(FOM) Mechanical Advantage Lesson

Mechanical Advantage

Simple machines are devices used to help make work easier or get done faster. How do machines help us perform work? Simple machines help to multiply our effort force or help to change the direction in which we apply the force. You should be familiar with the six types of simple machines as seen below learned in earlier science courses throughout your life.

Simple machines images: inclined plane, lever, wheel and axle, Screw, wedge, pulley

Simple Machines Video

As you saw in the video, mechanical advantage is a numerical value that indicates how beneficial it is for you to use the machine. In other words, mechanical advantage provides the ratio of resistance to effort force magnitudes for any simple machine. The higher the mechanical advantage means the more the machine multiplies the effort force. If the mechanical advantage is less than one it means that you are having to exert more force than if you were not to use the machine, but you have to exert the force over less distance.

It is also important to realize that the total amount of work is the same regardless of if you use a simple machine or not. For a machine to do work, an effort force must be applied over a distance. If you exert less force, then you will have to apply that less force over a greater distance. Thus, it might seem "easier" as you are exerting less effort force but you are having to go over a greater distance. Let's look at an example to help explain this concept. In the picture below, you will see two men loading barrels onto a platform. One man is using an inclined plane while the other man is lifting straight up onto the platform. Both men are doing the same amount of work. One man is using less force over a greater distance while the other man is using more force over a smaller distance.

Mechanical Advantage image of man with ramp and barrel: Using an inclined plane requires less effort over a longer distance.
with inclined plane
effort needed = 10 N
1 m
2.5 m
without inclined plane effort needed = 25 N n

equation: F * d = F * d

Whether or not you realize it, you utilize mechanical advantage on a daily basis. Maybe the lid of a pickle jar is tightly shut, so you use a flat-edged screwdriver as a lever to pry it open. Maybe one of your grandparents uses a wheelchair, so instead of taking the stairs up to the shopping center, you wheel him or her up the wheelchair ramp.

Mechanical advantage means putting a smaller force into a simple machine to output a larger one. It certainly makes our lives easier, but it actually gives us a little less extra output than we would expect. This is because factors such as friction and machine wear over time and cause a loss of energy. Actual mechanical advantage takes these factors into account, while ideal mechanical advantage does not.

Calculating Mechanical Advantage

In order to understand how to mathematically calculate mechanical advantage, there are a couple of terms that are important to know. The force applied to a simple machine is called the effort force, Fe. It can also be called input force, but for this course, we will focus on calling it the effort force. The force exerted by the machine is called the resistance force, Fr. The resistance force is the load or the weight of the object being moved. Resistance force can also be called the output force. The distance in which the effort force is applied over is called the effort distance, d_e. The distance in which the resistance of the object moves is called the resistance distance, d_r. The mechanical advantage is determined by using one of two equations:

mechanical advantage equations:
actual mechanical advantage (AMA) = Fr/Fe
ideal mechanical advantage (IMA) = de / dr

Did you notice that the force unit involved in the calculation, the newton (N) is present in both the numerator and the denominator of the fraction and the distance units in the other equation are present in both the numerator and the denominator? These units cancel each other, leaving the value for mechanical advantage unitless.

It is important that you be able to identify the effort force, resistance force, effort distance, and resistance distance on various types of simple machines. Please look at the pictures below to see where each variable is labeled on the simple machine.

Lever

1^st and 2^nd class levers help to multiply the effort force, but 3^rd class levers are used to multiply the distance. Therefore, 3^rd class levers will give a mechanical advantage that is less than 1. A mechanical advantage less than one doesn't mean a machine isn't useful. It just means that instead of multiplying force, the machine multiplies distance. For example, a broom is a 3^rd class lever. A broom doesn't push the dust with as much force as you use to push the broom, but a small movement of your arm pushes the dust a large distance.

