needed to compress the spring. Units: newtons per meter (N/m) Change in spring length (Δx): The distance between the at-rest length of the spring minus the compressed or extended length of the spring. Units: meters (m) Moment of inertia (I): Describes mass and its distribution. (kg•m2) Angular momentum (ω): Angular velocity times mass (inertia). (rad/s)