Weight and Free Body Diagrams

“Work expand to fill the time available for its completion”

C. Northcote Parkinson – Parkinson’s Law (1958)

• WEIGHT

• The weight of an object is the force with which a body is attracted to the earth.
• Weight = mass x acceleration due to gravity   (w = mg)
• Since "g" is approximately the same everywhere on the earth's surface, weight is often quoted in units of mass.
• Weight depends on location, mass is constant independent of location, e.g. an object on the moon weighs about one sixth of its weight on earth, but its mass is unchanged.
• Units:  Newtons (SI), Pounds (British)

• TENSION

• Strings, ropes, wires, cables etc. are used as a mechanism to transmit  force.
• The Tension in a rope is equal to the force being transmitted.
• Tension is always a "pulling" force.  A rope cannot push an object.  Therefore a rope under tension "pulls" in both directions.

• FREE BODY DIAGRAMS

• Introduction: A free body diagram is a picture of the forces which act on an object and is the first (and perhaps the most important) step in solving force problems.

• Purpose: The purpose of the free body diagram (FBD) is to help you identify and analyze the forces that act on a particular object or body.

o Types of Forces: The types of forces normally encountered in classical mechanics are

a. WEIGHT: As we have seen, weight is the gravitational force exerted on an object by the Earth (or any other celestial body). If an object is near the Earth's surface and has mass, then the object has a weight. The magnitude of its weight is w = mg and its direction is toward the center of the Earth.

b. APPLIED FORCE: Applied forces usually result from things physically touching and acting on a body. By this definition, normal, tension and friction forces are applied forces, but we usually categorize them separately. In our context, applied forces include such phenomena as pushing an object or the force exerted by a spring on an object.

c. NORMAL FORCE: The normal force is due to contact of an external surface on an object. The surface supplies the normal force to the object due to Newton's 3rd law. If two objects are in contact, normal forces exist. The direction of the normal force is perpendicular to the surfaces in contact and directed toward the body under consideration. The magnitude of the normal force can only be determined by analyzing all forces acting on the body.

d. TENSION FORCE: (See above) A tension force is a force applied to a body by a rope or string. Ropes and strings are incapable of pushing a body; they always must pull a body. Tension forces are directed away from the body being pulled and along the direction of the rope or string.

e. FRICTIONAL FORCES: Frictional forces occur because of surface adhesion between two objects in contact. We will consider frictional forces in detail at a later date.

o Drawing a FBD. In summary, the steps (discussed in more detail below) for drawing a FBD are:

a. Isolate the body. This step requires that you separate the body (block, wheel, cart, etc.) that you are investigating from everything else and draw it.

b. Draw and label all forces acting on the body. This is the most difficult step. Here is a suggested method to use:

1) List the possible types of forces. A handy mnemonic is WANT FORCES:

W W eight

A A pplied

N N ormal

T T ension

Forces F riction

2) Answer the following questions about the types of forces:

a) Does the body have mass? If it does, weight must be included.

b) Is there an applied force described in the problem?

c) Is there a normal force? Does the body touch another surface? If so, there must be a normal force.

d) Is there a rope or string attached to the body? If so, there is a tension force directed in such a way as to pull the body.

e) Is there a friction force? If the surface is frictionless or smooth, then no friction force is present. Otherwise a friction force is present and is directed parallel to the surface and opposing motion or impending motion.

3) Determine where to draw the forces on the FBD. The positions of the force vectors on the diagram will not affect our analysis.  However, if we were to discuss rotational motion, position would become important. Use the following rules:

a) Identify the point of application of the force.

1) For a homogeneous mass, the weight acts at the geometric center of the object.

2) The point of application of applied and tension forces is normally specified in the problem statement.

3) Normal and friction forces involve areas of surfaces in contact. By convention we will choose the center of the surface in contact as the point of application of these two forces.

4) The normal force is drawn perpendicular to the surface of contact through the center of the body.

5) The friction force is drawn parallel to the contact surface.

b) The line of action for a force is defined as a straight line in the direction of the force and passing through the point of application.

c. Choose a coordinate system (and direction of positive torque). Virtually any coordinate axes can be used in a problem; however, certain selections will make the application of Newton's Second Law much easier mathematically. Some rules to find a good set of axes:

1) Object experiencing acceleration.Choose one axis along the direction of acceleration. Choose the other axis perpendicular to the first.

2) Object moving a constant velocity.Choose one axis along the direction of motion. Choose the other axis perpendicular to the first.

3) Object at rest. Choose one axis parallel to the surface upon which the object rests. Choose the other axis perpendicular to the first.

The direction of positive torque will be discussed later in the course when we cover rotational motion. At this stage in the course, we need not concern ourselves with torque.

d. Include critical angles and dimensions. In many cases, it will be necessary to break force vectors into their components. Identifying critical angles allows the simple application of trigonometric relationships.

• PROBLEM SOLVING

For all problems involving the application of Newton's 2nd Law follow the prescription below:

1. Draw a diagram describing the situation described in the problem.

2. Make sure you know which object(s) the possible motion refers to.

3. Draw free body diagrams (FBD) for each object under consideration, separately from your original diagram.  This step is critical.  Unless you can identify and correctly add all the forces acting on each object you will be unable to solve the problem.

4. Choose an appropriate co-ordinate system for each object.  In most situations for each object you  choose axes such that any motion takes place along an axis.  Note that the co-ordinate systems for each object do not have to be the same.

5. Finally, apply Newton’s 2^nd law for each object in each direction (this will involve resolution of forces along the chosen co-ordinate axes) and use some common sense. This is often described as writing down the “equations of motion”. Once you have the equations of motion, in principle, you can solve them for the desired unknowns.

Example Problem

“Vote for the man who promises least; he’ll be the least disappointing”

Bernard Baruch

Dr. C. L. Davis
Physics Department
University of Louisville
email: c.l.davis@louisville.edu