1) A cart with mass m0=20 kg is being pulled across a horizontal plane by a force F=20 N at an angle of α=60°. Using equations of motion, the time taken to travel a distance d=300 m is calculated as 8.66 minutes, and the velocity at this point is 17.32 m/s.
2) The normal force N acting on the cart is calculated as 182.7 N. Since N<weight, the cart is being lifted by the vertical component of the pulling force.
3) If the coefficient of friction is μ=0.5, the time and velocity are recalculated using this value in the equations of motion.
This document discusses kinematics and projectile motion in two dimensions. It provides examples of calculating the displacement, velocity, acceleration, time and maximum height of objects moving horizontally and vertically. It also examines the effects of throwing objects at different angles, such as determining where a bullet fired in the air from a moving vehicle would land.
The document discusses forces and equilibrium examples in physics. It covers textbook sections on Newton's laws of motion, free body diagrams, friction, gravity, springs, tension, and two-dimensional examples. Examples include calculating the force needed to keep a block from sliding down an incline and finding the tension in a string suspending a mass.
The document discusses forces and equilibrium examples in physics. It covers textbook sections on Newton's laws of motion, friction, gravity, springs, tension, and two-dimensional examples. Key concepts include drawing free body diagrams, analyzing forces in different directions, and using equations like the force of a spring being proportional to its displacement.
Here are the steps to solve this problem using virtual work:
1. Define the system of interest as the platform, worker and supplies with mass m.
2. The virtual displacements are δx and δy in the x and y directions.
3. The external forces are the pressures p in the cylinders.
4. Write the virtual work equation:
-pAδx - pAδy = 0
5. Note that δx and δy are not functions of θ.
6. Solve for p:
p = mg/2A
Therefore, the required pressure p is independent of θ as required to support the mass m.
1. The document provides solutions to 7 miscellaneous physics problems involving kinematics, dynamics, and rotational motion. The problems involve calculating quantities like angular speeds, forces, and times using principles like conservation of energy, angular momentum, and equations of motion. Complex algebraic and calculus solutions are shown.
2. Key steps involve setting up and solving equations derived from applying relevant physics principles to diagrams of the systems. Calculus techniques like differentiation and numerical integration are used.
3. Emphasis is placed on being able to instantly solve equations of the form f(x)=0 that arise, with references provided to resources on techniques like Newton-Raphson iteration.
This document summarizes key concepts from Physics 211 Lecture 15:
1) The lecture covered the parallel axis theorem, torque, angular acceleration, and applying concepts like torque and moment of inertia to problems involving objects rolling up and down ramps.
2) Examples were worked through to demonstrate how to calculate moment of inertia for composite objects and determine which axis would have the smallest moment of inertia.
3) Relationships between torque, force, moment of inertia, and angular acceleration were reviewed, showing similarities to linear motion concepts. Forces of different magnitudes required to produce the same angular acceleration on objects with different radii were explored.
Today's physics lecture covered torque and rotational equilibrium. Key points included definitions of torque as the rotational effect of a force and rotational inertia. Examples were worked through to demonstrate torque, equilibrium conditions requiring the sum of forces and sum of torques to equal zero, and applications to problems involving objects in static equilibrium situations. An exam review and homework problems related to these concepts were also discussed.
This document discusses kinematics and projectile motion in two dimensions. It provides examples of calculating the displacement, velocity, acceleration, time and maximum height of objects moving horizontally and vertically. It also examines the effects of throwing objects at different angles, such as determining where a bullet fired in the air from a moving vehicle would land.
The document discusses forces and equilibrium examples in physics. It covers textbook sections on Newton's laws of motion, free body diagrams, friction, gravity, springs, tension, and two-dimensional examples. Examples include calculating the force needed to keep a block from sliding down an incline and finding the tension in a string suspending a mass.
The document discusses forces and equilibrium examples in physics. It covers textbook sections on Newton's laws of motion, friction, gravity, springs, tension, and two-dimensional examples. Key concepts include drawing free body diagrams, analyzing forces in different directions, and using equations like the force of a spring being proportional to its displacement.
