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Stability

Module by: OpenStax College. E-mail the author

Summary:

  • State the types of equilibrium.
  • Describe stable and unstable equilibriums.
  • Describe neutral equilibrium.

It is one thing to have a system in equilibrium; it is quite another for it to be stable. The toy doll perched on the man’s hand in Figure 1, for example, is not in stable equilibrium. There are three types of equilibrium: stable, unstable, and neutral. Figures throughout this module illustrate various examples.

Figure 1 presents a balanced system, such as the toy doll on the man’s hand, which has its center of gravity (cg) directly over the pivot, so that the torque of the total weight is zero. This is equivalent to having the torques of the individual parts balanced about the pivot point, in this case the hand. The cgs of the arms, legs, head, and torso are labeled with smaller type.

Figure 1: A man balances a toy doll on one hand.
In the figure a man is shown balancing a child on his hand. The child is enjoying the activity.

A system is said to be in stable equilibrium if, when displaced from equilibrium, it experiences a net force or torque in a direction opposite to the direction of the displacement. For example, a marble at the bottom of a bowl will experience a restoring force when displaced from its equilibrium position. This force moves it back toward the equilibrium position. Most systems are in stable equilibrium, especially for small displacements. For another example of stable equilibrium, see the pencil in Figure 2.

Figure 2: This pencil is in the condition of equilibrium. The net force on the pencil is zero and the total torque about any pivot is zero.
A pencil is balanced vertically on its flat end. The weight W of the pencil is acting at its center of gravity downward. The normal reaction N of the surface is shown as an arrow upward. A free body diagram is shown at right of the pencil. The midpoint of the flat base of the pencil is marked as pivot point.

A system is in unstable equilibrium if, when displaced, it experiences a net force or torque in the same direction as the displacement from equilibrium. A system in unstable equilibrium accelerates away from its equilibrium position if displaced even slightly. An obvious example is a ball resting on top of a hill. Once displaced, it accelerates away from the crest. See the next several figures for examples of unstable equilibrium.

Figure 3: If the pencil is displaced slightly to the side (counterclockwise), it is no longer in equilibrium. Its weight produces a clockwise torque that returns the pencil to its equilibrium position.
A pencil is tilted slightly toward left. The left end point of its flat surface is marked as the pivot point. The weight W of the pencil is acting at the center of gravity of the pencil. The normal reaction N of the pencil is acting upward at the pivot point. The line of action of the normal reaction is toward left of the line of action of the weight of the pencil.

Figure 4: If the pencil is displaced too far, the torque caused by its weight changes direction to counterclockwise and causes the displacement to increase.
A pencil is tilted toward left so that the line of action of its weight is toward left of the pivot point which is the left end of the flat end of the pencil.

Figure 5: This figure shows unstable equilibrium, although both conditions for equilibrium are satisfied.
A vertical pencil balanced at its sharp end is shown. The weight of the pencil is acting at its center of gravity and is in the line with the normal reaction N at the pivot point of the pencil.

Figure 6: If the pencil is displaced even slightly, a torque is created by its weight that is in the same direction as the displacement, causing the displacement to increase.
A vertical pencil tilted toward left is shown. The sharp end of the pencil is down and labeled as pivot point. The weight of the pencil is acting at its center of gravity and the line of action of the weight is toward left of the pivot point.

A system is in neutral equilibrium if its equilibrium is independent of displacements from its original position. A marble on a flat horizontal surface is an example. Combinations of these situations are possible. For example, a marble on a saddle is stable for displacements toward the front or back of the saddle and unstable for displacements to the side. Figure 7 shows another example of neutral equilibrium.

Figure 7: (a) Here we see neutral equilibrium. The cg of a sphere on a flat surface lies directly above the point of support, independent of the position on the surface. The sphere is therefore in equilibrium in any location, and if displaced, it will remain put. (b) Because it has a circular cross section, the pencil is in neutral equilibrium for displacements perpendicular to its length.
In figure a,  a ball is lying on a flat surface and the point of contact with the surface is labeled pivot point. The weight of the ball is acting at the center of gravity of the ball. The normal force N is in the same line as the weight of the ball. The torque on the ball is zero. In figure b, a side view of a pencil lying flat on a table is shown. The sharp end of the pencil is toward right. The weight of the pencil is acting at the center of gravity of the pencil. The normal reaction N of the table surface is in the same line of action as the weight but in the upward direction.

