BIOPHYSICS FALL SEMESTER DENTISTRY PROGRAM syllabus ...
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Mechanics I. 1
BIOPHYSICS โ FALL SEMESTER
DENTISTRY PROGRAM
syllabus
CLASSICAL MECHANICS I.
Dr. Beรกta Bugyi
associate professor
Andrea Vig
assistant professor
University of Pรฉcs, Medical School
Department of Biophysics
2016
Mechanics I. 2
CLASSICAL MECHANICS I.
KINEMATICS: VELOCITY, ACCELERATION
DYNAMICS: NEWTONโS LAW OF MOTION, FORCE
LINEAR MOMENTUM
WORK, ENERGY
INTRODUCTION
Classical mechanics concerns with the physical laws, which describe the motion of bodies under the
action of a system of forces.
Classical mechanics includes:
Kinematics: study of motions of object without regard of its causes
Dynamics: study of motion of objects and its relation to forces
Statics: study of equilibrium of objects and its relation to forces
KINEMATICS
Kinematics describes the motion of objects without considering its causes.
LINEAR QUANTITIES DESCRIBING MOTION
For simplicity
we analyze the motion of objects without any change in direction, along a straight line (1D, linear
motion)
we assume that the objects are rigid bodies: an ideal solid body in which deformation is neglected,
i.e. the distance between any two points of the body remains constant in time regardless of
external forces acting on it.
Reference (observational) frame
physical concept: set of physical reference points that uniquely fix (locate and orient) the frame
and standardize measurements (the motion is described relative to the reference frame)
mathematical concept: an abstract coordinate system
1D line: ๐ฅ, 2D plane: ๐ฅ, ๐ฆ, 3D: ๐ฅ, ๐ฆ, ๐ง
Example
Describe the motion of the driver considering the car or the road as the reference frame.
Mechanics I. 3
(Elapsed) Time [๐ก] = ๐
change (โ) or the interval over which motion occurs
difference between the ending time and beginning time
โ๐ก = ๐ก๐๐๐๐๐ โ ๐ก๐๐๐๐ก๐๐ = ๐ก โ ๐ก0
Position
where the object is in space at a particular time
relative position to a convenient reference frame (e.g. Earth, lecture room, blackboard, etc.)
1D line: ๐ฅ, 2D plane: ๐ฅ, ๐ฆ, 3D: ๐ฅ, ๐ฆ, ๐ง
Displacement [๐ฅ] = ๐
change (โ) in position of the object due to its movement relative to the reference frame
has both direction and magnitude (vector quantity1)
negative and positive displacement
โ๐ฅ = ๐ฅ๐๐๐๐๐ โ ๐ฅ๐๐๐๐ก๐๐ = ๐ฅ โ ๐ฅ0
Distance
magnitude of the displacement
Distance traveled
total path length traveled between two positions
distance traveled is not necessarily equals to the magnitude of the displacement
Velocity [๐ฃ] =๐
๐ = ๐๐ โ1
the rate at which displacement changes, i.e. displacement divided by the time of travel
has both direction and magnitude (vector quantity)
can be caused by either a change in the magnitude or the direction of the displacement
negative and positive velocity
average velocity: displacement divided by time over which displacement occurs
o ๐ฃ๐๐ฃ๐๐๐๐๐ =โ๐ฅ
โ๐ก=
๐ฅโ๐ฅ0
โ๐ก
instantaneous velocity: velocity at a specific instant, or the average velocity over an
infinitesimal time interval
ADVANCED
๐ฃ๐๐๐ ๐ก๐๐๐ก๐๐๐๐๐๐ข๐ = limโ๐กโ0
โ๐ฅ
โ๐ก=
๐๐ฅ
๐๐ก
1 APPENDIX 1
Mechanics I. 4
Acceleration [๐] =๐
๐ 2= ๐๐ โ2
the rate at which velocity changes, i.e. change in velocity divided by the time
has both direction and magnitude (vector quantity)
can be caused by either a change in the magnitude or the direction of the velocity
average acceleration: change in velocity divided by time over which change occurs
o ๐๐๐ฃ๐๐๐๐๐ =โ๐ฃ
โ๐ก=
๐ฃโ๐ฃ0
โ๐ก
instantaneous acceleration: acceleration at a specific instant
Deceleration always refers to acceleration in the direction opposite to the direction of the
velocity. Deceleration always reduces speed. Negative acceleration, however, is acceleration in
the negative direction in the chosen coordinate system. Negative acceleration may or may not be
deceleration, and deceleration may or may not be considered negative acceleration.,
ADVANCED
๐๐๐๐ ๐ก๐๐๐ก๐๐๐๐๐๐ข๐ = limโ๐กโ0
โ๐ฃ
โ๐ก=
๐๐ฃ
๐๐ก
Examples
Source: College Physics Page 82. Example 12.