Here are the steps to solve this problem using virtual work:
1. Define the system of interest as the platform, worker and supplies with mass m.
2. The virtual displacements are δx and δy in the x and y directions.
3. The external forces are the pressures p in the cylinders.
4. Write the virtual work equation:
-pAδx - pAδy = 0
5. Note that δx and δy are not functions of θ.
6. Solve for p:
p = mg/2A
Therefore, the required pressure p is independent of θ as required to support the mass m.
1. The document provides solutions to 7 miscellaneous physics problems involving kinematics, dynamics, and rotational motion. The problems involve calculating quantities like angular speeds, forces, and times using principles like conservation of energy, angular momentum, and equations of motion. Complex algebraic and calculus solutions are shown.
2. Key steps involve setting up and solving equations derived from applying relevant physics principles to diagrams of the systems. Calculus techniques like differentiation and numerical integration are used.
3. Emphasis is placed on being able to instantly solve equations of the form f(x)=0 that arise, with references provided to resources on techniques like Newton-Raphson iteration.
This document summarizes key concepts from Physics 211 Lecture 15:
1) The lecture covered the parallel axis theorem, torque, angular acceleration, and applying concepts like torque and moment of inertia to problems involving objects rolling up and down ramps.
2) Examples were worked through to demonstrate how to calculate moment of inertia for composite objects and determine which axis would have the smallest moment of inertia.
3) Relationships between torque, force, moment of inertia, and angular acceleration were reviewed, showing similarities to linear motion concepts. Forces of different magnitudes required to produce the same angular acceleration on objects with different radii were explored.
Today's physics lecture covered torque and rotational equilibrium. Key points included definitions of torque as the rotational effect of a force and rotational inertia. Examples were worked through to demonstrate torque, equilibrium conditions requiring the sum of forces and sum of torques to equal zero, and applications to problems involving objects in static equilibrium situations. An exam review and homework problems related to these concepts were also discussed.
The document summarizes key points from Physics 101 Lecture 14 on torque and equilibrium:
1. The lecture covers rotational kinetic energy, rotational inertia, torque, and static equilibrium conditions.
2. Torque is defined as the rotational effect of a force and is calculated as τ = rF × sinθ. Static equilibrium requires the sum of all torques and forces to equal zero.
3. Examples are provided to demonstrate calculating torque for different situations and using equilibrium conditions to solve problems involving forces and torques.
This document summarizes key concepts from a physics lecture on kinematics, including definitions of position, displacement, velocity, acceleration, and relative velocity. Examples of graphs such as position vs. time, velocity vs. time, and acceleration vs. time are provided to illustrate these concepts. The document also provides examples of calculating relative velocity when objects are moving relative to both stationary and non-stationary reference frames.
This document summarizes key concepts from a physics lecture on kinematics, including definitions of position, displacement, velocity, acceleration, and relative velocity. Examples of graphing position vs time, velocity vs time, and acceleration vs time are provided, as well as examples of calculating relative velocity when objects are moving relative to each other in different reference frames.
This document discusses various types of trusses and methods for analyzing truss structures. It begins by describing common types of trusses used in roofs and bridges. It then covers topics such as classifying trusses as simple, compound, or complex, and determining their stability and determinacy. The document introduces analytical methods like the method of joints and method of sections for calculating member forces in statically determinate trusses. It provides examples of applying these methods to solve for unknown member forces.
Resolução.física sears zemansky 12ª edição young e freedman (todos os...ASTRIDEDECARVALHOMAG
1) The document applies Newton's laws of motion to analyze various physical situations involving forces, masses, and accelerations. It considers tension forces in ropes and chains, as well as normal and frictional forces.
2) Key steps include drawing free-body diagrams, identifying all forces, and writing the appropriate force equations. Forces are resolved into components parallel and perpendicular to surfaces.
3) Solutions involve calculating tensions, normal forces, angles, and other variables by setting the force equations equal to mass times acceleration and solving.
This document discusses determining forces in members of truss structures. It provides examples of analyzing trusses by applying the method of joints. Key steps include:
1) Analyzing force equilibrium at individual joints by setting sums of forces in x and y directions to zero.
