Chapter 8

Rotational Equilibrium and Dynamics

Chapter 8 Objectives• Define torque• Identify the lever arm associated with the torque

of an object• Identify the center of mass of an object• Define second condition of equilibrium• Recognize the moment of inertia of several

objects• Identify the six types of simple machines.• Explain how simple machines effect work• Calculate mechanical advantage

Definition of Torque• Torque is a quantity that measures the ability of a

force to rotate an object around some axis.• Mr. Lent’s Definition – Torque is work in a circle.• Torque depends on a force, a lever arm, and the

angle at which the force is applied.• The lever arm is the perpendicular distance from

the axis of rotation to the line drawn along the direction of the force.

• The sign on torque is based on the direction in which the force causes the object to rotate.

• Rotation in a clockwise direction is negative and counterclockwise is positive.

Picture of Torque

Axis of rotation is the center of the object being rotated.

Force

ΘLever Arm

Formula for Torque

Θ

τ = Fd sin Θ

This creates your lever arm so it is perpendicular between the force and the axis of rotation.

Be aware that Θ is always less than or equal to 90o.

Greek “tao”

How to Find Θ• Θ is an angle less than or

equal to 90o

• Extend the force vector in either direction so that it will create an acute angel with the original lever arm.

• You only need extend it so you can draw a line from the axis of rotation to the force vector such that the line is perpendicular.

• That perpendicular line is now your new lever arm.

Θ

Θ

Center of Mass• Center of mass is the point at which all the mass of a

body can be considered to be concentrated when analyzing translational motion.

• The center of gravity is the point at which the force of gravity acts on an object.

• In the higher levels of physics, center of mass and center of gravity are two different concepts and therefore can exist at two different locations of an object.

• For our purposes, we will consider them to be the same point in an object.

• For regularly shaped objects, such as a sphere, cube or solid rod, the center of mass is located in the geometric center of the object.

Moment of Inertia• Remember back to Newton’s 1st Law of Motion,

Objects tend to stay in motion, or at rest, unless acted upon by a net force.

• Notice it says Motion, but does not specify whether the motion is linear or rotational.

• We also said that Newton’s 1st Law describes the term inertia, or the the resistance of a change in motion.

• The tendency of a body rotating about a fixed axis to resist a change in rotational motion is called the moment of inertia of an object.

• The moment of inertia of an object depends on the mass of an object and how far away it is from the axis of rotation.

• The general rule is that the further the mass is from the rotating axis, the larger the magnitude for the moment of inertia.

Calculating the Moment of Inertia of Common Shapes

Object Picture Formula Characteristics

Thin hoop about the symmetry axis

MR2 R is the radius of the hoop and M is the total mass. Wedding ring.

Thin hoop about the diameter

½MR2 This is like a ring standing on its end and spinning about the diameter.

Point mass about axis MR2 This is spinning a weight on the end of a string. Acts like thin hoop.

Disk or cylinder about symmetry axis

½MR2 Flat plate or any solid cylinder, no matter how long.

Thin rod through center axis

1/12ML2 L is the total length of the rod and it rotates like spinning a ruler on finger.

Thin rod about end of rod

1/3ML2 Hold ruler at end and spin in a circle.

Solid sphere about the diameter

2/5MR2 Spinning a bowling ball on your finger.

Spherical shell about the diameter

2/3MR2 Spinning a basketball on your finger.

Rotational Equilibrium•Recall the 1st Condition of

Equilibrium is that the net force on an object is equal to 0.•That is called translational equilibrium.•The word translation means the refers to the translation of an object which only happens in a straight line, or linear path.•Translational equilibrium assures that the net horizontal and the net vertical forces sum to 0.

•We can now add the 2nd Condition of Equilibrium.•Rotational equilibrium is achieved when the net torque is equal to 0.•So the sum of all clockwise torques should be equal to the sum of all counterclockwise torques.

Newton’s 2nd Law of Rotation• Recall Newton’s 2nd Law of Motion said

that F = ma.• Since F can go in infinite directions

around a circle, we cannot use that formula for rotational motion.

• But Torque is work directed in a circle, which does account for force and its direction.

Translational v Rotational • We saw from section 8.2 that if the net

force was not equal to 0, rotation would occur. And more torque would mean faster rotation.

• If the speed of rotation changes that’s called angular acceleration.

• But all of this depends on the object’s shape and its mass.

• The only way to account for the mass of the rotating object is to account for its moment of inertia

So τNET= Iα

Comparing Translational and Rotational •Translational is a straight line• F=ma•Sign shows direction •Positive means right/up •Negative means left/down

•Rotational is circular

•τNET= Iα

•Sign shows direction •Positive – counterclockwise •Negative – clockwise

Angular Momentum• Recall that translational momentum was

p=mv• Objects still have momentum while

rotating • We account for rotating mass by using

moment of inertia• And velocity in a circle is called angular

velocity

L=IωAngular momentum

Conservation of Angular Momentum• Translational momentum is conserved,

thus angular momentum is conserved• That is why a figure skater pulls his/her

arms in order to spin faster.• Be sure to match the proper moment of

inertia to the rotating object(s) in the system

Rotational Kinetic Energy• Imagine a bowling ball rolling down

an alley

• We know it has kinetic energy in a straight line…

• But it also has rotational kinetic energy as well as translational kinetic energy

• The formulas look alike just use the correct variables for the correct situation. KROT= 1/2Iω2

Translational and Angular Convservation• If ever asked to find the net kinetic

energy or net momentum on an object, you must– Sum the translational and angular

kinetic energys.

