Physics 125, Fall 1995 -- Lecture 12: Static Equilibrium

© Timothy E. Chupp, 1995

12.1 Equilibrium

In our experience, any object at rest is nevertheless subject to a combination (or sum) of forces that balance. (Implicit in our considerations of "objects" up to now has been the simplifications that igonre the extent of the object, the exact position where the force is applied, and others that will be considered here.) This balance of forces is often true only at a special position of the object. Consider a our mass suspended from a cable, we'll call it Bob. The forces on Bob balance only when Bob is "at rest" and the cable is vertical. Only when the cable is vertical is the force of the cable on Bob exactly opposite the force of gravity on Bob. What happens when Bob is slightly displaced, so that the cable makes a small angle with the vertical? Well in order to maintain Bob at that displacement, an additional force must be applied to balance the forces. When the force is removed, there is an imbalance of forces on Bob, and it accelerates in a direction that will return it to the equilibrium position. However the momentum of Bob, when it reaches the equilibrium position, carries it past vertical to a slight displacement on the other side. It eventually reverses direction and the swings back etc. These are the oscillations of a pendulum about its equilibrium position. The equilibrium position is that where the forces on Bob are balanced.

In this example of Bob, the slight displacement from equilibrium leads to a force that pushes the object back toward its equilibrium position. (Note that the force is not directly toward the equilibrium position. See Figure 1.) This kind of equilibrium is called stable equilibrium because 1) the forces are balanced and 2) small displacements lead to restoring forces.

Figure 1: Bob suspended by a cable at its equilibrium position and displaced by and angle . The restoring forces push Bob toward the equilibrium position which is stable.

As another example, consider Bob supported by two pieces of wood connected at the appex of a triangle. It is in principle possible, if difficult, to "balance" Bob in this position. Once again the forces are balanced, however in this case a small displacement in either direction leads to a force that pushes Bob away from the equilbrium point. This is called unstable equilibrium.

Figure 2: Bob supported by a triangle. The restoring forces push Bob away from the equilibrium position which is unstable.

These two examples both ignore the possibility that Bob can rotate without moving. To set Bob rotating without moving while at the end of the cable or at the apex of the triangle requires an additional pair of forces that are in opposite directions with equal magnitudes, BUT do not act along the same line. Such a pair of forces exerts a net torque, a vector quantity that will be defined below. For the case of Bob hanging from the cable, the nature of the cable resists the rotation and a restoring torque acts to rotate Bob toward the equilibrium rotational position.

Figure 3: The forces and are balanced, and the forces of gravity and the cable tension (into and out of the page) are also balanced. However, since and are not colinear, a net torque arises that can rotate Bob away from the equilibrium rotational position.

It is therefore most accurate to describe an object in equilibrium, not at rest. In equilibrium, the forces applied to the object are balanced and the torques applied to the object are balanced. For stable equilibrium, the restoring forces and/or torques always push the object toward the equilibrium position. For unstable equilibrium, the forces and/or torques push the object away from the equilibrium position.

A final example is an object that will be neither restored toward nor pushed away from an equilibrium position, such as a puck on an air table. This is sometimes called a position of neutral equilibrium because a displacement causes no change in the balance of forces and torques.

12.2 Torque

We will encounter many examples for which the forces balance, but are not colinear. Consider a see saw balancing two people of very different mass (for example Prof. Chupp (70 kg) and Theo (10 kg)). We'll be as accurate as possible here, so let's note that the see-saw itself has a mass of 20 kg, 10 kg on each side of the fulcrum. The figure shows how the forces on the see-saw balance: the masses push downward and the fulcrum pushes up.

Figure 4: The forces on a seesaw due to 70 kg and 10 kg masses, the mass of the seesaw and the fulcrum.

You know well that Theo can balance me, but only if we are different distances from the fulcrum. The reason is that the torque caused by the force of Theo must equal the torque caused by me. Torque depends on both the force and the position at which the torque is applied, called the position of the line of force. Both force and torque are vectors. An imbalance of torques leads to a change of rotational motion around an axis that is perpendicular to the force and to the displacement of the line of force from the axis of rotation. Changes of rotational motion caused by torque will be discussed more later. For now, we'll consider only cases for which the torques balance. The torque about an axis caused by a force displaced from that axis by is

This is a new kind of vector equation that combines two vectors into a third vector in three dimensions. Such a product of vectors is called the cross product. The magnitude of the cross product of and is

Where is the component of perpendicular to , that is , where is the angle between and . So

The direction of is along the axis around which the object would rotate that is perpendicular to the plane containing and . The rule is this: consider the direction the object at rest would begin to rotate and curl the fingers of the right hand in that direction. The thumb points in the direction of . This is called the right handed convention or the right hand rule for defining the direction the result of a vector cross product.

12.3 An object in equilibrium

The seesaw with me and Theo is in equilbrium as long as the forces and torques balance. Here's how it works:

a.) = 0

The forces are due to the force of gravity on each half of the seesaw (10 kg 9.8 N/kg each), the downward force of me on the left side (70 kg 9.8 N/kg = 686 N), the downward force of Theo (10 kg 9.8 N/kg and the upward force on the seesaw by the fulcrum ():

b.) = 0

The torques are due to the force of gravity on each half of the see saw which is directed at the center of mass of each half, and the torques due to me and due to Theo. Note that the direction of the torque of the right side of the seesaw and of Theo would rotate the seesaw clock-wise and the torque due to me and the left hand side of the seesaw would rotate the seesaw counter clock-wise. Thus the torques due to Theo () and me ( are opposite (by the right hand convention: is into the page and is out of the page). The torques along an axis perpendicular to the page and passing through the point of intersection of the seesaw and fulcrum balance as follows (notice that and are perpendicular for each force so ):

Everywhere you look, you will see objects in equilibrium. Hold your arm out. Sit, leaning back, in a chair. The awning of a cafe. Everywhere! Practice looking and analyzing the balance of forces and torques in things you see around you.



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