2) Determining member forces and whether each member is in tension or compression.
3) Examples show this process applied to various truss configurations with given forces or member angles. Diagrams clearly show member forces.
This document discusses determining forces in members of truss structures. It provides examples of analyzing trusses by applying the method of joints. Key steps include:
1) Analyzing force equilibrium at individual joints by setting sums of forces in x and y directions to zero.
2) Determining member forces and whether each member is in tension or compression.
3) Examples show this process applied to various truss configurations with given force/loading conditions. Diagrams illustrate the joint labels and member forces.
This document discusses the composition of forces and moments. It defines key terms like resultant force and moment of a force. It describes the parallelogram, triangle and polygon laws for combining concurrent coplanar forces into a single resultant force. It also explains Varignon's principle of moments, which states that the algebraic sum of the moments of individual forces equals the moment of the resultant force about the same point. Several example problems are provided to illustrate how to use these principles to find the magnitude and direction of resultant forces and moments in systems of coplanar concurrent and non-concurrent forces.
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The document discusses four problems related to forced vibrations of mass-spring systems. Problem 1 derives the solution to a forced vibration differential equation and applies initial conditions. Problem 2 applies the solution from Problem 1 to different initial conditions and explores the case where the driving and natural frequencies are similar. Problem 3 derives the solution for pure resonance where the amplitude grows linearly with time. Problem 4 explores maximizing the amplification factor by taking the derivative and solving for the frequency.
1) The document discusses equilibrium of particles and rigid bodies through examples of forces and free body diagrams.
2) It provides an example of determining the tension in a supporting cable and reaction force on a pin for a jib crane. Forces and moments are summed about the pin to solve for tension, which is then used to find the horizontal and vertical reaction forces.
3) The tension in the cable is found to be 19.61 kN and the horizontal and vertical reaction forces on the pin are 17.77 kN and 6.37 kN respectively.
The document discusses torque and its relationship to force and moment arm. Torque is defined as the tendency to produce rotational motion and is calculated as the product of a force and its moment arm. Several examples are provided to illustrate calculating torque based on given forces and moment arms. The importance of moment arm in producing torque is that torque depends on both the magnitude of force and its distance from the axis of rotation.
The document summarizes key points from Physics 101 Lecture 14 on torque and equilibrium:
1. The lecture covers rotational kinetic energy, rotational inertia, torque, and static equilibrium conditions.
2. Torque is defined as the rotational effect of a force and is calculated as τ = rF × sinθ. Static equilibrium requires the sum of all torques and forces to equal zero.
3. Examples are provided to demonstrate calculating torque for different situations and using equilibrium conditions to solve problems involving forces and torques.
This document summarizes key concepts from a physics lecture on kinematics, including definitions of position, displacement, velocity, acceleration, and relative velocity. Examples of graphs such as position vs. time, velocity vs. time, and acceleration vs. time are provided to illustrate these concepts. The document also provides examples of calculating relative velocity when objects are moving relative to both stationary and non-stationary reference frames.
This document summarizes key concepts from a physics lecture on kinematics, including definitions of position, displacement, velocity, acceleration, and relative velocity. Examples of graphing position vs time, velocity vs time, and acceleration vs time are provided, as well as examples of calculating relative velocity when objects are moving relative to each other in different reference frames.
This document discusses various types of trusses and methods for analyzing truss structures. It begins by describing common types of trusses used in roofs and bridges. It then covers topics such as classifying trusses as simple, compound, or complex, and determining their stability and determinacy. The document introduces analytical methods like the method of joints and method of sections for calculating member forces in statically determinate trusses. It provides examples of applying these methods to solve for unknown member forces.
Resolução.física sears zemansky 12ª edição young e freedman (todos os...ASTRIDEDECARVALHOMAG
1) The document applies Newton's laws of motion to analyze various physical situations involving forces, masses, and accelerations. It considers tension forces in ropes and chains, as well as normal and frictional forces.
2) Key steps include drawing free-body diagrams, identifying all forces, and writing the appropriate force equations. Forces are resolved into components parallel and perpendicular to surfaces.
