Chapter 8: Rotational Equilibrium and Dynamics

8-1: Torque

Torque: a quantity that measures the ability of a force to rotate an object around some axis.
Torque can be measured in the situation of a hinged door where the door is free to rotate around a line that passes through the hinge. This is an axis of rotation. The perpendicular distance from the axis of rotation to a line drawn along the direction of the force is called the lever arm.
· Torque depends on a force and a lever arm. It varies on the strength of the force and the area of which it is exerted. The farther the force is from the axis of rotation, the easier it is to rotate the object and the more torque is produced.
· Torque depends on the angle between a force and a lever arm. Force does not have to be perpendicular to rotate an object. It can be exerted at other angles, but doing so makes rotating it more difficult.

F is the force exterted on the rotating object, d is the distance, or the lever arm, and θ is the angle from which the force is exerted. d(sinθ) is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force, so it is the lever arm.
Torque is a vector quantity but we measure it as a scalar. All forces and displacements are assigned a positive or negative sign according to the sign convention. If there are multipletorques working on an object being rotated, the net torque can tell you which way it rotates.

8-2: Rotation and Inertia

Center of Mass: the point at which all the mass of the body can be considered to be concentrated when analyzing translational motion. This means if the object is rotating, it will rotate around the center of mass.
Center of Gravity: the position at which the gravitational force acts on the extended object as if it were a point mass. In many cases, center of mass and center of gravity are equivalent.
Moment of Inertia: the tendency of a body rotating about a fixed axis to resist a change in rotational motion. It is similar to mass because they are both forms of inertia. Mass resists change in translational motion and moment of inertia resists change in rotational motion.
· Calculating the moment of inertia depends on the type, or shape, of the object, as well as its mass and area. There are differing equations for different shapes. Here are some examples.

Moment of Intertia
Point mass at a distance r from the axis

Solid disk or cylinder of radius r about the axis

Solid sphere of radius r about its diameter

Thin spherical shell of radius r about its diameter

Thin hoop of radius r about the axis

Thin hoop of radius r about the diameter

Thin rod of length l about its center

Thin rod of length l about its end

Translational equilibrium: when there is no net force acting on an object.
Rotational equilibrium: when there is no net torque acting on the object.

8-3: Rotational Dynamics

The net force on an object is related to the angular acceleration given to the object, according to Newton's second law. Thus:

This rotational formula correlates with this translational formula:

Because a rotating object has intertia, it also has momentum associated with its rotation, called angular moment (L).

The translational counterpart to this equation is:

Law of Conservation of Angular Momentum: when the net external torque acting on an object is zero, the angular momentum of the object does not change.

Rotating objects have rotational kinetic energy, the energy of an object due to its rotational motion. This is associated with the angular speed, as shown in this equation:

This is analagous to the translational kinetic energy of a partical, shown below:

Mechanical energy may be conserved because gravity is the only external force acting on the objects. This mechanical energy includes the translational and rotational kinetic energies as well as the gravitational potential energy:

8-4: Simple Machines

Machines: devices used to make your task easier.
Simple Machines: fundamental types of machines that are used in combination to create all machines. These include the lever, inclined plane, wheel and axel, wedge, pulley, and screw.
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Mechanical Advantage (MA): having a greater output than input force, the goal of machines. This number is unitless.

output force/input force

Fout/Fin = din/dout
A machine can increase (or decrease) the force acting on an object athte expense (or gain) of the distance moved, but the product of the two--the work done on the object--is constant.

Efficiency: a measure of how much input energy is lost compared with how much enegery is used to perform work on an object. It is the ratio of work done by the machine to work put in to the machine:
eff = Wout/Win
In a frictionless machine, the amount of input work compared to the amount of output work would be equal, making the efficiency 1, or 100%. This is an ideal and unlikely situation, so the efficiency of real machine is less than one.

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