The speed of propagation of the action potential (an electrical signal) in a nerve cell depends (inversely) on
the diameter of the axon (nerve fiber). If the nerve cell connecting the spinal cord to your feet is 1.1 m long,
and the nerve impulse speed is 18 m/s, how long does it take for the nerve signal to travel this distance? Give
the velocity in km/h.
๐ฅ = ๐ฅ0 + ๐ฃ0๐ก โ ๐ก =๐ฅ
๐ฃ๐= 0.0611 = 61 โ 10โ3๐ = 61 ๐๐
๐ฃ = 18๐
๐ = 18
10โ3 ๐๐
13600โ
= 64.8๐๐
โ
Source: College Physics Page 83. Example 26.
Blood is accelerated from rest to 30 cm/s in a distance of 1.8 cm at a constant
acceleration by the left ventricle of the heart. How long does the acceleration
take? Give the acceleration of the blood. Is it reasonable when compared with
the time for a heartbeat (consider 70 hbs in 1 min)? What is the bloodโs
acceleration, expressed in multiples of g?
๐ฃ๐ =๐ฃ + ๐ฃ0
2= 15
๐๐
๐
๐ฅ = ๐ฅ0 + ๐ฃ0๐ก โ ๐ก =๐ฅ
๐ฃ๐= 0.12 ๐ = 120 ๐๐
1 heartbeat = 60/70 = 0.85 s = 850 ms, reasonable
Mechanics I. 5
๐ฃ๐ = ๐ฃ0 +๐๐๐ก
2โ ๐ =
2๐ฃ๐๐ก
= 500๐๐
๐ 2= 5
๐
๐ 2
SUMMARY โ MOTION EQUATIONS IN 1D
Elapsed time: โ๐ก = ๐ก๐๐๐๐๐ โ ๐ก๐๐๐๐ก๐๐๐ = ๐ก (๐ก๐๐๐๐ก๐๐๐ = 0) (Eq. 1)
Displacement: โ๐ฅ = ๐ฅ๐๐๐๐๐ โ ๐ฅ๐๐๐๐ก๐๐๐ = ๐ฅ โ ๐ฅ0 (Eq. 2)
Change in velocity: โ๐ฃ = ๐ฃ๐๐๐๐๐ โ ๐ฃ๐๐๐๐ก๐๐๐ = ๐ฃ โ ๐ฃ0 (Eq. 3)
Average velocity: ๐ฃ๐ =โ๐ฅ
โ๐ก=
๐ฅโ๐ฅ0
๐ก (Eq. 4)
Average acceleration: ๐๐ =โ๐ฃ
โ๐ก=
๐ฃโ๐ฃ0
๐ก= ๐๐๐๐ ๐ก๐๐๐ก (Eq. 5)
Eq. 4 โ final position:
๐(๐) = ๐๐ + ๐๐๐ (Eq. 6)
Eq. 5 โ final velocity:
๐(๐) = ๐๐ + ๐๐๐ (Eq. 7)
If ๐๐ = ๐๐๐๐ ๐ก๐๐๐ก โ ๐๐ =๐+๐๐
๐ (Eq. 8)
Eq. 7 & 6 โ
๐ + ๐๐
๐=
๐๐๐ + ๐๐๐
๐
๐๐ = ๐๐ +๐๐๐
๐
๐(๐) = ๐๐ + ๐๐๐ +๐๐๐
๐
๐
Mechanics I. 6
DYNAMICS
Dynamics describes the motion of objects and its relation to forces.
NEWTONโS LAWS, FORCES
1st law (law of inertia)
A body at rest remains at rest, or, if in motion, remains in motion at a constant velocity unless acted
on by a net external force.
