9.3 Stability

9.3 Stability

• Torque has an importance beyond static equilibrium.
• Torque is the same as force in linear motion.
• We will look at this in the next chapter.

• Take a piece of modeling clay and put it on a table, then mash a cylinder down into it so that a ruler can balance on the round side of the cylinder.
• A penny is 8 cm away from the pivot.

• One thing to have a system in equilibrium is another thing to be stable.
• Stable, unstable, and neutral are the types of equilibrium.
• The figures show various examples.

• The toy doll on the man's hand has its center of gravity directly over the pivot so that the total weight is zero.
• This is similar to balancing the individual parts about the pivot point.
• The arms, legs, head, and torso are labeled with smaller type.

• A man is balancing a toy on one hand.

• When a marble is displaced from its equilibrium position, it will experience a restoring force.
• The force moves it back to the equilibrium position.
• Stable equilibrium is the case for most systems.
• The pencil in is an example of stable equilibrium.

• The pencil is in equilibrium.
• The total force on the pencil is zero and the total force on the pivot is zero.

• If a system is displaced even slightly, it will move away from its equilibrium position.
• A ball rests on top of a hill.
• It goes away from the crest after being displaced.
• See the next figures for unstable equilibrium.

• The pencil is no longer in equilibrium if it is moved to the side.
• The pencil is returned to its equilibrium position by its weight.

• If the pencil is displaced too far, the weight of the pencil causes it to change direction and cause the displacement to increase.

• Both conditions for equilibrium are satisfied in this figure.

• The displacement of the pencil causes a Torque to be created by its weight that is in the same direction as the displacement.

• A marble on a horizontal surface is an example.
• There are combinations of these situations that are possible.
• A marble on a saddle is stable for either the front or back of the saddle, and unstable for the side.

• The position on the surface does not affect the cg of the sphere above the point of support.
• If the sphere is displaced, it will remain put.

• Some systems in stable equilibrium are more stable than others.
• The critical point is when the cg is no longer above the base of support.
• The pivots in the hips must be quickly controlled since the cg of the person's body is above them.
• When we learn to hold our bodies erect as infants, this control is developed.
• The feet should be spread apart to give a larger base of support.
• When a football player prepares to receive a ball or braces themselves for a tackle, the center of gravity is lowered by bending the knees.
• As the base of support widens, a cane, a crutch, or a walker increases the stability of the user.
• The cg of a female is usually lower than that of a male.
• The challenge of learning to walk is increased by the fact that young children have their center of gravity between their shoulders.

• The person is stable in relation to sideways displacements, but small displacements make him unstable, like a pencil on its eraser.
• Humans are less stable because their feet are not long.
• The muscles are used to balance the body.
• If the base is expanded the stability will be increased.

• Chickens have easier systems to control.
• 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.
• The balance systems of some birds are similar to those of humans.

• Figure 9.17 shows that the cg of a chicken is below the hip joints and lies above a broad base of support.
• The chicken is stable since a large displacement is needed to make it unstable.
• The chicken's body is supported by the hips and acts as a pendulum between them.

• The chicken is stable for both front-to-back and side-to-side displacements.

• The hip joints of a chicken are below the center of gravity.
• The chicken is stable.
• The chicken's body is supported by the hips and acts as a pendulum between them.

• Engineers and architects strive to achieve extremely stable equilibriums for buildings and other systems that must endure wind, earthquakes, and other forces.
• The basic conditions for equilibrium are the same for all types of forces.
• The net external force and net Torque must be zero.

• Raising a drawbridge, bad posture, and back strain are some of the situations that statics can be applied to.
• There are problem-solving strategies used for statics.
• The general problem-solving strategies and the special strategies forNewton's laws still apply since statics is a special case.

• When the system's acceleration is zero and rotation does not occur, this is the case.

• It's important to draw a free body diagram for the system of interest.
• Whenever these are known, label all forces and note their relative magnitudes, directions, and points of application.

• Pick the pivot point to simplify the solution if the second condition is involved.
• The most useful pivot points cause unknown forces to be zero.
• It's always a good idea to choose a convenient coordinate system.

• To find out if the solution is reasonable, look at the magnitude, direction, and units of the answer.
• It is more difficult to judge reasonableness in unfamiliar applications, but the importance of this last step never decreases.
• Judgements become easier with experience.

• The pole vaulter is shown in the three figures below.
• The pole has a mass of 5000 grams.
• It seems reasonable that the force exerted by each hand is equal to half the weight of the pole.
• The second condition is also satisfied, as we can see by the choice of the pivot point.
• The lever arm of the weight is zero so it doesn't exert any Torque at the pivot point.
• The hands exert equal and opposite forces because they are equidistant from the pivot.
• Similar arguments exist for other systems where the supporting forces are symmetrical.

• Each hand exerts a force equal to half the weight of the pole.
• If the pole is held to the left of the person, he must push down with his right hand and up with his left.
• The forces he exerts are larger here because they are in opposite directions.

• Similar observations can be made using a meter stick.

• A pole vaulter is holding a pole with both hands.

• A pole vaulter is holding a pole with both hands.
• His right hand is near the center of gravity.

• A pole vaulter is holding a pole with both hands.
• The left side of the vaulter is the center of gravity.

• The force he exerts is equal to the weight of the pole, but it is not evenly divided between his hands.
• The right hand should support more weight since it is closer to the cg.
• All the weight will be supported if the right hand is moved directly under the cg.
• The situation is similar to two people carrying a load and one carrying more of it.

• The direction of the force applied by the right hand of the vaulter reverses its direction if the pole vaulter holds the pole from near the end of the pole.

• The pole's cg is from the left hand to the right.

• Figure 9.19 has a free body diagram for the pole.
• There is not enough information to use the first condition for equilibrium, since two of the three forces are unknown and the hand forces cannot be assumed to be equal in this case.
• If the pivot point is chosen to be at either hand, the second condition for equilibrium will be used.
• We decided to locate the pivot at the left hand in this part of the problem, to eliminate the Torque from the left hand.

• The push or pull of the right hand is the only nonzero Torque.

• The weight of the pole creates a counterclockwise Torque and the right hand Counters with a clockwise Torque.

• The first condition for equilibrium is based on the free body diagram.

• One force reverses direction.

• This is an experiment to perform in public.
• Stand upright.

• There are objects on a teeter totter.
• Try the Balance Challenge game to see what you've learned.