Writing Equations for a Free Body Diagram

In most cases, it is a simple task to determine which forces are acting on a body, which way they point, and which way the acceleration points. A free body diagram (FBD) is simply a sketch of this information. What we usually don't know is the magnitude of one or more of the forces acting on the body or the acceleration. Apply Newton's Second Law to the FBD to find a set of equations that can be solved to find those magnitudes. You will find two equations for each FBD of each object you are interested in. Newton's Second Law is actually two laws F_{net x} = ma_x and F_{net y} = ma_y. You will get one equation for each direction. In this note we summarize how to write the set of equations.

Steps

• Choose a set of x and y axes.
• To make your equations as simple as possible, choose either the x or y axis to lie in the same direction as the acceleration, if there is one.
• Never rotate a FBD so that the x and y axes are horizontal and vertical.
• Break each force in the FBD up into its x and y components. Note that the component next to the given angle will have the cosine while the component opposite the given angle will have sine (see diagram below).

• Write the components using exactly the same symbols and subscripts as used in the FBD.
• The force components and acceleration can point up or down, right or left. Up or right are indicated by + in equations. Down or left are indicated by − in equations.
• Forces must be kept on the left hand side of the equation. Acceleration can only go on the right hand side. You can rearrange the equations any way you want after you have first written them in conventional form.
• Never put numbers in the equations. You need to find the symbolic form first.
• Solve the set of equations using whatever technique you wish.

Note

• If you find that your normal force is a negative number, you have made a mistake somewhere. Normals push they do not pull.
• If you find that your tension force is a negative number, you have made a mistake somewhere. You can pull an object by a string but you cannot push it with one.
• If you find that acceleration is a negative number, this indicates that the acceleration is in the opposite direction to your initial guess.

Example

Solution

Choose the x axis to lie along the direction of the acceleration. Also making use of geometry to find the angle that the weight makes with the y axis.

Finding the components of each force and of the acceleration. You do not have to draw the table if you don't want to.

Thus the equations are found from each column.

Tsinφ - Mgcosθ + N = 0

Solve the equations as you wish to find the quantities of interest.