There must be a cause (net external force, force acts from outside the system of interest) for there to be
any change in velocity (either a change in magnitude or direction). If no net external force (Fnet = 0) acts
on a body its velocity cannot change, the object cannot accelerate.
What do we mean by a change in motion? A change in motion is equivalent to a change in velocity. A
change in velocity means that there is an acceleration. Newtonโs 1st law says that a net external force
causes a change in motion; thus, a net external force causes acceleration.
net external force โ change in motion ~ change in velocity (magnitude, direction) โ acceleration
Inertia
The property of a body to remain at rest or to remain in motion with constant velocity is called inertia.
The resistance to any change in the state of motion. The inertia of an object is measured by its inertial
mass. As we know from experience, some objects have more inertia than others. It is obviously more
difficult to change the motion of a large truck than that of a basketball.
It is easy to show experimentally that acceleration is linearly proportional to the net external forceโฆ
๐~๐น๐๐๐ก and inversely proportional to the mass ๐~1
๐
Mechanics I. 7
2nd law
The acceleration of a system is directly proportional to and in the same direction as the net external
force acting on the system, and inversely proportional to its mass.
๐ =๐ญ๐๐๐
๐ ๐ญ๐๐๐ = ๐๐
Unit of force: ๐ญ๐๐๐ = ๐๐ โ [๐น] = ๐๐๐
๐ 2= ๐(๐๐๐ค๐ก๐๐)
ADVANCED
The net force on an object is equal to the rate of change (that is, the derivative) of its linear momentum p
in an inertial reference frame:
Original form: ๐น๐๐๐ก =๐(๐๐ฃ)
๐๐ก= ๐
๐๐ฃ
๐๐ก+
๐๐
๐๐ก๐ฃ
Limitations: theory of relativity ๐ =๐0
โ1โ๐ฃ2
๐2
3rd law (symmetry in forces, one of the fundamental symmetry principles of the universe, law of equal
and opposite reactions):
Whenever one body (A) exerts a force on a second body (B), the first body experiences a force that is
equal in magnitude and opposite in direction to the force that the first body exerts.
๐ญ๐จ๐ฉ = โ๐ญ๐ฉ๐จ
Example
How many times do you realize Newtonโs laws on the videos?
http://video.mit.edu/watch/concept-vignettes-newtons-laws-28734/
https://www.youtube.com/watch?v=C_r5UJrxcck
Mechanics I. 8
LINEAR MOMENTUM
Linear momentum:
๐ = ๐๏ฟฝโ๏ฟฝ
[๐] = ๐๐๐
๐
Example
Source: College Physics Page 286. Example 8.1
Calculate the linear momentum of a 110 kg football player running at 8 m/s. Compare the playerโs momentum
with the momentum of a hard-thrown 0.4 kg ball that has a speed of 25 m/s.
Linear momentum and Newtonโs 2nd law:
๏ฟฝโ๏ฟฝ = ๐๏ฟฝโ๏ฟฝ = ๐โ๐ฃโโ โโ โ
โ๐ก=
โ๐โโโโโโ
โ๐ก
Net external force equals to the change in the linear momentum of a system over the time during which
the change occurs.
CONSERVATION OF LINEAR MOMENTUM
Net force results in a change in the linear momentum of the moving object. If the net force is zero, there
is no change in the linear momentum, it is constant.
๏ฟฝโ๏ฟฝ =โ๐โโโโโโ
โ๐ก= 0 โ โ๐โโโโโโ = 0 โ ๐ = ๐๐๐๐ ๐ก๐๐๐ก
Example
Source: College Physics Page 310. Example 8.
A car moving at 10 m/s crashes into a tree and stops in 0.26 s. Calculate the force the seat belt exerts on a
passenger (m = 70 kg) in the car to bring him to halt.
๐น =โ๐
โ๐ก=
๐โ๐ฃ
โ๐ก= 2690๐
Source: College Physics Page 310. Example 12.
One hazard of space travel is debris left by previous missions. There are several thousand objects orbiting
Earth that are large enough to be detected by radar, but there are far greater numbers of very small objects,
such as flakes of paint. Calculate the force exerted by a 0.1 mg chip of paint that strikes a spacecraft window
at a relative speed of 4*103 m/s, given the collision lasts 6*10-8 s.
