Equilibrium of Coplanar- Concurrent force systems

Equilibrium of two force body Equilibrium law: Two forces can be in equilibrium only if they are equal in magnitude, opposite in direction and collinear in action. P

Equilibrium of three force body Lami’s theorem If three concurrent forces are acting on a body kept in equilibrium, then each force is proportional to the sine of angle between the other two forces and the constant of proportionality is same.

Conditions of Equilibrium Coplanar Concurrent force system Coplanar Non-Concurrent force system Fx = 0 Fy = 0 Fx = 0 Fy = 0 M = 0

LAW OF SUPERPOSITION The law of superposition states that Rest P P Q Q R Rest Rest P P R Q Q The law of superposition states that “the state of rest of the body does not change even if we add to or subtract a system of equilibrium forces from any system in equilibrium”

Law of transmissibility P P A A A P P B B B P Law of transmissibility states that “the state of rest of the body does not change even if the force is transferred from one point to another point along its line of action”

Free Body Diagram

Free Body Diagram

Free Body Diagram

Free body diagram Free body diagram (FBD) is a sketch of the isolated body, which shows the external forces on the body and the reactions exerted on it by the removed elements.

FBD-Examples

FBD-Examples

FBD-Examples

FBD-Examples

Reactions at Various Types of Supports and Connections 1. Rollers

Reactions at Various Types of Supports and Connections 1. Rollers  F.B.D.  Rc  Frictionless Surface Single reaction force at 90 0 to supporting surface

2. Hinged Joints

2. Hinged Joints F.B.D. or  Fx  R Fy  Rough surface Two components resolved into x and y axis or a single reaction of unknown direction Frictionless pin or hinged joint Rough surface 

3. Cable & Short Link F.B.D.  T  Short Rope/ Short Link Cable Single force following direction of rope or link  F.B.D.  Short Rope/ Cable Short Link

4. Fixed Support

3. Block P = 5kg and block Q of mass m kg is suspended through the chord is in the equilibrium position as shown in figure. Determine the mass of block Q. Q= 5.67 kg

4. A smooth sphere of mass 2kg is supported by a chain as shown in fig 4. A smooth sphere of mass 2kg is supported by a chain as shown in fig. The length of chain AB is equal to the radius of the sphere. Draw free body diagram of each element and find the tension in the chain and reaction of the wall. T(AB) or S(AB)= 22.65 N Rc = 11.32 N

5. A sphere of weight 1000 N rests on a angled trough, as shown in Fig 5. A sphere of weight 1000 N rests on a angled trough, as shown in Fig. Determine the reactions exerted on the sides of the trough at A and B if all surfaces are perfectly smooth. Ra= 732 N Rb = 896.57 N

6. A roller of weight W=4450 N rests on a smooth inclined plane and is kept from rolling down by a string as shown in the figure. Find the tension S in the string and the reaction Rb at the point of contact B. W A 15° C B 45°

7. A system of connected flexible cables shown in fig 7. A system of connected flexible cables shown in fig. is supporting two vertical forces 200N and 250N at points B and D. Determine the forces in various segments of the cable. T(DE) = 224.14 N, T(BD)= 183.01 N T(BC)= 336.6 N, T(AB)= 326.79 N

8. The top end of vertical boot is connected by two cables having tension T1=500N and T2=1500N as shown in fig. The third cable AB is used as a guy wire. Determine the tension in cable AB if resultant of the three concurrent forces acting at A is vertical. Also find the resultant. R= 2028.37 N (DOWNWARD)

9. A pulley A is supported by two bars AB and AC which are hinged at points B &C to a vertical mast EF. Over the pulley hangs a flexible cable DG Which is fastened to the mast at D and carries at the other end G a load Q=20kN.Neglecting friction in the pulley, determine the forces produced in the bars AB and AC. The angles between the various members are shown in figure. S1= 0 S2= -34.64 N

10. Two identical rollers, each of weight Q=445 N, are supported by an inclined plane and a vertical wall as shown in figure. Assuming smooth surfaces, find the reactions induced at the points of support A, B and C. Q Q A C B 30°