Notes, solutions, and commentary on Theoretical Mechanics A Short Course by Semen Targ
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1.1 The Subject of Statics
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Basic Concepts : :
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Statics is a branch of mechanics which studies the laws of compositions of forces and the conditions of equilibrium of material bodies under the action of forces.
Equilibrium is the state of rest of a body relative to other material bodies.
If the frame of reference a body is sitting in is fixed, the body is in absolute equilibrium.
If the frame of reference a body is sitting in is not fixed, the body is in relative equilibrium.
Statics is the study of bodies in absolute equilibrium, that is bodies at rest in a fixed reference frame.
Conditions of equilibrium depend on the state of matter (solid, liquid, gaseous).
General mechanics deals with the study of equilibrium of solids.
All bodies change their shape when subjected to external forces, this is called deformation.
The amount of deformation of a body depends on the material, shape, dimensions, and forces acting on the body.
Under specific conditions we can treat bodies as absolutely rigid (cannot be deformed).
An absolutely rigid body is one whose particles are all at a constant distance from one another.
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Primary Problems of Statics : :
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(1) The composition of forces and reduction of force systems acting on rigid bodies to as simple form as possible.
(2) The determination of the conditions for the equilibrium of force systems acting on rigid bodies.
The problems of statics can be solved via geometrical constructions or by mathematical calculus.
1.2 Force
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Forces and Equilibrium : :
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The state of equilibrium or motion of a body depends on its mechanical interactions with other bodies.
In mechanics the quantitative measure of the mechanical interactions of material bodies is called force.
Quantities in mechanics are either scalar or vectors.
Force is a vector quantity because its action on a body is characterized by its magnitude, direction, and point of application.
The magnitude of a force is expressed in terms of a physical unit, typically SI.
Fundamental units of forces are the Newton (N) and the Kilogram Force (kgf).
The direction and point of application of a force on a body depends on the nature of the interaction between bodies.
Forces graphically are represented by your typical graphical representation of a vector.
The length of a vector AB is given by |AB| and this is the magnitude of the force given in some standard unit (SI).
The direction of the line, often given measure of degrees, radians, or orientation, is the direction of the force.
Any set of forces acting on a rigid body is known as a force system.
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Important definitions : :
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(1) A body not connected with other bodies and which from any given position can be displaced in any direction in space is called a free body.
(2) If a force system acting on a free rigid body can be replaced by another force system without disturbing the body's initial condition of rest or notion, the two systems are called equivalent.
(3) If a free rigid body can remain at rest under the action of a force system, that system is said to be balanced or equivalent to zero.
(4) If a given force system is equivalent to a single force, that force is the resultant of the system. Thus, the resultant is a single force capable of replacing the action of a system of forces on a rigid body.
(5) Forces acting on a rigid body can be divided into two groups: the external forces and the internal forces. External forces represent the action of other material bodies on the particles of a given body. Internal forces are those with which the particles of a given body act on each other.
(6) A force applied to one point of a body is called a concentrated force. Forces acting on all points of a given volume or given area of a body are called distributed forces.
last edit on
5/29/2022 1:44:59 AM
1.3 Fundamental Principals
All theorems and equations in statics are deduced from a few fundamental principals.
These principals are treated as axioms, as such they are not proven.
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1st Principal : :
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A free rigid body subjected to the action of two forces can be in equilibrium if , and only if, the two forces are equal in magnitude (F₁ = F₂), collinear, and opposite in direction.
A body is in equilibrium if it is at rest in a force system the forces must be balanced so that they equate to zero. The principal is stating the conditions necessary for forces to be balanced in order for them to be equivalent to zero.
Two forces must be equal because F₁ - F₂ = 0, only forces of equivalent magnitude can cancel each other out.
Forces must be collinear, meaning they lay on the same straight line, because if they were not they may be applied to different points which would make them unbalanced in their application despite having the same magnitudes.
Forces must be opposite directions, all forces are vectors which have direction. If two forces had equal magnitude and where collinear but being applied to an object in the same direction they would not be balanced. Only when opposite directions can two forces cancel each other out.
There are the minimum conditions so that the simplest force system is balanced, a system of greater requirements may require additional conditions.
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2nd Principal : :
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The action of a given force system on a rigid body remains unchanged if another balanced force system is added to, or subtracted from, the original system.
If additional forces are added to a force system on a rigid body but all conditions of 1st principal are satisfied then the rigid body will still be at rest and in equilibrium.
