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Chapter 4 Vectors. The Cardinal Directions. Vectors. An arrow-tipped line segment used to represent different quantities. Length represents magnitude. Arrow head represents direction. Vector Addition in 1 - Dimension.
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Chapter 4 Vectors The Cardinal Directions
Vectors • An arrow-tipped line segment used to represent different quantities. • Length represents magnitude. • Arrow head represents direction.
Vector Addition in 1 - Dimension • When vectors point in the same direction we add them just as we would add any two numbers.
Vector Addition in 1 - Dimension • When vectors point in opposite directions we subtract them just as we would with any two numbers.
Vector Addition in 2-Dimensions • Vectors in 2-dim are added by placing the tail of one to the head of another.
Addition of Several Vectors • The order of addition is not important. • R is called the resultant.
Independence of Vector Quantities • Perpendicular vectors can be treated independently of each other.
Analytical Method of Vector Addition • The sum of any two vectors can be determined using trigonometry.
Angle θ is = • 25 deg • 14 deg • 35 deg • 45 deg
Angle θ is = • 25 deg b) 14 deg c) 35 deg d) 45 deg
Vector Components • We can take two vectors and replace them with a single vector that has the same effect. This is vector addition. • We can start with a single vector and think of it as a resultant of two perpendicular vectors called components. • This process is called vector resolution.
Problem Solving Strategy • In resolving vectors choose the most convenient axis according to the specifics of the problem. • Choose the axis that simplifies the solution. • Axis may be up-down, left-right, east-west or north-south. • Be sure to specify the positive direction for each.
Adding Vectors at Any Angle • Vector resolution is the method used. • Resolve all vectors into x and y components. • Add all x’s and all y’s together. • Use xtot and ytot to create a right triangle. • Use Pythagorean formula to calculate resultant and trig to find angle.
R is = ? • 15 N • 12 N • 20 N • 11N
R is = ? • 15 N • 12 N • 20 N • 11N
Θ is = ? • 53 deg • 35 deg • 25 deg • 45 deg
Θ is = ? • 53 deg • 35 deg • 25 deg • 45 deg
Applications of Vectors Vectors can be used to represent: • displacement • velocity • acceleration • force
Equilibrium • When the net force is zero, the object is in equilibrium. • When the vector sum of the forces is not zero, a force can be applied that will produce equilibrium. This force is called the equilibrant. • It is equal in magnitude but opposite in direction to the resultant.
3 Forces in Equilibrium: • produce a net force. • produce a triangle for a vector diagram. • are called an equilibrant. • produce an acceleration.
3 Forces in Equilibrium: • produce a net force. • produce a triangle for a vector diagram. • are called an equilibrant. • produce an acceleration.
Gravitational Force and Inclined Planes • Gravitational force always points towards center of Earth. • This is weight. • Choose one axis parallel to the plane and the other perpendicular to it.
Formulas • R2 = A2 + B2 – 2AB cos Θ • Ax = A cos Θ • AY = A sin Θ • A = Ax + AY