ROTATIONAL EQUILIBRIUM AND DYNAMICS

Section 8-1: Torque

Like a net external force acting on an object produces linear acceleration, a net torque produces angular acceleration. The torque caused by a force acting on an object is represented by the equation:



where F is the force causing the torque, r is the distance from the center to the point where the force acts on an object, and theta is the angle between the force and a line from the object's center through the point where the force is acting. Torque is measured in Newton meters, written as:

The convention used in our class is that torque in a counterclockwise direction is positive and torque in a clockwise direction is negative (corresponding to right-hand rule). If more that one force is acting on an object, the net torque can be found by adding up all of the torques:



Torque practice:
In a canyon between two mountains, a spherical boulder with a radius of 1.4 m is just set in motion by a force of 1600 N. The force is applied at an angle of 53.5 degrees measured with respect to the vertical radius of the boulder. What is the magnitude of the torque on the boulder? (Page 265, Holt Physics 2006)


Section 8-2: Rotation and Inertia

The center of mass of an object is the point at which all the mass of an object is concentrated. A freely rotating object will rotate about the center of mass.
The center of gravity of an object is the point through which a gravitational force acts on the object. For most objects the center of mass and center of gravity will be the same point.
The moment of inertia (I) of an object is the object's resistance to changes in rotational motion about an axis. Moment of inertia in rotational motion is similar to mass in translational motion.
The moments of inertia for some common shapes are:

Shape

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


An object is in rotational equilibrium when there is no net torque acting on the object. If there is also no net force acting on the object (which is referred to as translational equilibrium) then the object is in equilibrium.

Type
Equation
Meaning
Translational
Equilibrium

The net force on the object is zero
Rotational
Equilibrium

The net torque on the object is zero

Moment of Inertia Practice:
A force of 25 N is applied to the end of a uniform rod that is 0.50 m long and has a mass of 0.75 kg. Find the moment of inertia of the rod, and the torque applied to the rod. (Pg 54, Review Packet)


Section 8-3: Rotational Dynamics

Newton's second law can be applied to angular motion using the equation:



This is related to the equation for translational motion as follows:

Type of Motion
Equation
Translational

Rotational


Just as moment of inertia was similar to mass, the angular momentum (L) in rotational motion is similar to the momentum of an object in translational motion.
This is related to the equation for translational motion as follows:

Type of Motion
Equation
Translational

Rotational


The angular momentum of an object is conserved when there is no acting external force or torque.
Rotating objects have rotational kinetic energy according to the following equation:



Just as other types of mechanical energy can be conserved, rotational kinetic energy is also conserved when there is no friction.

Rotational Kinetic Energy Practice:
A satellite in orbit around Earth is initially at a constant angular speed of 7.27 x 10^-5 rad/s. The mass of the satellite is 45 kg, and it has an orbital radius of 4.23 x 10^7 m. Find the rotational kinetic energy of the satellite around Earth. (Pg 54, Review Packet)


Section 8-4: Simple Machines

In physics, there are six basic types of machines, referred to as simple machines, which are used as the tools to build other complex machines. The six simple machines are levers, inclined planes, wheels, wedges, pulleys, and screws.
The main purpose of a machine is to maximize the ratio of the output of the machine and the input force. That ratio is called the mechanical advantage (MA) of a machine. MA is a unitless number according to the following equation:



Sadly, when frictional forces are brought into the mix, some output force is lost, causing less work to be done by the machine than the original force. The ratio of work done by the machine to work put in to the machine is called the efficiency (eff) of the machine:



If a machine is perfectly efficient (eff = 1) then the ideal mechanical advantage (IMA) can be found by comparing the input and output distances:



Combining the equations for MA and IMA provide us with another equation to find the efficiency of a machine:



Effiency Practice:
A pulley system is used to lift a piano 3.0 m. If a force of 2200 N is applied to the rope as the rope is pulled in 14 m, what is the efficiency of the machine? Assume the mass of the piano is 750 kg. (Pg 266, Holt Physics 2006)





Practice Problems (Methods/Answers):

Torque:

r = 1.4 m F = 1600 N Theta = 53.5 degrees


In accordance with the laws of sig figs, your answer should be



Moment of Inertia:

F = 25 N l = 0.50 m m = 0.75 kg








Rotational Dynamics Practice:

Angular speed = 7.25 x 10^-5 rad/s m = 45 kg r = 4.23 x 10^7 m







Efficiency Practice:

Piano distance = 3.0 m Applied force = 2200 N Rope distance = 14 m Piano mass = 750 kg











All information was gathered from the Honors Physics Review Notes 2008-2009, distributed by Tom Strong.