ENGR 1100 Introductory Mechanics

The branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements and the subsequent effect of the bodies on their environment. Who needs it/Why use it? Mechanical/Civil/Aerospace/Biomedical/ Manufacturing/Mechatronics Almost all engineers utilize a mathematical interpretation of force/displacement problems in their required analysis

Mechanics Statics: bodies are stationary or moving at a constant velocity Rigid Bodies Deformable bodies-material mechanics Fluids-fluid mechanics Dynamics: bodies are accelerating/rotating/vibrating

Vectors A physical quantity that is completely described by a real number is called a scalar A physical quantity that is described by a nonnegative real number and a direction direction describes a vector quantity

Vectors A vector is represented graphically by an arrow The direction represents the direction of the vector, the length of the arrow represents its magnitude Note the opposite direction would be considered -ve Eg. Forces, displacements -FF

Vector Addition We add vectors by placing the tip of one to the tail of the other or vice versa Eg. Move your text along your table to two different locations, examine resulting displacement V U+ V U U V

Vector Addition We may also add vectors U and V by making a parallelogram, the diagonal gives us the resultant vector U V U+V V U

Polygon Rule To add more than two vectors we can form a polygon U V W U V W U+V+W Note: order is irrelevant! i.e. commutative

Vector Addition Vectors do not add as scalars unless they are in the same direction

Vector Addition/subtraction Same direction Opposite direction P - Q = P + (-Q ) P-Q P - Q PQ P + Q

Vector Subtraction The difference between two vectors U and V is obtained by adding U to the vector -V U+ (-V) -V U U V

Vector Subtraction We may also add vectors U and -V by using the same parallelogram, the other diagonal gives us the resultant vector U V U-V V U

Components in two dimensions If is // to the x-y plane, we can write it as the sum of two vector components parallel to the x and y axes U UxUx UyUy U x y

Components in two dimensions We can introduce unit vector, defined to point in the positive direction of the x-axis and defined to point in the positive direction of the y-axis U therefore, we can write UxUx UyUy U x y j i i j

Components in two dimensions Ux and Uy are called the scalar components of U from Pythagorean theorem we see that the magnitude of U is given by UxUx UyUy U x y j i

Components in two dimensions Also, if the angle  is as shown, we may write UxUx UyUy U x y j i 

Question/Example See second example in class If you know the scalar components of a vector how can you determine its magnitude?

Particle Equilibrium or Recall: When the resultant of all the forces acting on a particle is zero, the particle is in equilibrium

Free Body Diagrams A free body diagram of a body shows all the forces directly applied to it, it is the first step in the solution of conditions of equilibrium A free body diagram may be of a particle or a rigid body or components of a rigid body 10N 5N 8N Particle Rigid Body

FBD (rigid bodies) All external forces should be indicated Magnitudes and directions of all known external forces should be clearly marked Reactions are exerted at the points where the free body is supported or connected to other bodies Include dimensions Detach body from all contacts when sketching an FBD of it. You may sketch several FBD’s of connected bodies