Force Is a Vector
Force, like displacement, velocity, and acceleration, is a vector quantity, which is why Newton’s Second Law is written as sigmaF = ma. Put into words, it says that the vector sum of the forces acting on an object is equal to its mass (a scalar) multiplied by its acceleration (a vector).
Because force is a vector quantity, you add forces together as vectors. That fits right into Newton’s Second Law.
Sample question

Suppose that you have two forces as shown: A = 5.0 N at 40 degrees, and B = 7.0 N at 125 degrees. What is the net force, sigmaF?
The correct answer is magnitude 8.9 N, angle 91 degrees.

Convert force A into vector component notation. Use the equation A_{x} = A cos theta to find the x coordinate of the force: 5.0 cos 40 degrees = 3.8.

Use the equation A_{y} = A sin theta to find the y coordinate of the force: 5.0 sin 40 degrees, or 3.2. That makes the vector A (3.8, 3.2) in coordinate form.

Convert the vector B into components. Use the equation B_{x} = B cos theta to find the x coordinate of the acceleration: 7.0 cos 125 degrees = –4.0.

Use the equation B_{y} = B sin theta to find the y coordinate of the second force: 7.0 sin 125 degrees, or 5.7. That makes the force B (–4.0, 5.7) in coordinate form.

Perform the vector addition to find the net force: (3.8, 3.2) + (–4.0, 5.7) = (–0.2, 8.9).

Convert the vector (–0.2, 8.9) into magnitude/angle form. Use the equation theta = tan^{–1}(y/x) to find the angle: tan^{–1}(–44.5) = 91 degrees.

Apply the equation
to find the magnitude of the net force, giving you 8.9 N.

Practice questions

Add two forces: A is 8.0 N at 53 degrees, and B is 9.0 N at 19 degrees.

Add two forces: A is 16.0 N at 39 degrees, and B is 5.0 N at 125 degrees.

Add two forces: A is 22.0 N at 68 degrees, and B is 6.0 N at 24 degrees.

Add two forces: A is 12.0 N at 129 degrees, and B is 3.0 N at 225 degrees.
Following are answers to the practice questions:

Magnitude: 16 N; Angle: 35 degrees

Convert force A into vector component notation. Use the equation A_{x} = A cos theta to find the x coordinate of force A: 8.0 cos 53 degrees = 4.8 N.

Use the equation A_{y} = A sin theta to find the y coordinate of force A: 8.0 sin 53 degrees = 6.4 N. That makes force A (4.8, 6.4)N in coordinate form.

Convert the vector B into components. Use the equation B_{x} = B cos theta to find the x coordinate of force B: 9.0 cos 19 degrees = 8.5 N.

Use the equation B_{y} = B sin theta to find the y coordinate of the second force: 9.0 sin 19 degrees = 2.9 N. That makes force B (8.5, 2.9)N in coordinate form.

Perform vector addition to find the net force: (4.8, 6.4)N + (8.5, 2.9)N = (13.3, 9.3)N.

Convert the force vector (13.3, 9.3)N into magnitude/angle form. Use the equation theta = tan^{–1}(y/x) to find the angle: tan^{–1}(0.70) = 35 degrees.

Apply the equation
to find the magnitude of the net force, giving you 16 N.


Magnitude: 17 N; Angle: 56 degrees

Convert force A into vector component notation. Use the equation A_{x} = A cos theta to find the x coordinate of force A: 16.0 cos 39 degrees = 12.4 N.

Use the equation A_{y} = A sin theta to find the y coordinate of force A: 16.0 sin 39 degrees = 10.0 N. That makes force A (12.4, 10.0)N in coordinate form.

Convert force B into components. Use the equation B_{x} = B cos theta to find the x coordinate of force B: 5.0 cos 125 degrees = –2.9 N.

Use the equation B_{y} = B sin theta to find the y coordinate of the second force: 5.0 sin 125 degrees = 4.1 N. That makes force B (–2.9, 4.1)N in coordinate form.

Perform vector addition to find the net force: (12.4, 10.0)N + (–2.9, 4.1)N = (9.5, 14.1)N.

Convert the force vector (9.5, 14.1)N into magnitude/angle form. Use the equation theta = tan^{–1} (y/x) to find the angle: tan^{–1}(1.5) = 56 degrees.

Apply the equation
to find the magnitude of the net force, giving you 17 N.


Magnitude: 27 N; Angle: 59 degrees

Convert force A into vector component notation. Use the equation A_{x} = A cos theta to find the x coordinate of force A: 22.0 cos 68 degrees = 8.24N.

Use the equation A_{y} = A sin theta to find the y coordinate of force A: 22.0 sin 68 degrees = 20.4 N. That makes force A (8.24, 20.4)N in coordinate form.

Convert force B into components. Use the equation B_{x} = B cos theta to find the x coordinate of force B: 6.0 cos 24 degrees = 5.5 N.

Use the equation B_{y} = B sin theta to find the y coordinate of force B: 6.0 sin 24 degrees = 2.4 N. That makes force B (5.5, 2.4)N in coordinate form.

Perform vector addition to find the net force: (8.24, 20.3)N + (5.5, 2.4)N = (13.7, 22.7)N.

Convert the force vector (13.7, 22.7)N into magnitude/angle form. Use the equation theta = tan^{–1}(y/x) to find the angle: tan^{–1}(1.66) = 59 degrees.

Apply the equation
to find the magnitude of the net force, giving you 27 N.


Magnitude: 12 N; Angle: 143 degrees

Convert force A into vector component notation. Use the equation A_{x} = A cos theta to find the x coordinate of force A: 12.0 cos 129 degrees = –7.6.

Use the equation A_{y} = A sin theta to find the y coordinate of force A: 12.0 sin 129 degrees = 9.3 N. That makes force A (–7.6, 9.3)N in coordinate form.

Convert force B into components. Use the equation B_{x} = B cos theta to find the x coordinate of force B: 3.0 cos 225 degrees = –2.1 N.

Use the equation B_{y} = B sin theta to find the y coordinate of force B: 3.0 sin 225 degrees = –2.1 N. That makes force B (–2.1, –2.1)N in coordinate form.

Perform vector addition to find the net force: (–7.6, 9.3)N + (–2.1, –2.1)N = (–9.7, 7.2)N.

Convert the force vector (–9.7, 7.2)N into magnitude/angle form. Use the equation theta = tan^{–1}(y/x) to find the angle: tan^{–1}(–0.74) = 143 degrees.

Apply the equation
to find the magnitude of the net force, giving you 12 N.
