Static Equilibrium; Elasticity & Fracture

The Conditions for Equilibrium

An object with forces acting on it that is not moving, is said to be in equilibrium.

!! The first condition for equilibrium is that the forces along each coordinate axis add to zero.

!! The second condition of equilibrium is that there be no torque around any axis; the choice of axis is arbitrary.

F! = 0

!" = 0

Statics is concerned with the calculation of the forces acting on and within structures that are in equilibrium.

Solving Statics Problems !! Choose one object at a time, and make a free-body diagram showing all the forces

on it and where they act

!! Choose a convenient coordinate system, and resolve the forces into their

components.

!! Write down the equilibrium equations for the forces

!! Write down the torque equilibrium equation. Choose any axis perpendicular to

the xy plane that might make the calculation easier. Pay careful attention to

determining the lever arm for each force correctly. Give each torque a + or ! sign

to indicate torque direction.

!! Solve these equations for the unknowns.

Problem(1) A board of mass M = 4.0 kg serves as a seesaw for two children. Child A has a mass of 30 kg and sits 2.5 m from the pivot point, P (his center of gravity is 2.5 m from the pivot). At what distance x from the pivot must child B, of mass 25 kg, place herself to balance the seesaw? Assume the board is uniform and centered over the pivot.

Stability and Balance An object in static equilibrium, if left undisturbed, will undergo no translational or rotational acceleration since the sum of all the forces and the sum of all the torques acting on it are zero. However, if the object is displaced slightly, three outcomes are possible: (1)! the object returns to its original position: stable equilibrium (2)! the object moves even farther from its original position: unstable equilibrium (3)! the object remains in its new position: neutral equilibrium.

Examples of Stability and Balance

Stable : the object returns to its original position

Unstable: the object moves even farther from its original position

Balance: the object readjusts its center of gravity

Problem (2)

A 1 kg ball is hung at the end of a 1 m long uniform rod. If the system balances at a point on the rod 0.25 m from the end holding the mass, what is the mass of the rod?

a)! 1/4 kg b)! 1/2 kg c)! 1 kg d)! 2 kg e)! 4 kg 1kg

1m

1 kg

X

CM of rod

same distance mrod = 1 kg

Elasticity; Stress and Strain A model of a common physical system on which the force varies with position is a spring. If the spring is either stretched or compressed a small distance, "#, from its unstretched (equilibrium) configuration, the force exerted on the spring can be written as:

This equation is called Hooke's law after Robert Hooke. The Hooke’s law is found to be valid for almost any solid material from iron to bone but it is valid only up to a point. If the force is too large, the object stretches excessively and eventually breaks.

F = k!l

Elasticity; Stress and Strain

The strain is the changes in length, "#, of a sample per unit of undistorted length, #, of the sample,

The stiffness of a rod (its ability to resist stretching) is also directly proportional to its cross-sectional area. Stress is defined to be the total force applied to an object divided by the cross-sectional area of the object.

Strain = change in lengthoriginal length

= !ll0

Stress = force area

= FA

!! Tensile stress: when sample is under tension (is being stretched)

!! Compressive stress: when a sample is under compression (is being squeezed)

!! Sheer stress: when a pair of parallel forces with opposite directions are applied to the surface of the object (It is the kind of stress applied by a pair of shears when it is cutting)

If we compare rods made of the same material but of different lengths and cross-sectional areas, it is found that for the same applied force, the change in length is proportional to the original length and inversely proportional to the cross-sectional area. So for a given force, the longer the object, the more it elongates and the thicker it is, the less it elongates.

!l " FAl0

or# $# !l = 1EFAl0

E is a constant of proportionality known as the elastic modulus, or Young's modulus; its value depends only on the material. The value of Young's modulus for various materials.

Elasticity; Stress and Strain

Young modulus ! Elastic modulus = StressStrain

= F A" l l0

#$# Y = E = FAl0"l

Fracture The ultimate strengths of materials under tensile stress, compressional stress, and shear stress have been measured. If the stress is too great, the object will fracture.

When designing a structure, it is a good idea to keep anticipated stresses less than 1/3 to 1/10 of the ultimate strength.

Problem (3)

A 1.60-m-long steel piano wire has a diameter of 0.20 cm. How great is the tension in the wire if it stretches 0.25 cm when tightened? Young’s modulus for steel is 200 x 109 (N/m2).