3) Solutions involve calculating tensions, normal forces, angles, and other variables by setting the force equations equal to mass times acceleration and solving.
This document discusses determining forces in members of truss structures. It provides examples of analyzing trusses by applying the method of joints. Key steps include:
1) Analyzing force equilibrium at individual joints by setting sums of forces in x and y directions to zero.
2) Determining member forces and whether each member is in tension or compression.
3) Examples show this process applied to various truss configurations with given forces or member angles. Diagrams clearly show member forces.
This document discusses determining forces in members of truss structures. It provides examples of analyzing trusses by applying the method of joints. Key steps include:
1) Analyzing force equilibrium at individual joints by setting sums of forces in x and y directions to zero.
2) Determining member forces and whether each member is in tension or compression.
3) Examples show this process applied to various truss configurations with given force/loading conditions. Diagrams illustrate the joint labels and member forces.
This document discusses the composition of forces and moments. It defines key terms like resultant force and moment of a force. It describes the parallelogram, triangle and polygon laws for combining concurrent coplanar forces into a single resultant force. It also explains Varignon's principle of moments, which states that the algebraic sum of the moments of individual forces equals the moment of the resultant force about the same point. Several example problems are provided to illustrate how to use these principles to find the magnitude and direction of resultant forces and moments in systems of coplanar concurrent and non-concurrent forces.
Hdhdhshsissdddd
D
D
Dd
D
D
D
C
Cg
G
G
H
H
H
Ffjsjshshyfyxyztsufifiv
Cicucuxyxhcuxyc
Cuxuxjxyxhcuxyc
Cducjxyxhxuxyc
Nxyxhxhxhcj
Mcuxh jcy
Cyxhxjxyx
Jcyxhxhxhcd
D
Sjvsvsvivskvksbosbis
Smsusuvisvisvivs
Ns vusvisvisvusvuvs
Cuxjxyxyxh
The document discusses four problems related to forced vibrations of mass-spring systems. Problem 1 derives the solution to a forced vibration differential equation and applies initial conditions. Problem 2 applies the solution from Problem 1 to different initial conditions and explores the case where the driving and natural frequencies are similar. Problem 3 derives the solution for pure resonance where the amplitude grows linearly with time. Problem 4 explores maximizing the amplification factor by taking the derivative and solving for the frequency.
1) The document discusses equilibrium of particles and rigid bodies through examples of forces and free body diagrams.
2) It provides an example of determining the tension in a supporting cable and reaction force on a pin for a jib crane. Forces and moments are summed about the pin to solve for tension, which is then used to find the horizontal and vertical reaction forces.
3) The tension in the cable is found to be 19.61 kN and the horizontal and vertical reaction forces on the pin are 17.77 kN and 6.37 kN respectively.
The document discusses torque and its relationship to force and moment arm. Torque is defined as the tendency to produce rotational motion and is calculated as the product of a force and its moment arm. Several examples are provided to illustrate calculating torque based on given forces and moment arms. The importance of moment arm in producing torque is that torque depends on both the magnitude of force and its distance from the axis of rotation.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
Beyond Degrees - Empowering the Workforce in the Context of Skills-First.pptxEduSkills OECD
Iván Bornacelly, Policy Analyst at the OECD Centre for Skills, OECD, presents at the webinar 'Tackling job market gaps with a skills-first approach' on 12 June 2024
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
2. PROBLEMĂ
COMBINATĂ –
DINAMICĂ -
CINEMATICĂ
Un cărucior de masă m=20 kg,
este tractat pe plan orizontal
cu F = 20 N, sub unghi de
α=600 . A) În cât timp
parcurge d = 300 m; b) care
este viteza în acest punct ; c)
cât este normala la plan; d)
dacă μ=0,5 care sunt valorile
cerute la punctele a) și b) ?