When we consider how far a system in stable equilibrium can be displaced before it becomes unstable, we find that some systems in stable equilibrium are more stable than others. The pencil in Figure 2 and the person in Figure 8(a) are in stable equilibrium, but become unstable for relatively small displacements to the side. The critical point is reached when the cg is no longer above the base of support. Additionally, since the cg of a person’s body is above the pivots in the hips, displacements must be quickly controlled. This control is a central nervous system function that is developed when we learn to hold our bodies erect as infants. For increased stability while standing, the feet should be spread apart, giving a larger base of support. Stability is also increased by lowering one’s center of gravity by bending the knees, as when a football player prepares to receive a ball or braces themselves for a tackle. A cane, a crutch, or a walker increases the stability of the user, even more as the base of support widens. Usually, the cg of a female is lower (closer to the ground) than a male. Young children have their center of gravity between their shoulders, which increases the challenge of learning to walk.

Figure 8: (a) The center of gravity of an adult is above the hip joints (one of the main pivots in the body) and lies between two narrowly-separated feet. Like a pencil standing on its eraser, this person is in stable equilibrium in relation to sideways displacements, but relatively small displacements take his cg outside the base of support and make him unstable. Humans are less stable relative to forward and backward displacements because the feet are not very long. Muscles are used extensively to balance the body in the front-to-back direction. (b) While bending in the manner shown, stability is increased by lowering the center of gravity. Stability is also increased if the base is expanded by placing the feet farther apart.
Part a of the figure shows a man standing on the ground. The feet are a shoulder-width apart from each other. The weight W of the man is acting at the center of gravity of the body of the man. Two normal reactions N each are shown acting on the feet of the man. The distance between the feet of the man is marked as the base of support. A free body diagram is also shown on the left side of the figure.  Part b of the figure shows a man standing upright with his knees bent. The feet are a distance apart from each other. The weight W of the man is acting at the center of gravity of the body of the man. Two normal reactions N each are shown acting on the feet of the man. The distance between the feet of the man is marked as the base of support.

Animals such as chickens have easier systems to control. Figure 9 shows that the cg of a chicken lies below its hip joints and between its widely separated and broad feet. Even relatively large displacements of the chicken’s cg are stable and result in restoring forces and torques that return the cg to its equilibrium position with little effort on the chicken’s part. Not all birds are like chickens, of course. Some birds, such as the flamingo, have balance systems that are almost as sophisticated as that of humans.

Figure 9 shows that the cg of a chicken is below the hip joints and lies above a broad base of support formed by widely-separated and large feet. Hence, the chicken is in very stable equilibrium, since a relatively large displacement is needed to render it unstable. The body of the chicken is supported from above by the hips and acts as a pendulum between the hips. Therefore, the chicken is stable for front-to-back displacements as well as for side-to-side displacements.

Figure 9: The center of gravity of a chicken is below the hip joints. The chicken is in stable equilibrium. The body of the chicken is supported from above by the hips and acts as a pendulum between them.
A chicken is shown standing on the ground. The weight of the chicken is acting at the center of gravity of the chicken’s body. The distance between the feet of the chicken is labeled as base of support. The normal forces N each are acting at the feet of the chicken. A free body diagram is shown at the right side of the figure.

Engineers and architects strive to achieve extremely stable equilibriums for buildings and other systems that must withstand wind, earthquakes, and other forces that displace them from equilibrium. Although the examples in this section emphasize gravitational forces, the basic conditions for equilibrium are the same for all types of forces. The net external force must be zero, and the net torque must also be zero.