๐น =โ๐
โ๐ก=
๐โ๐ฃ
โ๐ก=
0.1 โ 10โ3๐๐ โ 4 โ 103๐/๐
6 โ 10โ8๐ = 6.67 โ 106 ๐
Mechanics I. 9
WORK, ENERGY
WORK
The work done on a system by a constant force (๐น):
๐ = ๏ฟฝโ๏ฟฝ โ ๐
magnitude: ๐ = ๐น๐๐๐๐ ๐ผ, where ๐น๐๐๐ ๐ผ is the component of the force in the direction of the
motion and ๐ is the distance through which the force acts
[๐] = ๐๐ = ๐๐๐2
๐ 2= ๐ฝ (Joule)
POWER
the rate at which the work is done
๐ =๐
โ๐ก
[๐] =๐ฝ
๐ = ๐ (Watt)
1 horsepower = 735.5 W (needed to lift 75 kg - the average body mass of a person โ by 1 meter
within 1 second)
James Watt determined that a horse could turn a mill wheel 144 times in an hour (or 2.4 times a
minute). The wheel was 3.65 meters in radius; therefore, the horse travelled 2.4ยท2ฯยท3.65 m =
55.0128 m in one minute. Watt judged that the horse could pull with a force of 800.67 N.
๐~800 ๐ โ 55๐
60 ๐ ~733 ๐
๐ =๐นโ๐
โ๐ก= ๐น๐ฃ
Examples
Source: College Physics Page 242. Example 7.1
Investigate how the relative direction of the force and displacement affects the work. (parallel: ๐๐๐ 0๐ = 1,
perpendicular: ๐๐๐ 90๐ = 0).
How much work is done on the lawn mower by the person if he exerts a constant force of 75 N at an angle 35ยบ
below the horizontal and pushes the mower 25 m on level ground? Convert the amount of work from joules
to kilocalories and compare it with this personโs average daily intake of 10000kJ (about 2400 kcal) of food
energy. One calorie (1 cal) of heat is the amount required to warm 1 g of water by 1ยบC, and is equivalent to
4.184 J, while one food calorie (1 kcal) is equivalent to 4184 J.
๐ = ๐น๐๐๐๐ ๐ผ = 75 ๐ โ 25 ๐ โ ๐๐๐ 35๐ = 1536 ๐ฝ
Mechanics I. 10
๐(๐๐๐) =1536 ๐ฝ
4.184 ๐ฝ= 367 ๐๐๐ = 0.367 ๐๐๐๐
The ratio of the work to the daily consumption:
0.367 ๐๐๐๐
2400 ๐๐๐๐= 1.53 โ 10โ4 = 0.0153 %
This ratio is a tiny fraction of what the person consumes, but it is typical. Very little of the energy released
in the consumption of food is used to do work.
less than 10 % of our food energy intake is used to do work
more than 90 % is converted to thermal energy or stored as chemical energy in fat
Source: College Physics Page 283. Example 56.
The swimmer exerts an average horizontal backward force
of 80.0 N with his arm during each 1.80 m long stroke.
What is his work in each stroke? Calculate the power output
of his arms if he does 120 strokes per minute.
๐ = ๐น โ ๐ = 80 ๐ โ 1.8 ๐ = 144 ๐ฝ
1 stroke is done in 0.5 seconds
๐ =๐
๐ก=
144 ๐
0.5 ๐ = 288 ๐
Work as the area under the F(d) curve.
Mechanics I. 11
Work as the area under the p(V) curve.
Assume constant force, which is parallel to the direction of movement.
๐ = ๐น๐๐๐๐ ๐ผ = ๐น๐ = ๐๐ดโ๐ = ๐โ๐
This holds generally, the work can be derived graphically from the area under the p(V) curve.
Example โ work of the heart
Source: College Physics Page ?. Example ?.
Give the work done by the heart in one cardiac cycle, considering the p-V diagram of the left ventricle during
a single cardiac cycle.
Compare the amount of work done by the heart in 1 day with the personโs average daily intake of 10000 kJ
(about 2400 kcal) of food energy. One calorie (1 cal) of heat is the amount required to warm 1 g of water by
1ยบC, and is equivalent to 4.184 J, while one food calorie (1 kcal) is equivalent to 4184 J.
