Set 5 Problem number 7

• Problem

• Solution

• Generalized Solution

• Explanation in terms of Figure(s), Extension

• Figure(s)

Problem

What are x and y the components of the vector obtained when we add force vector A, with magnitude 5 Newtons and angle 227.6 degrees, to the force vector B whose angle and magnitude are 48.4 degrees and 17.8 Newtons?

• What are the magnitude and angle of this resultant vector?

Solution

The x and y components of A are easily found to be 5 Newtons cos( 227.6 deg) = -3.38 Newtons and 5 Newtons sin( 227.6 deg) = -3.7 Newtons.

The x and y components of B are found in the same manner to be 11.81 Newtons and 13.31 Newtons.

The x component of the resultant is simply the sum -3.38 Newtons + 11.81 Newtons = 8.43 Newtons of the x components of A and B.

The y component is similarly the sum -3.7 Newtons + 13.31  Newtons= 9.610001 Newtons of the y components of A and B.

The magnitude of the resultant vector, by the Pythagorean Theorem, is therefore

• magnitude of resultant = `sqrt( ( 8.43 Newtons) ^ 2 + ( 9.610001 Newtons) ^ 2) = 12.78 Newtons.

The angle of the resultant vector to the x axis is found from the inverse tangent

• tan^-1( R_y /  R_x ) = tan^-1( 9.610001 Newtons/ ( 8.43 Newtons)) = 48.74 degrees.

The final step depends on whether the x component 8.43 is greater or less than zero:

• If 8.43 is negative, then

• The vector is in quadrant 2 or 3 and the correct angle is therefore ( 48.74 + 180) degrees = 228.74 degrees.

• If 8.43 is positive, then

• Since the x component of this vector is positive, this is the correct angle.

Generalized Solution

If we have vectors A and B, at angles `theta<sub>1</sub> and `theta₂ as measured from the direction of the positive x axis, their sum is found by  first finding the x and y component of each (multiplying the magnitude of each vector by the cosine of its angle to obtain its x component, and multiplying the magnitude of each vector by the sine of its angle to obtain its y component).

The sum of the two x components will then be the x component of the resultant, and the sum of the two y components will be the y component of the resultant.

We obtain

A_x = | A | cos(`theta₁),

A_y = | A | sin(`theta₁),

B_x = | B | cos(`theta₂),

B_y = | B | sin(`theta₂).

We obtain the resultant vector R = A + B by first adding the x components of A and B:

R_x = A_x + B_x = | A | cos(`theta<sub>1</sub>) + | B | cos(`theta₂)

and

R_y = A_y + B_y = | A | sin(`theta<sub>1</sub>) + | B | sin(`theta₂).

We then find the magnitude and angle of R, using the Pythagorean Theorem and the tan^-1gent:

|R| = `sqrt(R_x^2 + R_y^2)

and

`theta = tan^-1(R_y / R_x).

Explanation in terms of Figure(s), Extension

The figure below shows two vectors A and B, and their components A_x, A_y, B_x and B_y.

• The y components A_y and B_y are seen to add up to R_y, as
• the x components A_x and B_x are seen to at up to R_x.

• The magnitude and angle of the vector R are obtained by using the Pythagorean Theorem and the tan^-1to the components R_y and R_x in the usual manner.

Figure(s)