Take-Home Experiment:

Stand straight with your heels, back, and head against a wall. Bend forward from your waist, keeping your heels and bottom against the wall, to touch your toes. Can you do this without toppling over? Explain why and what you need to do to be able to touch your toes without losing your balance. Is it easier for a woman to do this?

Section Summary

  • A system is said to be in stable equilibrium if, when displaced from equilibrium, it experiences a net force or torque in a direction opposite the direction of the displacement.
  • A system is in unstable equilibrium if, when displaced from equilibrium, it experiences a net force or torque in the same direction as the displacement from equilibrium.
  • A system is in neutral equilibrium if its equilibrium is independent of displacements from its original position.

Conceptual Questions

Exercise 1

A round pencil lying on its side as in Figure 4 is in neutral equilibrium relative to displacements perpendicular to its length. What is its stability relative to displacements parallel to its length?

Exercise 2

Explain the need for tall towers on a suspension bridge to ensure stable equilibrium.

Problems & Exercises

Exercise 1

Suppose a horse leans against a wall as in Figure 10. Calculate the force exerted on the wall assuming that force is horizontal while using the data in the schematic representation of the situation. Note that the force exerted on the wall is equal in magnitude and opposite in direction to the force exerted on the horse, keeping it in equilibrium. The total mass of the horse and rider is 500 kg. Take the data to be accurate to three digits.

Figure 10
In part a, a horse is standing next to a wall with its legs crossed. A sleepy-looking rider is leaning against the wall. Part b is a drawing of the same horse from a rear view, but this time with no rider.  The horse is crossing its rear legs, and its rump is leaning against the wall. The reaction of the wall F is acting on the horse at a height one point two meters above the ground. The weight of the horse is acting at its center of gravity near the base of the tail. The center of gravity is one point four meters above the ground. The line of action of weight is zero point three five meters away from the feet of the horse.

Solution

F wall = 1.43 × 10 3 N F wall = 1.43 × 10 3 N size 12{F rSub { size 8{"wall"} } =1 cdot "43"` times `"10" rSup { size 8{3} } `N} {}

Exercise 2

Two children of mass 20 kg and 30 kg sit balanced on a seesaw with the pivot point located at the center of the seesaw. If the children are separated by a distance of 3 m, at what distance from the pivot point is the small child sitting in order to maintain the balance?

Exercise 3

(a) Calculate the magnitude and direction of the force on each foot of the horse in Figure 10 (two are on the ground), assuming the center of mass of the horse is midway between the feet. The total mass of the horse and rider is 500kg. (b) What is the minimum coefficient of friction between the hooves and ground? Note that the force exerted by the wall is horizontal.

Solution

a) 2.55×103 N, 16.3º to the left of vertical (i.e., toward the wall)2.55×103 N, 16.3º to the left of vertical (i.e., toward the wall) size 12{θ} {}

b) 0.292

Exercise 4

A person carries a plank of wood 2 m long with one hand pushing down on it at one end with a force F1F1 size 12{F rSub { size 8{1} } } {} and the other hand holding it up at 50 cm from the end of the plank with force F2F2 size 12{F rSub { size 8{2} } } {}. If the plank has a mass of 20 kg and its center of gravity is at the middle of the plank, what are the magnitudes of the forces F1F1 and F2F2 size 12{F rSub { size 8{2} } } {}?

Exercise 5

A 17.0-m-high and 11.0-m-long wall under construction and its bracing are shown in Figure 11. The wall is in stable equilibrium without the bracing but can pivot at its base. Calculate the force exerted by each of the 10 braces if a strong wind exerts a horizontal force of 650 N on each square meter of the wall. Assume that the net force from the wind acts at a height halfway up the wall and that all braces exert equal forces parallel to their lengths. Neglect the thickness of the wall.

Figure 11
A seventeen meter high wall is standing on the ground with ten braces to support it. At the base of the figure a brown colored ground is visible. Only one brace is visible from a side. A brace makes an angle of thirty five degree with the wall. The point of contact of the brace is eight point five meters high. You have to find the force exerted by this brace on the wall to support.