Considering 70 heartbeats in 1 minute, give the power of the heart.
๐ = โ๐โ๐ = (16 ๐๐๐ โ 1 ๐๐๐) โ (140 ๐๐ฟ โ 80 ๐๐ฟ) โ 10โ6๐3 = 0.9 ๐ฝ
1 minute 70 heartbeat, 1 day 70*24*60 = 100800 heartbeat
๐ = 0.9 ๐ฝ โ 100800 = 90.72 ๐๐ฝ
๐(๐๐๐) =90.72 ๐๐ฝ
4.184 ๐๐ฝ= 21.68 ๐๐๐๐
The ratio of the work to the daily consumption:
21.68 ๐๐๐๐
2400 ๐๐๐๐= 9 โ 10โ3 = 0.9 %
1 heartbeat 0.857 seconds
๐ =๐
๐ก=
0.9 ๐ฝ
0.857 ๐ = 1.1 ๐
Mechanics I. 12
ENERGY
Example
Source: College Physics Page 243. Example 7.3
A package is pushed horizontally with a
force of F = 120 N at a distance of d = 0.8 m.
Neglect the friction.
The work done by the force: ๐ = ๐น๐๐๐๐ ๐ผ = ๐น๐. According to Newtonโs laws net force results in a change
in the velocity of the package changes from ๐ฃ0 to ๐ฃ, consequently the package accelerates. Using
Newtonโs 2nd law, the work can be expressed as follows:
๐ = ๐น๐ = ๐๐๐
Using the motion equations for constant acceleration ๐ =๐ฃโ๐ฃ0
๐ก and ๐ =
๐ฃ+๐ฃ0
2๐ก, the following equation can
be derived:
๐ =1
2๐๐ฃ2 โ
1
2๐๐ฃ0
2
The ๐๐พ๐ธ =1
2๐๐ฃ2 quantity is defined as the translational kinetic energy:
the energy of an object due to translational motion
mechanical form of energy
The above example illustrates that the work done by the net force gives to a system energy of motion
(translational kinetic energy).
WORK-ENERGY THEOREM
The work of the net forces on a point-like object equals to the change in its translational kinetic
energy:
๐พ = โ๐ป๐ฒ๐ฌ =๐
๐๐๐๐ โ
๐
๐๐๐๐
๐
Example
Source: College Physics Page ?. Example ?.
A package of 30 kg is lifted through a height of d = 1 m. The is work done against the gravitational force
๐ = ๐น๐ = ๐๐โ
The ๐บ๐๐ธ = ๐๐โ is defined as the gravitational potential energy:
the energy of an object due to its position in a gravitational field
mechanical form of energy
The above example illustrates that the work done by the net force gives to a system energy due to the
change its position in the gravitational field (gravitational potential energy).
Mechanics I. 13
CONSERVATIVE FORCES
the work does not depend on the path, only on the starting and end points of the motion: ๐พ๐
e.g.: gravitational force
NONCONSERVATIVE FORCES
work depends on the path: ๐พ๐๐
e.g.: frictional force
changes mechanical energy into thermal energy
LAW OF CONSERVATION OF ENERGY
When only conservative forces act on a system the total mechanical energy of a system is constant:
๐ฎ๐ท๐ฌ + ๐ป๐ฒ๐ฌ = ๐๐๐๐๐๐๐๐
Total energy (gravitational potential energy, translational kinetic energy, other forms of energy) is
constant in any process; energy may change in form or be transferred from one system to another,
but the total remains the same:
๐ฎ๐ท๐ฌ๐ + ๐ป๐ฒ๐ฌ๐ +๐พ๐๐ +๐ถ๐ฌ๐ = ๐ฎ๐ท๐ฌ๐ + ๐ป๐ฒ๐ฌ๐ +๐ถ๐ฌ๐
Example
Source: College Physics Page 249. Example 7.6
A 60 kg person jumps onto the floor from a height of 3 m. Give the kinetic energy just before landing. If he
lands stiffly with his knee joints compressing by 0.5 cm, calculate the force on the knee joints.
