chapter 12 static equilibrium

Chapter 12

Static Equilibrium

Equilibrium

• We already introduced the concept of equilibrium in Chapter 7: dU(x)/dx = 0

• More general definition of equilibrium:

• Static equilibrium:

• Stable equilibrium: the body returns to the state of static equilibrium after having been displaced from that state. Unstable equilibrium: the state of equilibrium is lost after a small force displaces the body

constLconstP

;

0;0 LP

Equilibrium

• Static equilibrium:

• Stable equilibrium: the body returns to the state of static equilibrium after having been displaced from that state. Unstable equilibrium: the state of equilibrium is lost after a small force displaces the body

0;0 LP

Equilibrium

• Stable equilibrium:

• Unstable equilibrium:

02

2

dx

Ud

02

2

dx

Ud

Chapter 12Problem 25

A particle’s potential energy as a function of position is given by U = 2x3 – 2x2 – 7x + 10,with x in meters and U in joules. Find the positions of any stable and unstable equilibria.

Center of mass: stable equilibrium

• We consider the torque created by the gravity force (applied to the CM) and its direction relative to the possible point(s) of rotation

Center of mass: stable equilibrium

• We consider the torque created by the gravity force (applied to the CM) and its direction relative to the possible point(s) of rotation

Center of mass: stable equilibrium

• We consider the torque created by the gravity force (applied to the CM) and its direction relative to the possible point(s) of rotation

Center of mass: stable equilibrium

• We consider the torque created by the gravity force (applied to the CM) and its direction relative to the possible point(s) of rotation

The requirements of equilibrium

• For an object to be in equilibrium, we should have two requirements met

• Balance of forces: the vector sum of all the external forces that act on the body is zero

• Balance of torques: the vector sum of all the external torques that act on the body, measured about any possible point, is zero

0 ; netFdt

PdconstP

0 ; netdt

LdconstL

Equilibrium: 2D case

• If an object can move only in 2D (xy plane) then the equilibrium requirements are simplified:

• Balance of forces: only the x- and y-components are considered

• Balance of torques: only the z-component is

considered (the only one perpendicular to the xy plane)

0;0 ,, ynetxnet FF

0, znet

Examples of static equilibrium

Examples of static equilibrium

Chapter 12Problem 38

A crane in a marble quarry is mounted on the quarry’s rock walls and is supporting a 2500-kg marble slab as shown in the figure. The center of mass of the 830-kg boom is located one-third of the way from the pivot end of its 15-m length, as shown. Find the tension in the horizontal cable that supports the boom.

Indeterminate structures

• Indeterminate systems cannot be solved by a simple application of the equilibrium conditions

• In reality, physical objects are not absolutely rigid bodies

• Concept of elasticity is employed

Questions?

Answers to the even-numbered problems

Chapter 12

Problem 24:

(a) 47 m from the origin (b) unstable

Answers to the even-numbered problems

Chapter 12

Problem 36:

1.2 m

Answers to the even-numbered problems

Chapter 12

Problem 54:

L/6 of the bottom book can overhangthe desk