Solution

F B = 2.12 × 10 4 N F B = 2.12 × 10 4 N size 12{F rSub { size 8{B} } =2 cdot "12" times "10" rSup { size 8{4} } `N} {}

Exercise 6

(a) What force must be exerted by the wind to support a 2.50-kg chicken in the position shown in Figure 12? (b) What is the ratio of this force to the chicken’s weight? (c) Does this support the contention that the chicken has a relatively stable construction?

Figure 12
A chicken is trying to balance on its left foot, which is 9 point zero centimeters to the right of the chicken. The force of the wind is blowing from the left toward the chicken’s center of gravity c g, which is 20 cm above the ground. The weight of the chicken w is acting at the center of gravity.

Exercise 7

Suppose the weight of the drawbridge in Figure 13 is supported entirely by its hinges and the opposite shore, so that its cables are slack. (a) What fraction of the weight is supported by the opposite shore if the point of support is directly beneath the cable attachments? (b) What is the direction and magnitude of the force the hinges exert on the bridge under these circumstances? The mass of the bridge is 2500 kg.

Figure 13: A small drawbridge, showing the forces on the hinges (FF size 12{F} {}), its weight (ww size 12{w} {}), and the tension in its wires (TT size 12{T} {}).
A small drawbridge is shown. There is one vertical and one horizontal wooden plank. The left end of the horizontal plank is attached to the vertical plank near its middle. At the point of contact, a hinge is shown. A wire is tied to the right end of the horizontal end, is passed over the top of the vertical plank and is connected to a pulley. The angle made by the wire with the horizontal plank is forty degrees. The reaction F at the hinge is inclined at an angle theta.

Solution

a) 0.167, or about one-sixth of the weight is supported by the opposite shore.

b) F=2.0×104NF=2.0×104N size 12{F=2 "." 0 times "10" rSup { size 8{4} } N} {}, straight up.

Exercise 8

Suppose a 900-kg car is on the bridge in Figure 13 with its center of mass halfway between the hinges and the cable attachments. (The bridge is supported by the cables and hinges only.) (a) Find the force in the cables. (b) Find the direction and magnitude of the force exerted by the hinges on the bridge.

Exercise 9

A sandwich board advertising sign is constructed as shown in Figure 14. The sign’s mass is 8.00 kg. (a) Calculate the tension in the chain assuming no friction between the legs and the sidewalk. (b) What force is exerted by each side on the hinge?

Figure 14: A sandwich board advertising sign demonstrates tension.
A sandwich board advertising sign is in form of a triangle. The base of the triangle is one point one zero meters. The other two sides are connected with a hinge at the top. A horizontal chain is connected to the two legs at zero point five zero meters below the hinge. The height of the hinge above the base is one point three zero meters. The centers of the gravity of the two legs are shown at their midpoints. The figure is labeled at uniform board with c g at the center.

Solution

a) 21.6 N

b) 21.6 N

Exercise 10

(a) What minimum coefficient of friction is needed between the legs and the ground to keep the sign in Figure 14 in the position shown if the chain breaks? (b) What force is exerted by each side on the hinge?

Exercise 11

A gymnast is attempting to perform splits. From the information given in Figure 15, calculate the magnitude and direction of the force exerted on each foot by the floor.

Figure 15: A gymnast performs full split. The center of gravity and the various distances from it are shown.
A gymnast with two pompoms in her hands is shown. One of the hand is horizontal toward left and the other is vertical. The gymnast is attempting to perform a full split. The span of her legs is one point eight meters, and the distance of one foot from the center of gravity is zero point nine meters. The weight of the girl is labeled as seven hundred newtons. The vertical distance of one foot from the center of gravity is zero point three zero meter.

Solution

350 N directly upwards

Glossary

neutral equilibrium:
a state of equilibrium that is independent of a system’s displacements from its original position
stable equilibrium:
a system, when displaced, experiences a net force or torque in a direction opposite to the direction of the displacement
unstable equilibrium:
a system, when displaced, experiences a net force or torque in the same direction as the displacement from equilibrium

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