2
A B
v0= 0m/s v
F
600
Fx
Fy
a
d
b) v = v0 + a . t →
(ox) Fx = m . a
Fx = F . cosα a =
F . cosα
m
a) v2 = v2
0 + 2 . a . d
v2 = 2 . a . d
v2 = 2 . . d
F . cosα
m
v = √ 2 . . d
F . cosα
m
t =
v
a
v= 17,32 m/s
t =
v . m
F . cosα
t = 519,2 s = 8,66 min
x
y
G
N
3. PROBLEMĂ
COMBINATĂ –
DINAMICĂ -
CINEMATICĂ
Un cărucior de masă m=20 kg,
este tractat pe plan orizontal
cu F = 20 N, sub unghi de
α=600 . a) În cât timp
parcurge d = 300 m; b) care
este viteza în acest punct și c)
cât este normala la plan?
3
A B
v0= 0m/s v
F
600
Fx
Fy
a
d
N = m . g -F . sinα
Fy = F . sinα
G= m . g
c) (oy) Fy +N - G=0
x
y
G
N
N = 20 . 10 -20 . √3/2= 182,7 N
Concluzie : N < G, deoarece corpul este ridicat
de componenta Fy a forței de tracțiune
4. PROBLEMĂ
COMBINATĂ –
DINAMICĂ -
CINEMATICĂ
Un cărucior de masă m=20 kg,
este tractat pe plan orizontal
cu F = 20 N, sub unghi de
α=600 . A) În cât timp
parcurge d = 300 m; b) care
este viteza în acest punct ; c)
cât este normala la plan; d)
dacă μ=0,5 care sunt valorile
cerute la punctele a) și b) ?
4
A B
v0= 0m/s v
F
600
Fx
Fy
a
d
(ox) Fx – Ff = m . a
Fx = F . cosα
a) v2 = v2
0 + 2 . a . d
x
y
G
N
Ff
(oy) Fy +N - G=0
Fy = F . sinα
G= m . g
N =m . g - F . sinα
F . cosα - Ff= m .a
Ff = μ . N
Ff = μ . N
F . cosα - μ . (m . g - F . sinα)= m .a
F ( cosα - μ . sinα) –μ . m g
m
a =
5. Un corp de masă m=20 kg aflat
pe suport orizontal poate fi deplasat
fie prin tracţiune F1 , fie prin
împingere F2 sub unghi α =300 .
Să se determine care este cea
mai eficientă acţiune, pentru a se
deplasa cu aceeaşi acceleraţie.
Se cunosc:
• coeficientul de frecare la alunecare
μ = 0,2
• acceleraţia, a = 20 m/s2,
• acceleraţia gravitaţională g=10 m/s2
Aplicăm principiul II generalizat,
pentru că sunt un număr mai mare de forţe :
R m a
Pentru cazul analizat, forţele sunt:
Continuare
f
R F G N F
Prin urmare, relaţia generalizată în acest caz
este:
f
F G N F m a
5
Folosim metoda analitică de compunere a vectorilor: proiectare pe cele două axe:
6. 6
x
y
G
1
N
1
F
1
F x
1
F y
a
1
Ff
.
I
x
G
2
N
2
F
2
F x
2
F y
2
Ff
a
y
.
II
1
1 0
y
o F
y G
N
1 1
f
F N
2 2
x f
F
o m
F
x a
2
2 0
y
o F
y G
N
2 2
f
F N
Continuare
G g
m
G g
m
1 1
x f
F
o m
F
x a
7. REZOLVARE
1
1 1
1
1
1
.
x
y
f
f y
F
G
ox m
I
F
N
m
a
oy
F
F g
F
1
1
1
1
1
1
. cos
sin
y
y
x
x F
m m
I
F
F g
F
F
F a
1 cos sin g
a
m
F
1
.
cos sin
g
a
m
I F
2
2 2
2
2
2
.
x
y
y
f
f
F
G
g
ox m
II oy
m
N
F
F
F
F a
2
2 1
2
1
2
. cos
sin
y
y
x
x F
m m
II
g
F
F
F
F
a
F
2 cos sin g
a
m
F
2
.
cos sin
m g
a
I F
I
2
2 1
1
cos sin
cos sin
F
F F
F
Efort mai mare la împingere decât la tragerea corpului !