• 1/2Iω2 + ½mv2

– Sum the translational and angular momentums.

• Iω + mv

Simple Machines

1. Lever – Teeter Totter, Baseball Bat, Broom, etc.

2. Pulley – A rope being pulled around a wheel.

3. Wheel and axle – A wheel attached to a smaller diameter rod.

1. Inclined Plane – A slanted surface like a wheel chair ramp.

2. Wedge – Two inclined planes back to back.

3. Screw – An inclined plane wrapped around a cylinder.

A machine is an object that transmits or modifies force, usually by changing the force applied to an object.

All machines that you may think of are actually combinations or modifications of 6 fundamental types of machines called simple machines. And those are broken down into 2 families.

Lever Family Inclined Plane

Mechanical Advantage• Because the purpose of simple

machines is to alter the direction or magnitude of an input force, then it must provide some type of advantage to produce the output force.

• That advantage is called mechanical advantage.

• The mechanical advantage of a machine is a ratio of the output force to the input force.

• It can also be a ration of the input distance to the output distance, which a little manipulation of the conservation of rotational equilibrium.

Fout

Fin

MA =

τ in = τout

Findin = Foutdout

Fout

Findout

din=

= dout

din

Lever• A lever is a type of arm that includes a pivot

point, or fulcrum.• There are 3 classes of levers based on the

orientation of the fulcrum, input, and output forces.

• Examples of levers are teeter totters, baseball bats, pry bars, bottle openers, golf clubs, hockey sticks, wrenches, etc.

Classes of Levers1st Class Lever – The input force and output force are on opposite sides of the fulcrum and are directed in opposite directions. (Teeter Totter)

2nd Class Lever – The fulcrum is at one end of the lever with the input force at the other end and the output force is in the middle. Also, the forces are directed in the same direction. (Wheel barrow)

3rd Class Lever – The fulcrum is at one end of the lever with the output force at the other end and the input force is in the middle. Again, the forces are directed in the same direction. (Baseball bat)

Mechanical Advantage of a Lever• The mechanical advantage of a any lever is

found by conserving rotational equilibrium.

τin = τout

Findin = Foutdout

Pulley

• A pulley is a part of the lever family because it uses a fulcrum to redirect the force.

Fulcrum

Fixed v Movable Pulleys•The behavior of pulleys can differ based on whether the pulley can move or not.•A fixed pulley is one in which the pulley itself is attached to a wall or pole and does not move when the load is moved.•These types of pulleys are often used to redirect the applied force.

•Therefore, a moveable pulley is one in which the pulley moves as the load moves.•These types of pulleys are used to spread the load out over an additional “rope” in the system.•This transforms the force of the load into 2 equal tensions that each carry half the weight.

Mechanical Advantage of a Pulley•For any fixed pulley, the mechanical advantage is 1.•It is 1 applied force (1 rope) redirected over the pulley to provide an advantage of pulling versus pushing, or vice versa.

•For any moveable pulley, the mechanical advantage is always 2.•There is now 2 applied forces that aid in lifting the load.

–So the input force is cut in half because it only has to lift one of the two lifting forces.

Complex Pulley Systems• Easiest way to calculate mechanical advantage of a

complex pulley system is to identify the following:– Load bearing ropes

• These are the ropes around a moveable pulley.

– Redirecting ropes• These are the ropes around a fixed pulley.

• The mechanical advantage is the reciprocal of the tension in the load bearing ropes.

MA = 2 MA = 2 MA = 3 MA = 3 MA = 4

Wheel and Axle• A wheel and axle is much like two pulleys of

different sizes that spin together.

Wheel

Axle

Mechanical Advantage of a Wheel and Axle

• The mechanical advantage is gained from the ratio of the radii of the wheel and axle.

• The best way to demonstrate the mechanical advantage is the ratio of tire size to axle size in the rear end of a car.

r = 1

R = 3

MA =

τin = τout

Findin = Foutdout

Fin(2πR) = Fout(2πr)

(2πr)(2πR) = Fout

Fin

13

= 3

Inclined Plane• An inclined plane is simply a flat surface that is

used to move an object up or down an incline, or elevation.

• The surface does not necessarily have to be flat, it must be able to be modeled flat. (ie: a set of stairs)

Mechanical Advantage of an Inclined Plane

• The mechanical advantage of an inclined plane is easiest to calculate using the input distance versus the output distance.

• You may need to use Pythagorean Theorem for this.

Input

Output

Fout

Findout

din=

Wedge• A wedge is two inclined planes placed back to

back.• The most common form of a wedge is an ax.

Mechanical Advantage of a Wedge

• The mechanical advantage of a wedge is the ratio of inclined surface to width of the wedge.

• It is very similar to the mechanical advantage of an inclined plane.

• You might think that since a wedge is two inclined planes, its mechanical advantage would be twice as large. But it is spread over two surfaces so it divides the two back out.

• Notice that to increase the mechanical advantage of the wedge, simply make the width smaller. That is why a sharp knife is better than a dull knife!

WidthInclined Surface

Screw• As simply put as possible, a screw is an inclined

plane wrapped around a cylinder.• It resembles a wheel and axle, except for the part

that is actually doing the work is the slope of the threads of the screw.

Mechanical Advantage of a Screw

• The mechanical advantage of a screw is hard to visualize, and therefore tricky to calculate.

• Avoiding all of the nasty calculation, it is a ratio of the circumference of the screw divided by its pitch.

• Pitch is the distance between threads.• If you think about it, one turn will make the screw

go into the wood the same depth that one thread of revolution. So that is how we get the mechanical advantage.

MA = Circumference of Screw

Pitch