Chapter 12: Static Equilibrium and Elasticity Reading assignment: Chapter 12.1 to 12.4 Homework : (due Thursday, Nov. 8): Problems: QQ1, QQ2, QQ3, QQ4, QQ5, QQ6, QQx3, AE2, AE3, AE4, 1, 5, 16, 27, 32, 63 Objects in static equilibrium don’t move. Of special interest to civil and mechanical engineers and architects. We’ll also learn about elastic (reversible) deformations. Plastic deformations are irreversible (like play dough)

Objects in static equilibrium don’t move or rotate. Conditions for static equilibrium Objects in static equilibrium don’t move or rotate.

Conditions for static equilibrium (For extended objects) The net force acting on the particle must be zero. The net torque about any axis acting on the particle must be zero. The angular and linear speeds must be zero.

Is this object in static equilibrium? Conditions for static equilibrium (For extended objects) Is this object in static equilibrium? A force couple is acting on an object. A force couple is a pair of forces of equal magnitude and opposite direction along parallel lines of action

It matters at which point the force is applied!! If equal and opposite forces are applied at different points  object is not in equilibrium, since there is a net torque. If equal and opposite forces are applied at the same point or along the same axis  object is in equilibrium

Conditions for static equilibrium (For extended objects) The net force acting on the particle must be zero. The net torque about any axis acting on the particle must be zero. The angular and linear speeds must be zero. We restrict ourselves to forces in the x-y plane. Thus:

Center of gravity How do we treat the gravitational force? Consider an extended object. The gravitational force Mg always acts on the center of gravity! The center of gravity is equal to the center of mass (see Chapter 9).

Black board example 12.1 A carpenter's square has the shape of an L, where d1 = 18.0 cm, d2 = 4.00 cm, d3 = 4.00 cm, d4 = 12.0 cm. Locate its center of gravity. (Hint: Take (x, y) = (0, 0) at the intersection of d1 and d4).

Examples of rigid objects in static equilibrium We will only consider objects that are homogeneous, symmetric, and in a uniform gravitational field. Look at examples in book!

Balanced rock Black board example 12.2 For this system to be in static equilibrium, the center of gravity must be directly over the support point. Why??

Black board example 12.3 A soda can rests on a table in the galley of a ship. As the ship rocks in the waves, the can tilts as the table tilts. Assuming the can does not slide, is the soda can most stable against tipping when it is full, empty or half-full? (Hint: Where is the center of mass? High is less stable). (_) sf > sh > se (_) se > sh > sf (_) none of these (_) sh > se > sf (_) sf > se > sh                                                              

Problem-solving hints: Objects in static equilibrium Draw a free body diagram. Show and label all the external forces acting on the object. Indicate where the forces are applied. Establish a convenient coordinate system for forces. Then apply condition 1: Net force must equal zero. Establish a convenient coordinate system for torque. Then apply condition 2: Net torque must equal zero.

Black board example 12.4 A uniform 40.0 N board supports a father (800 N) and daughter (350 N) as shown. The support is under the center of gravity of the board and the father is 1.00 m from the center. Determine the magnitude of the upward force n exerted on the board by the support. Determine where the child should sit to balance the system.

Black board example 12.5 A uniform plank with a length of 6.00 m and a mass of 30.0 kg rests horizontally across two horizontal bars of a scaffold. The bars are 4.50 m apart and 1.50 m of the plank hangs over one side of the scaffold. Draw a free body diagram for the plank. How far can a painter with a mass of 70.0 kg walk on the overhanging part of the plank before it tips?

Elastic properties of solids We will consider three types of deformations and define an elastic modulus for each. Change in length. YOUNG’S MODULUS measures the resistance of a solid to a change in its length. Shearing. SHEAR MODULUS measures the resistance to shearing. Change in volume. BULK MODULUS measures the resistance to changes in volume.

Elastic properties of solids Definitions of stress and strain. Stress: Force per unit cross sectional area. Strain: Measure of the degree of deformation.

Elastic properties of solids Young’s modulus:

Elastic properties of solids Shear modulus:

Elastic properties of solids Bulk modulus: P… pressure

Black board example 12.6 A 200 kg load is hung on a wire with a length of 4.00 m, a cross-sectional area of 2.00·10-5 m and a Youngs modulus of 8.00·1010 N/m2. What is its increase in length?