7
8. • Pe un plan înclinat de unghi , un corp lăsat
liber coboară cu accelaraţia .
• Acelaşi corp fiind aruncat de jos în sus de-a lungul
planului înclinat urca cu .
Determinaţi valoarea coeficientului de frecare dintre
corp şi planul înclinat.
0
45
c
a m / s2
1
u
a , m / s2
1 5
Rezolvare
Problema 3
8
9. f
F N
G
n
G
t
G
c
a
COBORÂRE
x
y
f c
m principi
F a
N ul
G II
0
sin
cos
f c
f
n
t
t
n
G
G
ox m
oy
metoda
G
G
analitică
m
F a
F N
G
G
G
N
g
1
sin cos
c
a g
Continuare
9
10. u
a
N
G
n
G
t
G
f
F
0
v
URCARE
x
y
f u
m principi
F a
N ul
G II
1
0
sin
cos
f u
n
f
n
t
t
G
G
ox m
oy
metoda
G
G
F a
an
G
a
N
N
litică
m
G
g
G
F
2
cos sin
u
a g
Continuare
10
11. (1) (2)
şi
u
c
i
a
D
a
g
n
( cos sin )
g
( cos sin )
(sin cos )
(sin cos )
u
c
a
a
cos sin cos sin
u u c c
a a a a
cos sin
c u u c
a a a a
sin
cos
u c
c u
a a
a a
;
1,5 1
1, 0 1,5
1
tg tg
0 0
45 45
0,5
2
1
,5
1
0,5 1
0,2
2,5 5
Enunţ
1
sin cos
c
a g
2
cos sin
u
a g
11
12. PROBLEMA TENSIUNE MECANICĂ – FRECARE –plan orizontal
• În sistemul din figura de mai jos se cunosc masele m 0 , m 1 şi m 2 şi coeficientul de frecare
dintre corpuri şi suprafaţa orizontală. Să se determine acceleraţia cu care se deplasează
sistemul şi tensiunea din firul care leagă corpurile 1 şi 2.
2
N 1
N
2
Ff
2
G
1
T
1
T
1
m 2
m
1
Ff
1
G
2
T
2
T
0
G
o
m
2
T
2
T
a
a
0 o 2 0
Corpul de masă m :(ox) G T m a
Împărţim sistemul pe subsisteme şi aplicăm metoda analitică.
Pe axa mişcării vom avea proiecţiile:
Rezolvare
Problema 2
12
1 2 1 f 1 1
Corpul de masă m :(ox) T T F m a
2 1 f 2 2
Corpul de masă m :(ox) T F m a
13. REZOLVARE
0 1 2 0 1 2
Din 1 :m g m g m g a (m m m )
f 2 2 2 2 f 2 2
F N şi pe oy N m g F m g
1 0 2
m m m
0 1 2
1 2 2
1 0 2
m (m m )
Din 3 ,( 5)şi 6 :T m g m g
m m m
0 2
1
0 1 2
m m
T 1 g
m m m
5
0 1 2
0 1 2
[m (m m )]
a g
m m m
4
Diagramă
13
f 1 1 1 1 f 1 1
Unde : F N şi pe oy N m g F m g
1 2 f 2
Dar T m a F
2 0 1 2 2 2 0 1 2
1
0 1 2
g m m g (m m ) m m g(m m m )
T
m m m
2 2
2 0 1 2 2 0 2 1 2
0 1 2
g m m g m m m m g m m g m g
m m m
14. DATELE PROBLEMEI
0
1
2
:
30 ;
0,35 ( )
0,65 ( )
v
A
A
Se dă
ct
m kg coborâre
m kg urcare
:
.
.
. ?
. ?