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Corollary of the 1st and 2nd Principal : :
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The point of application of a force acting on a rigid body can be transferred to any other point on the line of action of the force without altering its effect.
Consider a rigid body with a force F applied at a point A. Now take an arbitrary point B on the line of action of the force and apply to that point two equal and opposite forces F₁ and F₂ such that F₁ = F and F₂ = -F.
This does not affect the action of F on the rigid body by the 1st principal it follows that forces F and F₂ also form a balanced system and therefore cancel each other out.
The vector representing forces F can applied to any point along the line of action, that is a long the line all other forces are defined on that make them all collinear.
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3rd Principle (Parallelogram Law) : :
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Two forces applied at one point of a body have as their resultant a force applied at the same point and represented by the diagonal of a parallelogram constructed with two given forces as its sides.
Consider a vector R which is the diagonal of a parallelogram with vectors F₁ and F₂ as its sides. The vector R is called a geometrical sum of the vectors F₁ and F₂ if R = F₁ + F₂.
We see that two forces, F₁ and F₂, when applied to a single point sum to another vector R that too is applied to that same point.
The upshot is that forces can be treated individually or as their geometrical sum, the geometrical sum R contains information about vectors F₁ and F₂ and reveals how those vectors effect a rigid body overall.
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4th Principal : :
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To any action of on material body on another there is always an equal and oppositely directed reaction.
If a body A acts on a body B with a force F, body B simultaneously acts on body A with a force F' equal in magnitude, collinear, with opposite direction.
Forces F and F' do not form a balanced system because they are applied to different bodies.
Every body has internal forces, the 4th principal shows that all particles that make up a body must be interacting with all other particles in such a way that the system, which is the rigid body, is in equilibrium and at rest.
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5th Principle (Principle of Solidification) : :
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If a freely deformable body subjected to the action of a force system is in equilibrium, the state of equilibrium will not be disturbed if the body solidifies (becomes rigid).
This principal in practice allows one to treat a none rigid body as a rigid body.
If a deformable body is in equilibrium, the forces acting on it satisfy the conditions for the equilibrium of a rigid body.
Given the principle of solidification, which only holds on a nonrigid body when the conditions of the 1st principal are satisfied, we can apply to methods of rigid-body statics to flexible bodies.
i.e. We’re sticking to Newton (nothing about relativity, etc.).
That is correct, Targ's text is purely about what is typically called 'General Mechanics'. What differentiates his approach from others in the general mechanics book canon is how thorough his treatment of statics is. Theoretical Mechanics seems to be one of the great forgotten gems of the Soviet academic regime.
Physics for Mathematicians by Spivak does feature a section on the Schrodinger Wave Function while covering Hamilton-Jacobi Theory.
1.4 Constraints and Their Reactions
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Constraints : :
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A body whose displacement is restricted by other bodies is called a Constrained body.
A constraint is anything that restricts the displacement of a given body in space.
A body whose displacement is restricted by a constraint acts on that constraint with a force.
By the 4th principal a constraint reacts with a force of the same magnitude and opposite sense.
The force with which a constraint acts on a body, thereby restricting its displacement, is called the force of reaction of the constraint.
We call forces that are not reactions of constraints Applied or Active forces.
The magnitude and direction of an active force does not depend on the other forces acting on a body.
If there are no active forces acting on a body then forces of constraints vanish.
A common problem in general mechanics is computing the direction of forces of constraint.
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Smooth Surfaces : :
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A smooth surface is one whose friction can be neglected in first approximation.
The reaction N of a smooth surface is directed normal to both contacting surfaces at their point of contact and is app
lied at that point.
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String : :
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A constraint provided by a flexible inextensible string prevents a body M from receding from the point of suspension of the string in the direction AM.
The reaction T of the string is thus directed along the string towards the point of suspension.
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Cylindrical Pin (Bearing) : :
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When two bodies are joined by a pin passing through holds in them, the connection is called a pin joint or hinge.
The axial line of the pin is called the axis of the joint.
The reaction R of a pin can have any direction in the plane perpendicular to the axis of the joint.
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Ball-and-Socket Joint : :
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Prevents displacement in any direction.
The reaction R of a ball-and-socket joint can have any direction in space.
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Rod : :
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Let a rod AB be secured and be the constraint of a certain structure.
The wight of the rod compared to the load it carries may be neglected.
The only two forces applied are in equilibrium, the forces, according to the 1st principal, must be col-linear and directed along the axis of the rod.
In the case where forces are applied to its tips the reaction N will be directed along the rods axis.
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