Lecture #4 Equilibrium of a Particle and Intro to Free Body Diagram (ref: Chapter 3.1, 3.2) ΣFx = 0 ΣFy = 0 ΣFz = 0 Required for Static Equilibrium!! R. Michael PE 8/14/2012

Particle vs. Rigid Body Particle – Very small amount of matter compared to the system being analyzed. The volume of the particle and area it occupies is negligible to the problem and has no bearing on the solution. Can be modeled as a single point in space.

Particle vs. Rigid Body Rigid Body – A combination of a large number of particles occupying fixed positions with respect to each other. Must use a rigid body model when the area that the body of the object occupies has an affect on the outcome of the problem. Have additional requirement for equilibrium (moment)

Rigid Body Analysis Must be used: If the mass of the body comes into effect If there are forces with different points of application on the body If the size of the body has to be taken into account

Explanation of Rigid Body For Statics we define a Rigid Body as a body that has no deformation. A Rigid Body differs from a Particle because size comes into play: Both weigh 50 lb, which one is a rigid body and which one is a particle? A B W = 50 lb W = 50 lb

Concepts to be Introduced for Rigid Body Analysis Internal vs. External Forces Free Body Diagrams (applies to Particles and Rigid Bodies) Translational Equilibrium (applies to Particles and Rigid Bodies) Principle of Transmissibility Moments (Couples) acting on the bodies Force – Couple Systems on Rigid Bodies Equilibrium in Six degree of freedom situation (Rigid Body Equilibrium)

External vs. Internal Forces External Forces – Represent the action of other bodies on the Particle or Rigid Body. Entirely responsible for the motion of lack of motion of the Particle or Rigid Body as described by Newton’s First Law of Motion. Internal Forces – The forces holding the individual particles of a rigid member together. Can also be used to define the forces holding members of a composite structure together.

Small contact area; treat as a point Types of Forces: Small contact area; treat as a point FR is resultant of w(s) = area under curve, acts at centroid One body acting on another What’s a body? One body acting on another w/o contact Acting on narrow area

3.1 Free Body Diagram Introduction Free Body Diagram – A diagram of the Particle or Rigid Body of interest with all Forces acting ON the body and all measurements pertinent to the solution of the problem.

THE WHAT, WHY AND HOW OF A FREE BODY DIAGRAM (FBD) Free Body Diagrams are one of the most important things for you to know how to draw and use. What ? - It is a drawing that shows all external forces acting on the particle. Why ? - It is key to being able to write the equations of equilibrium—which are used to solve for the unknowns (usually forces or angles).

FBD for Rigid Body: This is a FBD of the truck!! = simplified

Truck External Forces The Forces can be shown by Vectors acting on the Truck W – Weight of the truck acting in the negative vertical direction and acting at the center of gravity of the truck R1 and R2 – Contact Force of the ground holding the truck from falling toward the center of the Earth Fr – Force of the Rope pulling the truck forward

COPLANAR FORCE SYSTEMS (Section 3.3) This is an example of a 2-D or coplanar force system. If the whole assembly is in equilibrium, then particle A is also in equilibrium. To determine the tensions in the cables for a given weight of the cylinder, you need to learn how to draw a free body diagram and apply equations of equilibrium. Explain the following terms : Coplanar force system – The forces lie on a sheet of paper (plane). A particle – It has a mass, but a size that can be neglected. Equilibrium – A condition of rest or constant velocity. Draw a FBD for the mass, C Draw a FBD of Ring A Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 3.1-3.3

Steps in Making the Free Body Diagram The most important part of making a Free Body Diagram is that a clear decision must be made as to what part of the Space Diagram or System is being analyzed VS.

External Force Vectors Represent the forces acting ON the Free Body BY the surroundings. This is very important. Must be applied to the body where they are acting on the body Must be represented with Direction and Magnitude. The weight of the object (if applicable) should be placed at the center of gravity of the object.

REACTION Forces Unknown external forces are known as reactions or constraining forces and must be shown on the Free body Diagram Reactions are forces that constrain one or more degrees of freedom of motion and must be indicated on the Free Body Diagram where it is supported or connected to an external body

Unknown External Forces

Unknown External Forces

Practice FBD’s!!!

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APPLICATIONS The crane is lifting a load. To decide if the straps holding the load to the crane hook will fail, you need to know the force in the straps. How could you find the forces? Straps

APPLICATIONS (continued) For a spool of given weight, how would you find the forces in cables AB and AC ? If designing a spreader bar like this one, you need to know the forces to make sure the rigging doesn’t fail.

APPLICATIONS (continued) For a given force exerted on the boat’s towing pendant, what are the forces in the bridle cables? What size of cable must you use?

Principle of Transmissibility The conditions of equilibrium or motion of a Rigid Body will remain unchanged by moving a force along its line of action provided the force magnitude and direction does not change

Equivalent Forces If F and F' have the same direction, magnitude, and line of action, they are said to be Equivalent Forces