T
B
Se cere
a reprezentare forte
b R scripete
c
d m
14
16. CONFIGURARE FORŢE LA URCARE
A2
G
T
T
T
T
x
B
G
n
G
t
G
N
f
F
v v
y REZOLVARE
16
17. 1
1
c.la urcare:
(1)
t f
A
t A f
B G T F
A T m g
G m g F
1
1 2
2
c.la coborâre:
(2)
t f
A
t f A
B G F T
A T m g
G F m g
2 1
(2) (1) : f A A f
Din F m g m g F
DIAGRAMĂ coborâre
DIAGRAMĂ urcare
2 1
2 1
2 (3)
2
A A
f A A f
m m g
F m m g F
:
0
R=0
0
B
A
f T
T
G
G
N F
În ambele cazuri
17
18. : cos
cos (4)
f B
f B
Dar F N unde N m g
F m g
2 1
1
2 1
(3),(4) (1) :
sin : sin
2
(5)
2 sin
A A
B A
A A
B
Introducem în
m m g
m g m g g
m m
m
18
19.
2 1 2 1
2 1 2 1
2 1 2 1
(3),(5) (4):
cos :
2 2 sin 2
sin
cos
0,65 0,35 1 0,3
0,17
0,65 0,35 1 1,7
3
A A A A
A A A A
A A A A
Introducem în
m m g m m g
g
m m m m
tg
m m m m
19
2 1
. (5) :
0,65 0,35
1,0
1
2 sin 2
2
A A
B B
d Din
m m
m m kg
20. • Un corp de masă m=150 g, cuplat cu
un resort elastic, este în repaus pe
suprafaţă orizontală. Se trage pe
verticală de capătul liber al resortului
cu viteza v=2 cm/s.
a) Cât este constanta elastică, dacă se
desprinde după t1 =5 s ?
b) Cu ce forţă apasă corpul pe suprafaţa
de sprijin după t2 =3 s ?
c) Se trage pe orizontală de resort cu
aceeaşi viteză. Care este valoarea
maximă a forţei de frecare dacă după
t = 3 s corpul ia startul ?
PROBLEMA FRECARE/ELASTICITATE
F
G
F
v
0
l
f
l
a b c
20
21. REZOLVARE
0
R=0 v=ct.
F G G
F
G
F
v
dar l t
a
v
m g
k
t
3
2
150 10 10
: 15
10 5
2
N
Numeric k
m
0
unde
f
l l l
21
Problema
1
1
k l k
m g
g
l
m
G
F
v
0
l
f
l
22. REZOLVARE
0
N
R=0 N
v=ct
F G G
F
b
F
G
F
v
0
l
2
l
N
2
N l
m k
g
2 2
l v t
2
N k v
g t
m
3 2
:
150 10 10 15 10 3
2
Numeric
N
N 0 6
, N
22
Problema
23. REZOLVARE
R=0 N+ 0
v=c
f
t
F G F
c
F
G
v
N
2
15 3 0,9
:
2 10
f
Numeric
F N
f
F
0
f f
F
F
) F
(ox F
N - N =
0
(oy) G G
3
F k l k v t
3
f k v t
F
23
Problema
24. În sistemul prezentat
scripetele este ideal, corpurile 1
şi 2 au greutăţile G1 = 4N,
respectiv G2 = 8N, iar firul de
care sunt legate este inextensibil
şi foarte uşor. Coeficientul de
frecare la alunecare între oricare
două suprafeţe este μ = 0,25.
Calculează valoare forţei
necesare pentru a deplasa
corpul 2 cu vitează constantă.
F
2
G
2
N
1
G
1
N
T T
T
T
2
v
2
f
F
1
Ff
Corpul de masă 1 este deplasat sub acţiunea tensiunii şi frânat prin
interacţiune cu corpul suport conform forţei normale datorate propriei
greutăţi
1
2
Corpul de masă 2 este deplasat sub acţiunea forţei de tracţiune şi frânat
de tensiune, forţa de frecare datorată interacţiunii cu corpul superior şi
respectiv cu planul, asupra căruia acţionează cu greutatea proprie dar şi
cu greutatea corpului 1
25. 1
G
1
N
T
1
Ff
1 1
0
f f
F
T T
ox
F
1 1
1 1
f f
F G
N F
1
1 1
1
0
N N
G
y G
o
2
G
1
G
2
N
F T
2
f
F
1
Ff
1 1
2 2
0
F
f f
f f
F
o F
F
F F
T
x T
2
1 2
1
2 2
0
o N N
G G
y G
G
2 2
2 2
1
f f
N G
F F G
1 2
25
1
T G
2
1 1 G
G T
F G
1 2
3 G G
F
5
F N
Explicaţiile sunt accesabile pe buline !