The person loses his GPE upon fall, while he gains TKE. The kinetic energy of the person upon reaching the
floor is:
๐ = โ๐พ๐ธ=โ๐๐โ = โ60๐๐ โ ๐ โ (0 ๐ โ 3 ๐) = 1766 ๐ฝ
The floor stops the person within the distance of the bending of the knee joints, the work done:
๐ = โ๐น๐
The floor brings his TKE to zero, the work done:
๐ = โโ๐พ๐ธ = ๐๐โ
๐๐โ = โ๐น๐ โ ๐น = โ๐๐โ
๐=
60๐๐ โ ๐ โ 3๐
5 โ 10โ3๐= 353000 ๐ = 3.53 โ 105๐
This large force over a short period of time is enough to break bones. A better way of landing is to bend
the legs or rolling on the ground, thereby increasing the time/distance over which the force acts.
As example bending motion of 0.5 m results in a force 100 times smaller.
Mechanics I. 14
ENERGY CONVERSION IN HUMANS
According to the conservation of energy, the chemical
energy stored in food is converted to work, thermal
energy and stored as chemical energy in fatty tissue. The
fraction of each energy form depends on how much we
eat and on the level of our physical activity.
Metabolic rate (MR)
characteristic to the energy conversion rate of a person
the rate at which our body uses energy to sustain life and do different activities.
Basal metabolic rate (BMR): energy conversion rate at rest
ORGAN POWER CONSUMED
AT REST (W) OXYGEN CONSUMPTION
(mL/min) BMR (%)
liver, spleen 23 67 27
brain 16 47 19
skeletal muscle 15 45 18
kidney 9 26 10
heart 6 17 7
other 16 48 19
total 85 250 100
Energy consumption is directly proportional to
oxygen consumption because the digestive process
is basically one of oxidizing food. We can measure
the energy people use during various activities by
measuring their oxygen use. Approximately 20 kJ of
energy are produced for each liter of oxygen
consumed, independent of the type of food. The
amount of oxygen in blood can be measured by a
pulse oximeter.
ACTIVITY POWER CONSUMED AT REST
(W) OXYGEN CONSUMPTION (L/min)
sleeping 83 0.24
walking (5 km/h) 280 0.8
cycling (13 -18 km/h) 400 1.14
shivering 425 1.21
swimming breaststroke 475 1.36
running cross country 740 2.12
sprinting 2415 6.90
Mechanics I. 15
Examples
Source: College Physics Page 269. Example 7.13
If a person who normally requires an average of 10000 kJ (2400 kcal) of food energy per day consumes 11000
kJ per day, he will steadily gain weight. How much bicycling per day is required to work off this extra 1000 kJ?
Consider that 400 w is used when cycling at a moderate rate.
๐ =๐
๐กโ ๐ก =
๐
๐=
1000 ๐๐ฝ
400 ๐= 2500๐ = 42 ๐๐๐
Source: College Physics Page 269. Example 7.13
If this person uses more energy than he or she consumes, the personโs body will obtain the needed energy by
metabolizing body fat. Give the amount of fat loss if the person uses 11000 kJ but consumes only 10000 kJ,
assuming the energy content of fat to be 39 kJ/g.
๐๐๐ก ๐๐๐ ๐ = 1000 ๐๐ฝ1 ๐
39 ๐๐ฝ= 25.6 ๐
Source: College Physics Page ?. Example ?.
Energy that is not utilized for work or heat transfer is converted to the chemical energy of body fat containing
about 39 kJ/g. How many grams of fat will you gain if you eat 10000 kJ (about 2400 kcal) corresponding to
the average daily consumption. Give the amount of fat loss due to 8 h shivering (P = 425 W), or due to sleeping
(P = 83 W).
๐๐๐ก ๐๐๐๐ = 10000 ๐๐ฝ1 ๐
39 ๐๐ฝ= 256 ๐
๐ =๐
๐กโ ๐ = ๐๐ก = 425 ๐ โ 8 โ โ 60 โ 60 = 12240 ๐๐ฝ
๐๐๐ก ๐๐๐ ๐ = 12240 ๐๐ฝ1 ๐
39 ๐๐ฝ= 313.8 ๐
๐ =๐
๐กโ ๐ = ๐๐ก = 83 ๐ โ 8 โ โ 60 โ 60 = 2390 ๐๐ฝ
๐๐๐ก ๐๐๐ ๐ = 2390 ๐๐ฝ1 ๐
39 ๐๐ฝ= 61.3 ๐
Mechanics I. 16
APPENDIX 1
SCALAR AND VECTOR QUANTITIES
We measure the mass of an object to be 60 kilograms (kg). In this case, the physical quantity is the mass
and its value โ or magnitude โ is 60 kg; composed of a number (60) and a unit (kg). Note that without
knowing the exact unit the physical quantity is meaningless! These kind of physical quantities โ
composed of a number and a unit โ are called scalar quantities. 60 kg contain all the information about
the mass of the object. Letโs see another example. A particle moved 10 meters (m) from a given place.
