PHYS101: Introduction to Mechanics

8a. Define the conditions necessary for a rigid body to be in equilibrium

• Define equilibrium.
• What is the study of statics?
• Compare and contrast static and dynamic equilibrium.
• What are the two major conditions for a body to be in equilibrium?

When an object is in equilibrium, the forces acting upon the object are balanced. That is, the net force on the object is zero. For this to occur, the object must either not be moving, or it must be moving at a constant velocity.

There are two types of equilibrium: static equilibrium and dynamic equilibrium.

Review an illustration of static equilibrium in Figure 9.3 and an illustration of dynamic equilibrium in Figure 9.4. An object in static equilibrium is completely motionless. An object in dynamic equilibrium is moving at constant velocity. The study of statics is the study of objects that are in equilibrium.

Two important conditions must be met for an object to be in equilibrium. First, the net force on the object must be zero. Secondly, a rotating object does not experience rotational acceleration. That is, a rotating object can be in equilibrium if its rotational velocity does not change.

8b. Define torque, remembering that it is a vector physical quantity

• Define torque. What is the unit for torque? 
• How is torque related to the second condition necessary for an object to be in equilibrium? 
• How does the direction of applied force impact torque?

We define torque as the effectiveness of a force to turn or twist an object, thus changing its rotational velocity. We can write the definition of torque as:

$\tau = rF\sin\theta$

where $\tau$ (the Greek letter tau) is torque, $r$ is radius of force, $F$ is magnitude, and $\theta$ is the angle between the force and the pivot point.

The unit for torque is the Newton meter (Nm).

We can restate the second condition for an object to be in equilibrium in terms of torque. In learning outcome 8a above, we said that an object in equilibrium must have no rotational acceleration. We can restate this by saying that an object in equilibrium must have a torque of zero.

Review a diagram showing the torque on a rotating plank of wood, secured at a pivot point at one end in Figure 9.6. This diagram shows how the direction of the force impacts the rotation of the plank of wood. When the force is perpendicular to the length of the plank of wood, the plank experiences torque and it rotates. When the force is parallel to the length of the plank of wood, it does not experience a net force and therefore does not experience torque or rotate. When the force is at an angle other than 90° from the length of the plank, the plank experiences less torque than if the force was at 90° from the plank's length.

8c. Solve statics problems

• Briefly outline the steps needed to solve statics problems.
• Solve statics problems to determine the force needed to support a weight.
• Solve statics problems to determine the torque needed to produce rotational motion for a given system.

When performing statics calculations, the first step is to determine if the system is in fact in equilibrium. Recall from learning outcome 8a that for a system to be in equilibrium, two conditions must be met: the system must not be accelerating and the torque must be zero. The second step is to draw a free body diagram of the system. It is important to determine all the forces acting upon the system. The third step is to solve the problem by applying the relevant conditions of equilibrium: force is zero, and torque is zero.

Review more details in Problem Solving Strategy: Static Equilibrium Situations.

Example 9.1: She Saw Torques on a Seesaw shows a worked example of a statics problem. Here, children are balanced on a seesaw. We are given information about the masses of both children, and how far from the pivot point one child is sitting. We are asked to determine where the second child is sitting to balance.

In Figure 9.8 we see that the children are balanced and therefore are at equilibrium. The free body diagram shows that there is no net force, and no net rotational acceleration. To determine the distance of the second child from the pivot point, we use the torque equation, and set torque equal to zero. To determine the upward balancing force from the pivot point, we use the fact that net force equals zero to solve for the individual force at the pivot point.

Example 9.2: What Force is Needed to Support a Weight Held Near its CG? shows a similar worked example of a statics problem. Here, a pole vaulter holds a pole at one end and we are asked to calculate the forces from each of the pole vaulter's hands. We take the same approach as in Example 9.1: She Saw Torques on a Seesaw.

Unit 8 Vocabulary

• Dynamic equilibrium
• Equilibrium
• Newton meter (Nm)
• Static equilibrium
• Statics
• Torque