Math of Simple Machines Tuesday July 24, 2007 The goal of this activity is to introduce the mathematical concepts and vocabulary for the afternoon activity on simple machines. See the lesson plan for an outline of the units covered in this PowerPoint.

Ratio and Proportion Ratio is a fraction Proportion is the relationship between two quantities with constant ratios or fractions. Most simply a ratio is a quantity that denotes the proportional amount or magnitude of one quantity relative to another. Two quantities are described as proportional if they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio.

Work (W) = force (F) × distance (d) Anything that pushes or pulls to move an object across a distance. Work (W) = force (F) × distance (d) W = F*d Conservation of Energy implies Scientifically speaking, work means using a force—anything that pushes or pulls, to move an object across a distance. Mathematically, work equals force times distance. W = f x d According to conservation of energy, for an ideal simple machine, work into a machine equals work out of a machine. An ideal machine does not loose any work to friction or heat. This never happens. Work In = Work Out + Work Lost to Friction Ideal Machine does Ideal Work with no work lost to friction

Mechanical Advantage Consider the work in and work out of a simple machine like an inclined plane : Work in = F1 x d1 Work out = F2 x d2 F2 x d2 = F1 x d1 Mechanical advantage is the unit less ratio of the force exerted by the machine to the force applied to the machine. Mechanical advantage is accomplished by a sacrifice of distance. This means we can apply a large amount of force over a small distance or a small amount of force over a large distance. Applying a smaller amount of force over a larger distance gets the same amount of work done but is usually easier. Force in: 6 N. Distance moved: 2 m. Work in = 6 * 2 = 12 Joules Work out = 3*4 = 12 Joules Force out: 3 N. Distance moved: 4 m. Mechanical Advantage = F1 / F2 = 6 / 3 = 2

Efficiency of a Machine An Ideal Machine has 100% efficiency What is the relationship between mechanical advantage and efficiency? Mechanical Advantage is the ratio of Force in over Force out and efficiency is the ratio of work out over work in.

Simple Machines The Lever The Pulley The Wheel and Axle The Inclined Plan The Wedge (single or double) The Screw

Lever A rigid bar that rotates around a fixed point called a fulcrum. A Load or force is also the gravitational weight d2 d1 Load, F1 (force of the object desired to move) Force, F2 With a big enough lever you can move the world (Archimedes) What are some other kinds of levers? What are some examples? D2 is the effort arm and d1 is the resistance arm.

Pulley Fixed Movable ? Movable = Fixed d1 d2 Load, F1 Force, F2 With a single fixed pulley there is not a lot going on. However a movable pulley, while using the same formula has a different mechanical advantage. To explain the movable pulley, consider a two foot string. Hold one end with your right hand and the other with your left. Bring the two ends of the string together such that you have a two foot loop. Raise your right hand with the end of the string 4 inches. How high does the loop move? ? Movable = Fixed

Wheel and Axle Rwheel Raxle The wheel and axle makes it easier to move heavy objects along. A wheel is a disc-shaped object of which the fundamental operation is to transfer linear motion (that is, going along) into rotary motion (that is, going around). The wheel and axle can be considered to be a lever in which the radius of the wheel is the effort arm and the radius of the axle represents the resistance arm. While maintaining the law of conservation of energy, would turning the wheel or axle require more force to cover the same distance?

Inclined Plane height, h Force, F2 Load, F1 length, l Consider two inclined planes that achieve the same vertical displacement of the load, how could you lesson F1 or your input force?

Wedge l Load, F1 Force, F2 h DOUBLE WEDGE SINGLE WEDGE

Screw Modified version of the inclined plane. Picture unrolling a screw and what do you get.

Trigonometric Functions Sine and cosine functions Sine function Possible extension is to talk about angles instead of l vs. h and distances discussed ealier. This is a possible opportunity to talk about SOHCANTOA and the use of trig identities for those who want to know more. Cosine function