Experiment 2: Vectors

The aim of this experiment is to study addition of vectors graphically, analytically and experimentally. The equipment needed include
  1. protractor
  2. force table
  3. 3 pulleys
  4. string
  5. set of weights
  6. 3 hangers
  7. a ring

Theory

Vectors and Scalars

All physical quantities are classified into two based on how they are represented. The physical quantities that can be represented fully by means of a number and a unit are called scalars. For example, length is a scalar because a number and a unit such as 2m specifies it completely. The physical quantities that require the specification of direction in addition to a number and a unit are called vectors. For example, velocity is a vector quantity because we need to know which way it is going in addition to its speed such as 50 miles/hour.

Symbolically, vectors are represented by bold letters ( in exercise books vectors are represented by capital letters with arrow on top). For example we may write A = 40 miles per hour east. Graphically vectors are represented by means of an arrow. The arrow is drawn in such a way that its length is proportional to the magnitude (absolute value of the numerical value--with a unit of course) and its direction is in the direction of the vector.

Negative of a vector is the vector with the same magnitude but opposite in direction. For example the negative of the vector 2 m east is 2 m west.

The effect of multiplying a vector by a constant is to multiply its magnitude (length) by the constant. If the constant is positive the direction remains the same while if the constant is negative the direction becomes opposite.

To add vectors graphically, first join the vectors tail to head. Then the sum vector (also called the resultant) is the vector whose tail is at the tail of the first vector and whose head is at the head of the last vector. The equilibrant of a number of vectors is the negative of their resultant (sum) vector.

Any vector can be represented as a sum of a horizontal vector and a vertical vector. The horizontal vector is called the horizontal component or the x-component of the vector. The vertical vector is called the vertical component or the x-component of the vector. x-components are taken to be positive if they are to the right and negative if they are to the left. y-components are taken positive if they are directed upwards and negative if they are directed downwards. If a vector has magnitude A and makes an angle θ with the positive x-axis (horizontal line to the right) in a counterclockwise direction then its horizontal component, Ax , and vertical component, Ay , are given by

Ax = A cos ( θ )

Ay = A sin ( θ )

If a vector has x-component, Ax ,and y-component, Ay , then its magnitude, A, and its direction, θ, are given by

A = √ ( Ax 2 + A 2 )

θ = arctan ( Ay ⁄ Ax )

f Ax is greater or equal to zero and

θ = arctan ( Ay ⁄ Ax ) + 180°

if Ay is less than zero. If Ax = 0 and Ay > 0, then θ = 90°. If Ax = 0 and Ay < 0, then θ = -90°. θ is angle with respect the positive x-axis measured in a counter clockwise direction. ( Negative anges are measured in a clockwise direction.

Vectors can be added analytically (algebraically) if their components are known. This is because the x-component of the sum vector is equal to the sum of the x-components of the vectors being added; and also the y-component of the sum vector is equal to the sum of the y-components of the vectors being added. That is, if R = A + B, then

Rx = Ax + Bx

Ry = Ay + By

and the magnitude ( R ) and the direction ( θ ) of the sum vector are given by

R = √ { ( Ax + Bx ) 2 + ( Ay + By ) 2 }

θ = arctan { ( Ay + By ) ⁄ ( Ax + Bx )}

If Ax + Bx > 0, and

θ = arctan { ( Ay + By ) ⁄ ( Ax + Bx )} + 180°

If Ax + Bx < 0

Subtracting vector B from vector A is equivalent to adding the negative of B to A.

The effect of multiplying a vector by a constant is to multiply its magnitude by the constant. If the constant is positive the direction remains the same if the constant is negative the direction becomes opposite.

Procedure

In the prcedures that follow, the following force vectors ( F1 and F2 ) (weights) will be added graphically, analytically and experimentally. Since mass and weight are proportional, grams can be used instead of Newtons. The angles are angles measured with repect to the positive x-axis (horizontal line to the right) in a counter clockwise direction.

F1 = 300 g at 30°

F2 = 200 g at 110°

  1. Adding vectors graphically

    Add these two forces graphically using a suitable scale and determine the magnitude and the direction (angle with respect to the positive x-axis in a counterclockwise direction) of the sum vector with the help of a ruler and a protractor. (Note: To add vectors graphically, first join the vectors head to tail and then the vector connecting the tail of the first vector to the head of the last vector is the sum vector). Use the graph paper at the last page of this manual.

    graphical magnitude ( F ) = g

    graphical direction ( θ ) = degree

  2. Adding vectors analytically

    To add the two vectors analytically, first the horizontal and vertical components of the vectors should be calculated.

    magnitude of F1 ( F1 ) = g

    direction of F1 ( θ1 ) = deg

    magnitude of F2 ( F2 ) = g

    direction of F2 ( θ2 ) = deg

    x-component of F1 ( F1x ) = F1 cos ( θ1 ) = g

    y-component of F1 ( F1y ) = F1 sin ( θ1 ) = g

    x-component of F2 ( F2x ) = F2 cos ( θ2 ) = g

    y-component of F2 ( F2y ) = F2 sin ( θ2 ) = g

    analytical magnitude of sum force ( F )= √ { ( F1x + F2x ) 2 + ( F1y + F2y ) 2 } = g

    analytical direction of sum force ( θ ) = arctan { ( F1y + F2y ) ⁄ ( F1x + F2x )} = deg

  3. Adding vectors experimentally

    Now determine the magnitude and the direction of the sum vector experimentally by means of the force table. Set the pulleys at the appropriate angles. (The positive x-axis is represented by the zero degrees line on the force table.) Tie a string into a ring (which is to be situated at the center of the force table) and then to a weight down a pulley for each force. Remember to put into consideration the weight of the hanger itself. After representing both forces F1 and F2 , by trial and error, find a third force (weight) that balances the two forces. (when balancing is achieved the center of the ring and the center of the force table should match.) Adjust both the magnitude (weight) and the direction of the third force until the three forces balance eachother. This third balancing force is called the equilibrant force and is opposite to the sum vector. That is the sum vector (resultant) and the equilibrant have the same magnitude but opposite direction.

    experimental magnitude of sum force = magnitude of balancing force = g

    experimental direction of sum force = direction of balancing force ± 180° = deg

    Pick the sign that gives an angle less than 360°.

  4. Error calculations

    1. Compare the magnitudes of the sum vectors obtained by the graphical method (procedure I.) and the analytical method (procedure II.) by calculating the percentage error.

      % error = |{analytical magnitude (II.) - graphical magnitude (I.)} ⁄ {analytical magnitude}| * 100% =

    2. Compare the directions of the sum vector obtained by the graphical method (procedure I) and the analytical method (procedure II) by calculating the percentage error.

      % error = |{analytical direction (II.) - graphical direction (I.)} ⁄ {analytical direction}| * 100% =

    3. Compare the magnitudes of the sum vector obtained by the analytical method (procedure II) and the experimental method (procedure III.) by calculating the percentage error.

      % error = |{analytical magnitude (II.) - experimental magnitude (III.)} ⁄ {analytical magnitude}| * 100% =

    4. Compare the directions of the sum vector obtained by the experimental method (procedure III.) and the analytical method (procedure II) by calculating the percentage error.

      % error = |{analytical direction (II.) - experimental direction (III.)} ⁄ {analytical direction}| * 100% =