Theoretical Mechanics STATICS KINEMATICS Technical University of Sofia Branch Plovdiv Theoretical Mechanics STATICS KINEMATICS * Navigation: Right (Down) arrow – next slide Left (Up) arrow – previous slide Esc – Exit Notes and Recommendations: ruschev@tu-plovdiv.bg

Lecture 3 Equilibrium of a Concurrent System of Forces Equilibrium refers to a condition in which an object (particle or body) is at rest if originally at rest, or has a constant velocity if originally in motion. Most often however, the term “equilibrium” or more specifically “static equilibrium” is used to describe an object at rest. Vector condition for equilibrium The necessary and sufficient vector condition for equilibrium of a concurrent force system is the resultant force of all forces acting on the particle to be zero. Analytical conditions for equilibrium Vector condition for equilibrium (after projecting it on three coordinate axes) may be written in scalar form as: Therefore, for a given concurrent force system to be in equilibrium it is necessary and sufficient the algebraic sums of components of all forces acting on the particle to be equal to zero. These scalar equations can be used for obtaining three scalar components of one unknown force equilibrating the system of concurrent forces.

Lecture 3 Free-Body Diagrams Free-Body Diagram: A sketch showing only the forces on the selected particle. Space Diagram: A sketch showing the physical conditions of the problem.

Lecture 3 Sample Problem 1 SOLUTION: Construct a free-body diagram for the particle at the junction of the rope and cable. Apply the conditions for equilibrium by creating a closed polygon from the forces applied to the particle. Apply trigonometric relations to determine the unknown force magnitudes. In a ship-unloading operation, a 3500-N automobile is supported by a cable. A rope is tied to the cable and pulled to center the automobile over its intended position. What is the tension in the rope?

Lecture 3 Sample Problem 1 Contd. SOLUTION: Construct a free-body diagram for the particle at A. Apply the conditions for equilibrium. Solve for the unknown force magnitudes.

Lecture 3 Sample Problem 2 As part of the design of a new sailboat, it is desired to determine the drag force which may be expected at a given speed. To do so, a model of the proposed hull is placed in a test channel and three cables are used to keep its bow on the centerline of the channel. Dynamometer readings indicate that for a given speed, the tension is 40 lb in cable AB and 60 lb in cable AE . Determine the drag force exerted on the hull and the tension in cable AC. 1 ft = 30.48 cm

Lecture 3 Sample Problem 2 Determination of the Angles

Lecture 3 Sample Problem 2 Free-Body Diagram Choosing the hull as a free body, we draw the freebody diagram shown. It includes the forces exerted by the three cables on the hull, as well as the drag force FD exerted by the flow. Equilibrium Condition We express that the hull is in equilibrium by writing that the resultant of all forces is zero: We resolve the forces into x and y components:

Lecture 3 Sample Problem 3 A sailor is being rescued using a boatswain’s chair that is suspended from a pulley that can roll freely on the support cable ACB and is pulled at a constant speed by cable CD. Knowing that a a=30° and b b=10° and that the combined weight of the boatswain’s chair and the sailor is 900 N, determine the tension in the support cable ACB, in the traction cable CD.