Equation of equilibrium
When a body is in equilibrium, resultant of all forces and moments acting on that body is zero.
Stated in another way, a body is in equilibrium if all forces and moments applied to it are in balance. These requirements are contained in the vector equations of equilibrium which in two dimensions may be written in scalar from as
∑FX = 0, ∑FY = 0, ∑MO =0
The third one represents the zero sums of the moments of all forces about any point O on or off the body.
Let's look at a truss
P Q S
A B
C D
FBD
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ΣMB Does not provide any new info. This is not an independent equation.
We can use ΣMB =0to replace one of the above 3.
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A simple free body diagram, shown above, of a block on a ramp illustrates this.
• All external supports and structures have been replaced by the forces they generate. These include:
• Mg: the product of the mass of the block and the constant of gravitation acceleration: its weight.
• N: the normal force of the ramp.
• Ff: the friction force of the ramp.
• The force vectors show direction and point of application and are labeled with their magnitude.
• It contains a coordinate system that can be used when describing the vectors.
Example 3. Two loads 400N and 500N are suspended in a vertical plane by three springs as shown in Figure. Find the tension in the strings
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Problem. 1 An electric light fixture weighing 15 Newton hangs from a point C, by two strings AC
and BC. AC is inclined at 60° to the horizontal and BC at 45° to the vertical as shown in Fig. /Prob.1.Using Lami’s theorem or otherwise determine the forces in the strings AC and BC.
400N
48 15N
Example2 Given: The loading car weight is 5500 lbs and its CG is at point G.
Find: tension in cable and reactions at wheels.
Multiple Choice Questions
1. Free body diagram can be applied only in a) Dynamic equilibrium problem
b) Static equilibrium problem
c) Both static & dynamic equilibrium problems d) None of these
2. If the body is in equilibrium ,we may conclude that a) No force acting in it
b) Moment of all forces about any point is zero c) The resultant of all forces acting on it is zero d) Both b&c
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3. The algebraic sum of the moments of two forces about any point in their plane is equal to the moment of their resultant about that point is known as
a) Principle of moments b) Varignon’s theorem c) Lamis theorem d) None of these
4. Two coplanar couples having equal and opposite moments a) Produce a couple and an unbalanced force
b) Are equivalent c) Balance each other d) None of these
5. A free body diagram of a body represents
a) With its surroundings and external forces acting on it
b) Isolated from its surroundings and all external forces acting on it c) Isolated from all external actions
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Lecture 14 : Friction
Friction is the force distribution at the surface of contact between two bodies that prevents or impedes sliding motion of one body relative to the other. This force distribution is tangent to the contact surface and has, for the body under consideration, a direction at every point in the contact surface that is in opposition to the possible or existing slipping motion of the body at that point.
Types of friction
• Dry friction resists relative lateral motion of two solid surfaces in contact. Dry friction is also subdivided into static friction between non-moving surfaces, and kinetic friction (sometimes called sliding friction or dynamic friction) between moving surfaces.
• Lubricated friction or fluid friction resists relative lateral motion of two solid surfaces separated by a layer of gas or liquid.
• Fluid friction is also used to describe the friction between layers within a fluid that are moving relative to each other.
• Skin friction is a component of drag, the force resisting the motion of a solid body through a fluid.
• Internal friction is the force resisting motion between the elements making up a solid material while it undergoes deformation.
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Consider a solid block of mass m resting on a horizontal surface as shown. Assume that the contacting surfaces are rough. As we gradually increase the load, the block remains static till the load reaches a threshold value. Since the force in the x-direction has to be balanced, it is apparent that as the magnitude of F increases from zero, the friction force also increases.
Friction force, is thus, self adjusting. However, the friction force cannot increase beyond a limit.
Thus there is a limiting value of friction. The maximum value of friction force, which comes into play, when the motion is impending, is known as limiting friction. When the applied force is less than the limiting friction, the body remains at rest and such frictional force is called static friction, which may have any value between zero and the limiting friction. If the value of the applied force exceeds the limiting friction, the body starts moving over the other body and the frictional resistance experienced by the body while moving is known as Dynamic friction. Dynamic friction is found to be less than limiting friction. See the following animation to understand the phenomenon of dry friction.
It is experimentally found that the magnitude of limiting friction bears a constant ratio to the normal relation between the two surfaces and this ratio is called coefficient of Friction.
Coefficient of friction = = µ
Where F is limiting friction and N is the normal reaction between the contact surfaces.
52 Columb Laws of friction:
(1) The force of friction always acts in a direction opposite to that in which the body tends to move.
(2) Till the limiting value is reached, the magnitude of friction is exactly equal to the force which tends to move the body.
(3) The magnitude of the limiting friction bears a constant ratio to the normal reaction between the two surfaces.
(4) The force of friction depends upon the roughness/smoothness of the surfaces.
(5) The force of friction is independent of the area of contact between the two surfaces.
Angle of static friction:
Consider the block on the following surface.
The free body diagram is shown. The direction of resultant R measured from the direction of N is specified by tan a=F/N. When the friction force reaches its limiting static value Fmax, the angle a reaches a maximum.
53 Value of fs. Thus
tan fs = ms
The angle fs is called the angle of static friction.
Angle of kinetic friction:
When slippage is occurring, the angle a has a value fR corresponding to the kinetic friction force.
tan fR = mR
Cone of friction:
When a body is having impending motion in the direction of P the frictional force will be the limiting friction and the resultant reaction R will make limiting friction angle a with the normal as shown in the following figure. If the body is having impending motion in some other direction, again the resultant reaction makes limiting frictional angle a with the normal in that direction.
Angle of Repose:
The maximum inclination of the plane on which a body, free from external forces, experiences repose (sleep) is called Angle of Repose.
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Now consider the equilibrium of the block shown above. Since the surface of contact is not smooth, not only normal reaction, but frictional force also develops. Since the body tends to slide downward, the frictional resistance will be up the plane.
∑forces normal to the plane =0, gives
N=Wcos θ …………1
∑forces normal to the plane =0, gives
F=Wsin θ ………2
Dividing equ (2) by equ (1), we get
If N is the value of normal force when motion is impending, frictional force will be µN and hence
Hence, to avoid free sliding, the inclination angle should be less than the friction angle.
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