Mechanics
Sometimes referred to as classical mechanics or Newtonian mechanics
is concerned with the effects of forces on material objects. A major development in the theory was provided by Isac Newton in 1687
The part of mechanics that describes motion without regard to its causes is called kinematics.
Here we will focus on one dimensional motion.
Position
To describe the motion of an object, one must be able to specify its position
atall time using some convinient coordinate system.For example, a particle might be located at x=+5 m, which means that it is 5 min the positive direction from the origin. If it were at x=-5 m, it would be just asfar from the origin but in the opposite direction.
Displacement
A change from one position to another position is called displacement
For examples, if the particles moves from +5 m to +12 m, then
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the + sign indicates that the motion is in the positive direction. The displacement is a vector quantity.
Average velocity
Which is the ratio of thedisplacement that occurs during aparticular time interval to that interval.A common unit of velocity is the meter persecond (m/s):
Average speed
Is a different way of describing “howfast” a particle moves, and involvesthe total distance covered independentof direction(m/s):
Average acceleration
When a particle’s velocity changes, the particle is said to accelerate. For motionalong an axis, the average acceleration over a time interval is
Acceleration is vector quantity.The common unit of acceleration is the meter per second per second (m/s²).
In many types of motion, the acceleration is either constant or approximately so.In that case the instantaneous acceleration and average acceleration are equal
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Also, let =(the initial velocity) and (the velocity at arbitrary time) with this notation we have
=
In a similar manner we can have
whereis the position of the particle at initial time.
Finally, from this two equations we can obtain expression that does not contain
Time
and if = 0 then
These equations may be used to solve any problem in one-dimensional motion
with constant acceleration.
All objects dropped near the surface of theearth in the absence of air resistance falltoward the Earth with the same nearlyconstant acceleration.
We denote the magnitude of free-fallacceleration as g. The magnitude of free-fall acceleration decreases with increasing altitude. Furthermore, slight variations occur with latitude. At thesurface of the Earth the magnitude isapproximately
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9.8 m/s². The vector is directeddownward toward the center of the Earth. Free-fall acceleration is an important exampleof straight-line motion with constantacceleration.When air resistance is negligible, even afeather and an apple fall with the sameacceleration, regardless of their masses.
Example:
= = 7 + 0.8x2 =8.6 m/s
Example:
d = 220 m, = 10.97 km/sec = 10.97 x103 m/sec, a ?
= 2 d ( = 0 , = 0 )
( g = 9.8 m/s2 )
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Exercises:
The Concept of Force
Classical mechanics describes the relationship between the motion of an object andthe force acting on it.There are conditions under which classical mechanics does not apply. Most oftenthese conditions are encountered when dealing with objects whose size iscomparable to that of atoms or smaller and/or which move at speed of light.If the speeds of the interacting objects are very large, we must replace Classical mechanicswith Einstein’s special theory of relativity, which hold at any speed, including those nearthe speed of light.
If the interacting bodies are on the scale of atomic structure, we must replace Classicalmechanics with quantum mechanics.Classical mechanics is a special case of these two more comprehensive theories. But, still itis a very important special case because it applies to the motion of objects ranging in sizefrom the very small to astronomical.
The concept of force: Force is an interaction that can cause deformation and/or change of motion of an object and it is a vector quantity.
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Fundamental forces are:
gravitational
electromagnetic
weak nuclear ( between leptons )
strong nuclear ( repulsion and attraction )
Newton’s second law:
“The net force on a body is equal to the product of the body’s mass and the
acceleration of the body.” In equation form:
represents the vector sum of all external forces acting on theobject, m is its mass, and is the acceleration of the object.
The SI unit of force is the newton (1 N = 1 kgm/s²).
The magnitude of the gravitational force is called the weight of the object
where g is the magnitude of the free-fall acceleration.
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There many forces:
1 - The Normal Force is perpendicular to the surface
2 - The Frictional Force
3 - The Tension Force FT
4 - The Spring Force F = - kx ( Hooks law ) The minus sign indicates that the spring force is always opposite in directionfrom the displacement of the free end.The constant k is called the spring constant
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Example:
Exercises:
An object has only one force acting on it. Can it be at rest? Can it have an
acceleration?
If a single force acts on it, the object must accelerate. If an object accelerates,
at least one force must act on it.
An object has zero acceleration. Does this mean that no forces act on it?
If an object has no acceleration, you cannot conclude that no forces act on it.
In this case, you can only say that the net force on the object is zero.
Is it possible for an object to move if no net force acts on it?
Motion can occur in the absence of a net force. Newton’s first law holds that an
object will continue to move with a constant speed and in a straight line if
there is no net force acting on it.
