Vectors

Physics

Vector Addition and Resolution

Vectors are any quantity that require a magnitude and a direction to describe them. The forces we have been studying are perfect examples of vector quantities. Many times we will be faced with multiple forces acting on a single object, often in differing directions. We must learn to work with these quantities- add them graphically and mathematically- and learn how they operate.

Graphically we can draw a vector where the length of a line represents the magnitude and the arrowhead at the end denotes the direction the force acts. If we do this we must select a scale, such as 1 cm = 50 N, and draw the line to scale. To add more than one of these vectors is easy if it operates in only one dimension. We simply draw these head to tail ( always draw vectors head to tail!!!) and treat it as motion along a number line. Right is positive and left is negative and the magnitude is simply the displacement from its origin.

Two dimensional vector addition can be done graphically as well. Through the use of a protractor and ruler we can construct any combination of vectors. Again we draw them head to tail and each arrowhead becomes the new "origin" for the next vector. The resultant again is the displacement from the origin to the terminal point of our final vector. If we establish a uniform scale and properly draw the vectors our resultant will be fairly accurate.

The most useful thing about graphically adding 2 dimensional vectors is for us to realize that the terminal point, denoted by the (x,y) cartesian coordinates is simply the sum of all the x, y values for each vector.

Also we realize that perpendicular vectors are independent of one another. The pull of gravity is the same whether the object is at rest or moving horizontally.

The most accurate way to add vectors is with the use of our trig functions ( sine, cosine, and tangent) in conjunction with the Pythagorean Theorem (a2 + b2 = c2). The trig function are simply ratios between the legs and hypotenuse of a right triangle and it becomes a simple algebra problem to use them to solve for a leg or the hypotenuse of a right triangle.

Sine0 = Opp/Hyp

Cos 0 = Adj/Hyp

Tan 0 = Opp/Adj

We can use the inverse function to find the angle or use them as they are to find a leg.

You need to practice the keystrokes with your particular calculator.

Perpendicular vectors are easiest solved with the use of the pythagorean theorem.

Vectors that aren't perpendicular present the first hurdle in this chapter. Remember the terminal point in vector addition represents the sum of all the x and y components. If we take each non perpendicular vector, resolve it into a single x and y component and add all the x and y values it will form a new triangle where the legs are known.

To resolve a given vector, we can establish the vector as the sum of 2 perpendicular legs and use our trig functions to solve for these legs. That is what vector resolution means - finding the perpendicular legs of an angular vector. It is important here to remember the signs as we break these legs down. To the right and up will be positive and down and to the left will be negative. Errors here are easily made and can sink you!

When we add several vectors and form a new single vector, this new vector is a "resultant" - a single vector with the same effect as those we combined.

If we apply the same magnitude force but 180 degrees in the opposite direction we have a vector that will exactly cancel out the resultant - leaving no net force. This is a condition of equilibrium where all forces cancel. This vector is an "equilibriant".

GRAVITY AND INCLINED PLANES

These situations require a careful analysis of the force vectors acting on the object resting at an incline. Gravity or weight always acts straight down, but only the part of gravity that acts down the plane will have any practical effect. We must resolve the weight into the force parallel to the plane and the force perpendicular to the plane (the normal force).

Friction will oppose this motion but its direction will depend upon whether the object is moving up or down the incline.

To solve these draw a force diagram, analyze the forces and determine the net force. This net force will decide what, if any, motion occurs

This page was last updated on: November 6, 2001



	Physics Vector Addition and Resolution Vectors are any quantity that require a magnitude and a direction to describe them. The forces we have been studying are perfect examples of vector quantities. Many times we will be faced with multiple forces acting on a single object, often in differing directions. We must learn to work with these quantities- add them graphically and mathematically- and learn how they operate. Graphically we can draw a vector where the length of a line represents the magnitude and the arrowhead at the end denotes the direction the force acts. If we do this we must select a scale, such as 1 cm = 50 N, and draw the line to scale. To add more than one of these vectors is easy if it operates in only one dimension. We simply draw these head to tail ( always draw vectors head to tail!!!) and treat it as motion along a number line. Right is positive and left is negative and the magnitude is simply the displacement from its origin. Two dimensional vector addition can be done graphically as well. Through the use of a protractor and ruler we can construct any combination of vectors. Again we draw them head to tail and each arrowhead becomes the new "origin" for the next vector. The resultant again is the displacement from the origin to the terminal point of our final vector. If we establish a uniform scale and properly draw the vectors our resultant will be fairly accurate. The most useful thing about graphically adding 2 dimensional vectors is for us to realize that the terminal point, denoted by the (x,y) cartesian coordinates is simply the sum of all the x, y values for each vector. Also we realize that perpendicular vectors are independent of one another. The pull of gravity is the same whether the object is at rest or moving horizontally. The most accurate way to add vectors is with the use of our trig functions ( sine, cosine, and tangent) in conjunction with the Pythagorean Theorem (a2 + b2 = c2). The trig function are simply ratios between the legs and hypotenuse of a right triangle and it becomes a simple algebra problem to use them to solve for a leg or the hypotenuse of a right triangle. Sine0 = Opp/Hyp Cos 0 = Adj/Hyp Tan 0 = Opp/Adj We can use the inverse function to find the angle or use them as they are to find a leg. You need to practice the keystrokes with your particular calculator. Perpendicular vectors are easiest solved with the use of the pythagorean theorem. Vectors that aren't perpendicular present the first hurdle in this chapter. Remember the terminal point in vector addition represents the sum of all the x and y components. If we take each non perpendicular vector, resolve it into a single x and y component and add all the x and y values it will form a new triangle where the legs are known. To resolve a given vector, we can establish the vector as the sum of 2 perpendicular legs and use our trig functions to solve for these legs. That is what vector resolution means - finding the perpendicular legs of an angular vector. It is important here to remember the signs as we break these legs down. To the right and up will be positive and down and to the left will be negative. Errors here are easily made and can sink you! When we add several vectors and form a new single vector, this new vector is a "resultant" - a single vector with the same effect as those we combined. If we apply the same magnitude force but 180 degrees in the opposite direction we have a vector that will exactly cancel out the resultant - leaving no net force. This is a condition of equilibrium where all forces cancel. This vector is an "equilibriant". GRAVITY AND INCLINED PLANES These situations require a careful analysis of the force vectors acting on the object resting at an incline. Gravity or weight always acts straight down, but only the part of gravity that acts down the plane will have any practical effect. We must resolve the weight into the force parallel to the plane and the force perpendicular to the plane (the normal force). Friction will oppose this motion but its direction will depend upon whether the object is moving up or down the incline. To solve these draw a force diagram, analyze the forces and determine the net force. This net force will decide what, if any, motion occurs





			This page was last updated on: November 6, 2001