Inclined Plane

Incline Plane image: Resistance is down
Force to the right (up the plane)
Resistance Distance-5 ft
Effort Distance-20 ft

Pulley

The mechanical advantage of a moveable pulley is equal to the number of ropes that support the moveable pulley. When calculating the mechanical advantage of a moveable pulley, count each end of the rope as a separate rope. You can also use the IMA and AMA equations to calculate the mechanical advantage, but using the number of support ropes is a shortcut.

three pulleys:each image has effort force, mechanical advantage indicated

Let's look at a couple of example problems:

You need to push a grand piano with a weight of 4900 N onto a stage that is 3 m above the ground. If you can only apply a maximum force of 1000 N, what is the mechanical advantage? What is the minimum distance from the stage that you should begin building your ramp?

Looking For:

Mechanical Advantage

Effort Distance

Solution:

Step 1:

$LaTeX: MA=\frac{F_r}{F_e}$ $MA=\frac{F_r}{F_e}$

$LaTeX: MA=\frac{4,900N}{1,000N}$ $MA=\frac{4,900N}{1,000N}$

MA = 4.9

Step 2:

$LaTeX: MA=\frac{d_e}{d_r}$ $MA=\frac{d_e}{d_r}$

$LaTeX: 4.9=\frac{d_e}{3m}$ $4.9=\frac{d_e}{3m}$

$LaTeX: d_e=4.9\times 3m$ $d_e=4.9\times 3m$

LaTeX: d_e=14.7m $d_e=14.7m$

Given:

piano moving up incline plane:
Fe = 1000 N
Fr = 4900 N
dr = 3 m

F_e = 1000 N

F_r = 4900 N

d_r = 3 m

Equation:

hand pulling lever: fulcrum to bottom: 1.2 m; .3m between hands; input force: hands

Actual Mechanical Advantage= equation image indicator (AMA)

Ideal Mechanical Advantage= (IMA)

For a broom, your upper hand is the fulcrum and your lower hand provides the effort force. Given the picture below, calculate the mechanical advantage of using the broom.

Looking For:

Mechanical Advantage

Solution:

$LaTeX: MA=\frac{d_e}{d_r}$ $MA=\frac{d_e}{d_r}$

$LaTeX: MA=\frac{0.3m}{1.2m}$ $MA=\frac{0.3m}{1.2m}$

MA = 0.25

Given:

Pulley: 500N from rope; 2000N at bottom of pulley

d_r = 1.2 m

d_e = 0.3 m

Equation:

Actual Mechanical Advantage = $LaTeX: \frac{F_r}{F_e}$ $\frac{F_r}{F_e}$ (AMA)

Ideal Mechanical Advantage = $LaTeX: \frac{d_e}{d_r}$ $\frac{d_e}{d_r}$ (IMA)

A block-and-tackle pulley is used to lift an object weighing 2000 N using an effort force of 500 N. Determine the mechanical advantage of the pulley.

Looking For:

Mechanical Advantage

Solution:

$LaTeX: MA=\frac{F_r}{F_e}$ $MA=\frac{F_r}{F_e}$

$LaTeX: MA=\frac{2000N}{500N}$ $MA=\frac{2000N}{500N}$

MA = 4

Given:

F_r = 2000 N

F_e= 500 N

Short Cut: The number of support ropes = 4

Equation:

Actual Mechanical Advantage = $LaTeX: \frac{F_r}{F_e}$ $\frac{F_r}{F_e}$ (AMA)

Ideal Mechanical Advantage = $LaTeX: \frac{d_e}{d_r}$ $\frac{d_e}{d_r}$ (IMA)

Suppose you need to remove a nail from a board by using a claw hammer. What is the input distance for a claw hammer if the resistance distance is 2.0 cm and the mechanical advantage is 5.5?

Looking For:

Effort Distance

Solution:

$LaTeX: MA=\frac{d_e}{d_r}$ $MA=\frac{d_e}{d_r}$

$LaTeX: 5.5=\frac{d_e}{2.0cm}$ $5.5=\frac{d_e}{2.0cm}$