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Mechanics is the science of describing the behavior of objects under the action of forces. By using mathematical operations scientists and engineers are able to predict the conditions of rest or motion of bodies.
Mechanics covers three main subjects:
- Mechanics of rigid bodies
- Mechanics of deformable bodies
- Mechanics of fluids
The first topic - mechanics of rigid bodies - is subdivided into Statics and Dynamics. Statics deals with bodies at rest and dynamics deals with bodies in motion. In this WIKI sample application, we will talk about STATICS.
A force acting on a body has a magnitude and a direction. Vectors describe the direction of a force along a two or three dimensional axis. If more than one force act on a body, the resultant direction of the force is obtained through vector addition.
For instance, consider two vectors a and b. The resultant vector r is obtained by the following operation: a = <3,0,-9>
b = <-2,-5,10>
a + b = r = <3+(-2),1+(-5),-9+10> = <1,-4,1>
The resultant vector will now have the coordinates <1,-4,1> in a three dimensional Cartesian coordinate system.
Forces are quantitative descrpitions of interactions of bodies that either push each other away or pull closer toghether. The magnitude of pulling or pushing is expresseed in terms of a quantity know as Newtons abbreviated as N.
According to Isaac Newton, there are three fundamental laws regarding forces:
- At equilibrium the sum of all forces equal to zero: ΣF = 0
- An external force acting on a body will accelerate the body at the same direction(vector) of the force. Thus, the sum of all forces is the product of the mass and the acceleration: ΣF = ma.
- If a body a excerts a force on body b, then body b will excert the same amount of force on body a: Fa = Fb.
The moments - also called torque - is a description of the tendency of a body to rotate about an axis when a force is applied. If the object is at equilibrium meaning that it is not rotating, the moment about any point on the object is zero.
The moment quantity is obtained by multiplying the force with the respective lever arm. The lever arm is the distance between the perpendicular applied force on the body and its axis. If the distance is zero, obviously, the moment is also zero. What if the force is not perpendicular? In this situation, the angle between the force and the member needs to be taken under consideration. The moment is obtained through the following mathematical operation:
where r is the distance between the applied force F and the axis, and θ is the angle formed between the member and the applied force. Note that when θ = 90º, sin90 º = 1 and thus, keeping the equation valid. If the force is parallel with the member, θ will be zero with sin0 º = 0 and therefore, the moment will be zero.
The unit of moment is in general expressed in Newton-meter (Nm). It can also be expressed as lb-ft, lb-in, etc. In case when more than one force is being applied to a member, the overall moment is the sum of all individual moments caused by each individual force. In equilibrium conditions, the sum of all moments equals to zero:
Knowing this, it will facilitate the computation in finding other unknown forces acting on the body.
Equilibrium is a condition of a body at rest. When a body is in equilibrium, either there are no forces acting on it or the sum of all forces and moments equal to zero.
ΣM = 0
For example, when placing a book on a table, the book will not fall on the floor but rather stay in equilibrium. Why? In this situation there are two different forces. The book is pressing vertically down on the table and the table is also pushing up on the book. Since both forces are the same in magnitude and in opposite direction, the forces acting on each other cancel out. That is why the sum of forces equal to zero, thus keeping the book in equilibrium.
The mass of an object can be imagined as many points spread throughout the entire surface area or volume. The gravitational pull a body experiences by the force of gravity, makes a body accelerate downwards. The object’s weight therefore is distributed over the entire volume of the body. To simplify calculations such as computing moments and forces, the body’s weight can also be imagined as acting on a single individual point – the center of its mass. It is as if the entire mass is concentrated at this infinitely small point. Moreover, at the point the body is at equilibrium.
The mathematical operation to obtain the centroid is expressed as
Moment of Inertia Edit
The quantity of moment of inertia is a very important subject in engineering when calculating the deflection of beams and analyzing distributed forces. In simple words, the moment of inertia is the resistance of a body to rotate, move, or deflect. The greater the moment of inertia, the greater is the resistance. This quantity can be either described in terms of area or in terms of mass. In either case, the rotation of a body occurs about the x-axis or the y-axis. In some instance the rotation can also occur about some angle between the x- and y-axis.
The moment of inertia about the x-axis is mathematically expressed as
The moment of inertia about the y-axis is expressed as
When the body rotates about the x- and y-axis at the same time, that is 45º from the x-axis, this quantity is expressed as the product of inertia
If the object rotates about the origin in a circular motion, the quantity is expressed as the polar moment of inertia.
The distances of x,y, and r are taken from the applied force to the rotational axis. This is when the concept of centroids has to be considered. Instead of computing each individual point of the area, the integral is the sum of all points within a boundary.
When sliding an abject on a surface, frictional forces need to be overcome. What is exactly friction? When an object is placed on a surface, electromagnetic forces are being formed between the object and the surface. These atomic bonds need to be detached for every instant during motion. Therefore, friction always acts in the opposite direction of the applied force and is dependent on two quantities. First, the magnitude of friction depends on the weight of the object. The heavier the object the stronger the frictional force. Second, the magnitude of friction depends on the properties of the surface. It is easier to slide ice on ice than on a piece of rug.
The magnitude of friction is expressed as
where f is the frictional force, μ is the coefficient of friction, and N is the weight of the object. Since the frictional force is different for bodies at rest and for bodies at motion, there are two different denotations for μ.
- μs is called coefficient of static friction and is used for bodies with an initial velocity of zero.
- μk is called coefficient of kinetic friction and is used for bodies with an initial velocity of greater than zero. Since the body is already moving, it easier to overcome the frictional force.
Metal on metal: μs = 0.15-0.2
Masonry on masonry: μs = 0.60-0.70
Wood on wood: μs = 0.25-0.5
Metal on masonry: μs = 0.30-0.70
Metal on wood: μs = 0.20-0.60
Rubber on concrete: μs = 0.50-0.90