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What force causes an automobile to move?
The force causing an automobile to move is the force of friction between the
tires and the roadway as the automobile attempts to push the roadway
backward.
The force of the wind on the sails of a sailboat is 390 N north. The water exerts
force of 180 N east. If the boat has a mass of 270 kg, what are the magnitude
and direction of its acceleration?
Work
?A force does no work on a object if the object does not move.
?The sign of the work depends on the angle θ between the force and
displacement.
? Work is a scalar quantity, and its units is joule (1 J = 1 Nm)
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Kinetic energy:
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Example:
Linear momentum: The linear momentum of an object of mass m moving
with a velocity is defined as the product of the mass
and velocity:
Momentum is a vector quantity, with its direction matching that of the velocity.
Newton didn’t write the second law as but as:
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Where is the time interval during which the momentum change . This expression is equivalent to F = ma for an object of constant mass.
Example:
Centripetal force:Considera ball of mass m tied to a string of length r and being whirled in a horizontal circular path. Let us assume that the ball move with constant speed. Because the velocity changes its direction continuously
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during the motion, the ball experiences a centripetal acceleration directed toward the center of motion, with magnitude:
The strings exerts a force on the ball that makes a circular path. This force is directed a long the length of the string toward the center of the circle with magnitude of:
This force is called centripetal force. It can be a frictional force, a gravitational force or any other force.
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Problem 1
A car travels up a hill at a constant speed of 37 km/h and returns down the hill at a constant speed of 66 km/h. Calculate the average speed for the whole trip.
Solution:
By definition the average speed is the ration of the total traveled distance and the total traveled time. Let us introduce the total traveled distance of the car as L. Then the time of the travel up the hill is
The time of the travel down the hill is
The total traveled time is
Then the average velocity is
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Problem 2
A car is accelerating at. Find its acceleration in.
Solution:
To find an acceleration in we need to use the relations:
,
Then we can write:
Problem 3
We drive a distance of 1 km at 16 km/h. Then we drive an additional distance of 1 km at 32 km/h. What is our average speed?
Solution:
By definition the average speed is the ratio of traveled distance and traveled time.
The traveled distance is 2 km.The traveled time is the sum of two contributions:
time of the motion a distance 1 km with speed 16 km/h. It is
time of the motion a distance 1 km with speed 32 km/h. It is
Then the average speed is
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Problem 4
A speedboat increases its speed uniformly from 20 m/s to 30 m/s in a distance of 200m. Find
(a) the magnitude of its acceleration and
(b) the time it takes the boat to travel the 200-m distance.
Solution:
(a) This is the motion with constant acceleration. We can use the following equation to find the magnitude of acceleration
Where
, , . Then
(b) We can find the time of the travel from the following equation:
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Problem 5
A car is moving with a velocity of 72 km/h. It's velocity is reduced to 36 km/h after covering a distance of 200 m. Calculate its acceleration.
Solution:
The first step is to check that we have correct units for all variables. The velocity should be measured in m/s and the distance in meters. Then initial and final speeds are
The covered distance is
.
To solve this problem we need to use the following equation, which relates initial speed, final speed, acceleration, and travelled distance
From this equation we can find acceleration
The acceleration is negative, which means that the direction of acceleration is opposite to the direction of velocity – the car is slowing down.
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Problem 6
A particle moves in a circle of radius 1 m. Its linear speed is given by, where t is in second and v in meter/second. Find the radial and tangential acceleration at
Solution:
The tangential acceleration is defined as the change of the speed (magnitude of the velocity) of the particle. Therefore the tangential acceleration is
The radial acceleration is centripetal acceleration, which changes the direction of velocity. The centripetal acceleration is determined by the speed (not the change of the speed) and the radius of circular orbit:
The velocity at
is , then
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Problem 7
The force required to stretch Hooke's Law spring varies from0 Nto65 Nas we stretch the spring by moving one end6.3 cmfrom its unstressed position. Find the force constant of the spring. Answer in units ofN/m.
Solution:
We understand from the formulation of the problem that the spring force is 65 N when the spring is stretched by
, which in SI units is
.
At the same time from the Hooke's law we know that the spring force is
Where
k :is a force constant (force constant) of the spring. We substitute the known values on the above expression and obtain
Then
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Problem 8
The work done by the heart is 1 joule per beat. Calculate the power of the heart if it beats 72 times in one minute.
Solution:
In this physics problem we need to use definition of the power. The power is the work done by unit time – per one second.
We know that the heart does 1 joule of work per beat and there are 72 beats per minute.
It means that the heart does work of
per minute – per 60 seconds.
Then we can easily find the work done per one second, which is power. We just need to divide the total work by the time:
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