Where is the particle at the end of its movement? Even if we know the magnitude of the displacement,
we do not know the exact new position of the particle. In this case, the magnitude of the physical quantity
does not contain all the information about the movement of the particle. What else do we have to know?
The direction of the movement in space. These kind of physical quantities โ composed of a number, a
unit and a direction โ are called vector quantities.
SCALAR AND VECTOR QUANTITIES
quantity
(notation) property how to treat examples
scalar
๐
only magnitude, no
direction ordinary algebra
mass, temperature, time,
energy, work, refractive
index, charge, pressure,
volume, areaโฆ
vector
๏ฟฝโโโ๏ฟฝ
both magnitude and
direction vector algebra
displacement, velocity,
acceleration, force, linear
momentum, torque,
angular momentum, โฆ
REPRESENTATION OF VECTORS, OPERATIONS WITH VECTORS
Representation of vector quantities
๏ฟฝโ๏ฟฝ โ๏ฟฝโ๏ฟฝ
Mechanics I. 17
๏ฟฝโ๏ฟฝ ๏ฟฝโโ๏ฟฝ
๏ฟฝโ๏ฟฝ + ๏ฟฝโโ๏ฟฝ = ๐ ๏ฟฝโ๏ฟฝ + โ๏ฟฝโโ๏ฟฝ = ๏ฟฝโ๏ฟฝ โ ๏ฟฝโโ๏ฟฝ = ๐
๏ฟฝโ๏ฟฝ
โ๏ฟฝโโ๏ฟฝ
๏ฟฝโ๏ฟฝ
๐ฅ
๐ผ
๐๐ฅโโโโโ
๐๐ฆโโ โโโ
๐ฆ
Components of a vector
Horizontal component: ๐๐ฅ = ๐ โ ๐๐๐ ๐ผ
Vertical component: ๐๐ฆ = ๐ โ ๐ ๐๐๐ผ
Magnitude of the resultant (Pythagorean theorem):โ๐๐ฅ2 + ๐๐ฆ
2 = ๐
Direction of the resultant: ๐๐ฆ
๐๐ฅ= ๐ก๐๐๐ผ
Adding and subtracting vectors
Graphical method (head-to-tail method)
๏ฟฝโ๏ฟฝ + ๏ฟฝโโ๏ฟฝ = ๐ (vector sum, resultant)
๏ฟฝโ๏ฟฝ + ๏ฟฝโโ๏ฟฝ = ๏ฟฝโโ๏ฟฝ + ๏ฟฝโ๏ฟฝ (commutative law)
๏ฟฝโ๏ฟฝ + ๏ฟฝโโ๏ฟฝ + ๐ = ๏ฟฝโ๏ฟฝ + (๏ฟฝโโ๏ฟฝ + ๐) (associative law)
๏ฟฝโ๏ฟฝ + โ๏ฟฝโโ๏ฟฝ = ๏ฟฝโ๏ฟฝ โ ๏ฟฝโโ๏ฟฝ (vector subtraction)
Mechanics I. 18
Analytical method (see the components of a vector section)
๐ = โ๐ ๐ฅ2 + ๐ ๐ฆ
2
๐ ๐ฆ
๐ ๐ฅ= ๐ก๐๐๐ผ
Multiplying vectors
MULTIPLYING VECTORS
Multiplying a vector by a scalar vector
๐ โ ๏ฟฝโ๏ฟฝ = ๏ฟฝโโ๏ฟฝ
magnitude:
scalar times the magnitude of the vector
direction:
same (positive scalar)
opposite (negative scalar)
force: ๏ฟฝโ๏ฟฝ = ๐ โ ๏ฟฝโ๏ฟฝ
linear momentum: ๐ = ๐ โ ๏ฟฝโ๏ฟฝ
electric force law: ๏ฟฝโ๏ฟฝ = ๐ โ ๏ฟฝโโ๏ฟฝ
Multiplying a vector by a vector โ scalar product (dot product) scalar
๏ฟฝโ๏ฟฝ โ ๏ฟฝโโ๏ฟฝ = ๐
magnitude:
๐ โ ๐ โ ๐๐๐ ๐ผ
direction:
no
work: ๐ = ๏ฟฝโ๏ฟฝ โ ๐
electric potential: ๐ = ๏ฟฝโโ๏ฟฝ โ ๐
Multiplying a vector by a vector โ vector product (cross product) vector
๏ฟฝโ๏ฟฝ ร ๏ฟฝโโ๏ฟฝ = ๐
magnitude:
๐ โ ๐ โ ๐ ๐๐๐ผ
direction:
torque: ๐ = ๐ ร ๏ฟฝโ๏ฟฝ
angular momentum: ๐ = ๐ ร ๐
magnetic force law: ๏ฟฝโ๏ฟฝ = ๐ โ ๏ฟฝโ๏ฟฝ ร ๏ฟฝโโ๏ฟฝ
Mechanics I. 19
forefinger
๏ฟฝโโ๏ฟฝ
middle finger ๐
thumb
๏ฟฝโ๏ฟฝ
right-hand rule
Right-hand rule.
Examples
Source: College Physics Page 171. Example 41.
Two muscles in the back of the leg pull upward on the Achilles tendon (the medial
and lateral heads of the gastrocnemius muscle). Find the magnitude and direction
of the total force on the Achilles tendon.
๐๐๐ 200 =๐น๐ด๐น
โ ๐น๐ด = 200 ๐ โ ๐๐๐ 200 = 188 ๐
Source: College Physics Page 342. Example 30.
A person working at a drafting board may hold her head as shown in Figure, requiring
muscle action to support the head. The three major acting forces are shown. Calculate
the direction and magnitude of the force supplied by the upper vertebrae (Fv) to hold the
head stationary, assuming that this force acts along a line through the center of mass
as do the weight and muscle force.
Vertical components: ๐น๐ฃ = 50 ๐ + 60๐ โ ๐ ๐๐330 = 82.6 ๐
Horizontal components: ๐นโ = 60๐ โ ๐๐๐ 330 = 50.32 ๐
Magnitude of the resultant: ๐น = โ๐น๐ฃ2 + ๐นโ
2 = 96.7 ๐
Direction of the resultant: ๐ก๐๐๐ผ =๐น๐ฃ
๐นโโ ๐ผ = 58.6๐
Mechanics I. 20
Source: College Physics Page 170. Example 33.
Braces are used to apply forces to teeth to realign them. The total force applied to
the tooth by the wire, points straight toward the back of the mouth. What force is
exerted on the tooth if the tension in the wire is T = 25 N? Note that the force applied
to the tooth is smaller than the tension in the wire, but this is necessitated by
practical considerations of how force can be applied in the mouth.
๐๐๐ 750 =๐น๐2๐
โ ๐น๐ = 2 โ 25 ๐ โ ๐๐๐ 750 = 2 โ 6.5 ๐
Source: College Physics Page 342. Example 28.
The upper leg muscle (quadriceps) exerts a force of 1250 N, which is carried by a
tendon over the kneecap (the patella) at the angles shown in Figure. Find the
direction and magnitude of the force exerted by the kneecap on the upper leg
bone (the femur).
Vertical components: ๐น๐ฃ = 1250 ๐ โ ๐ ๐๐550 โ 1250 ๐ โ ๐ ๐๐750 = โ183 ๐
Horizontal components: ๐นโ = 1250 ๐ โ ๐๐๐ 550 + 1250 ๐ โ ๐๐๐ 750 =
1040 ๐
Magnitude of the resultant: ๐น = โ๐น๐ฃ2 + ๐นโ
2 = 1560 ๐
Direction of the resultant: ๐ก๐๐๐ผ =๐น๐ฃ
๐นโโ ๐ผ = 9.98๐
CREDIT AND REFERENCE
https://openstax.org/details/college-physics