Întoarcere pe săgeţile din pagina enunţului !
26. PROBLEMA
1.8.22L
Bila de masă m = 200 g
este suspendată prin
intermediul a două
resorturi de constante
elastice k1 = 98 N/m și k2
= 147 N/m. Corpul se află
în stare de echilibru atunci
când resortul 1 este alungit
cu Δl1 = 0,5 cm.
Să se calculeze
deformarea celui de-al
doilea resort .
26
G
1
e
F
2
e
F
1
k
2
k
R=0 echilibru
1
e
F
2
e
F G 0
2
e
F
1
e
F G
1
e 1 1
F k l
2 2 2
e
F k l
G m g
2
l 1 1
2
m g k l
k
2 1 cm
l
2
0,2 10 98 0,005
147
l
27. PROBLEMA
1.8.23 L
Două resoturi de lungimi l1 = 10
cm și l2 =15 cm au constantele
elastice k1 = 100 N/m, respectiv
k2 = 40N/m.
I. Se leagă în serie , se
suspendă și de capătul liber
se prinde corpul de masă m1
= 500 g.
I. Să se calculeze noile
lungimi și constanta
elastică a sistemului .
II. Cât devin lungimile
resorturilor, dacă în sistemul
din fig. II, corpurile au
masele m1 = 300 g și m2=
200g , la starea de echilibru ?
27
1
k
2
k
1
k
2
k
2
e
F
G
1
e
F
I
ks l m g
𝛥l =Δl1 +Δl2
𝑘s
. (Δl1 +Δl2) = m . g
1
e
F m g
2
e 2 2
F k l
m g m g
1 1
k m
l g
1 2
1
ks
m g m
m g
k k
g
m g
2
1 2
1 100 40 200
k
100 40 7
28 7
s ,
k
k
k
k
N /m
28. PROBLEMA
1.8.23 L
Două resoturi de lungimi l1 = 10
cm și l2 =15 cm au constantele
elastice k1 = 100 N/m, respectiv
k2 = 40N/m.
I. Se leagă în serie , se
suspendă și de capătul liber
se prinde corpul de masă m1
= 500 g.
I. Să se calculeze noile
lungimi și constanta
elastică a sistemului .
II. Cât devin lungimile
resorturilor, dacă în sistemul
din fig. II, corpurile au
masele m1 = 300 g și m2=
200g , la starea de echilibru ?
28
1
k
2
k
1
k
2
k
2
e
F
G
1
e
F
I
,
1 1 1
,
1
1 1
1
m g
m
k l l
l
l
k
k
g
15cm
1 2 12 1
5 7 5
5
l
l , ,
l cm
,
2 2 2
, 2 2
2
2
m g
m
k l l
k l
l 2
k
g
7,5cm
1 15 10
l 5cm
2
l 27,5 15 12,5cm
200
0 17 5
5
7
s N greu
k l , tatea
29. PROBLEMA
1.8.23 L
Două resoturi de lungimi l1 = 10
cm și l2 =15 cm au constantele
elastice k1 = 100 N/m, respectiv
k2 = 40N/m.
I. Se leagă în serie , se
suspendă și de capătul liber
se prinde corpul de masă m1
= 500 g.
I. Să se calculeze noile
lungimi ?
II. Cât devin lungimile
resorturilor, dacă în sistemul
din fig. II, corpurile au
masele m1 = 300 g și m2=
200g , la starea de echilibru ?
29
1
k
2
k
II
1
k
2
k
1
G
1
e
F
2
e
F
2
G
Pentru corpul 1:
2
"
e 2 2 2
F k l l
m g m g
" 2 2
2
2
1 k l
l 22
m g
,5cm
k
1
e
"
1 1 2
1
1
1 2
F
k l m
l
g
m
m
m
g
1
" 2
1 1
1
1
m g k l
l
k
m
"
1
l 15cm