# 5: Torque and Angular Momentum

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How can we guarantee that a body is in equilibrium and what can we learn from systems that are in equilibrium? There are actually two conditions that must be satisfied to achieve equilibrium. These conditions are the topics of the first two sections of this chapter.

• 5.1: Prelude to Statics and Torque
Statics is the study of forces in equilibrium, a large group of situations that makes up a special case of Newton’s second law. We have already considered a few such situations; in this chapter, we cover the topic more thoroughly, including consideration of such possible effects as the rotation and deformation of an object by the forces acting on it.
• 5.2: The First Condition for Equilibrium
The first condition necessary to achieve equilibrium is the one already mentioned: the net external force on the system must be zero.
• 5.3: The Second Condition for Equilibrium
The second condition necessary to achieve equilibrium involves avoiding accelerated rotation (maintaining a constant angular velocity. A rotating body or system can be in equilibrium if its rate of rotation is constant and remains unchanged by the forces acting on it. To understand what factors affect rotation, let us think about what happens when you open an ordinary door by rotating it on its hinges.
• 5.4: Simple Machines
Simple machines are devices that can be used to multiply or augment a force that we apply – often at the expense of a distance through which we apply the force. Levers, gears, pulleys, wedges, and screws are some examples of machines. Energy is still conserved for these devices because a machine cannot do more work than the energy put into it. Machines can reduce the input force that is needed to perform the job. The ratio of output to input force magnitudes is called its mechanical advantage.
• 5.5: Forces and Torques in Muscles and Joints
Muscles, bones, and joints are some of the most interesting applications of statics. There are some surprises. Muscles, for example, exert far greater forces than we might think. Figure shows a forearm holding a book and a schematic diagram of an analogous lever system. The schematic is a good approximation for the forearm, which looks more complicated than it is, and we can get some insight into the way typical muscle systems function by analyzing it.
• 5.6: Prelude to Rotational Motion and Angular Momentum
Why do tornadoes spin at all? And why do tornados spin so rapidly? The answer is that air masses that produce tornadoes are themselves rotating, and when the radii of the air masses decrease, their rate of rotation increases. An ice skater increases her spin in an exactly analogous manner. The skater starts her rotation with outstretched limbs and increases her spin by pulling them in toward her body. The same physics describes the exhilarating spin of a skater and the wrenching force of a torna
• 5.7: Angular Momentum and Its Conservation
Angular momentum is completely analogous to linear momentum. It has the same implications in terms of carrying rotation forward, and it is conserved when the net external torque is zero. Angular momentum, like linear momentum, is also a property of the atoms and subatomic particles.

Thumbnails: Relationship between force (F), torque(τ), momentum (p), and angular momentum (L) vectors in a rotating system. (r) is the radius. (Public domain; Yawe).