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### Key Equations

$MA = \frac{d_1}{d_2} = \frac{F_{out}}{F_{in}}$

d1 is the distance of effort and d2 is the distance the object is moved

Guidance
• Mechanical Advantage is the ability to lift or move objects with great force while utilizing only a little force. The trade-off is that you must operate the smaller input force for a large distance. This is all seen through the work Equation. Work equals force times distance. Energy is conserved. Thus one can get a large force for a small distance equal to a small force for a large distance.
• Mechanical advantage equals the distance of effort divided by the distance the object moves. It is also equal to the output force divided by the input force.

#### Example 1

You need to push a 500 kg grand piano onto a stage that is 3 m above the ground. If you can only apply a maximum force of 1000 N, what is the minimum distance from the stage that you should begin building your ramp?

##### Solution

We should start this problem by determining the mechanical advantage required to move the piano based on the weight of the piano and the force you can apply. We'll define $F_{out}$ to be the force it would take to lift the piano straight up and $F_{in}$ to be the force you can apply.

$MA&=\frac{F_{out}}{F_{in}}\\MA&=\frac{500\;\text{kg} * 9.8\;\text{m/s}^2}{1000\;\text{N}}\\MA&=\frac{4900\;\text{N}}{1000\;\text{N}}\\MA&=4.9\\$

Now we can use this to find how long the ramp needs to be.

$MA&=\frac{d_{in}}{d_{out}}\\d_{in}&=MAd_{out}\\d_{in}&=4.9*3\;\text{m}\\d_{in}&=14.7\;\text{m}$

Now we can just use the Pythagorean theorem to determine how far away from the stage the ramp should start.

$a^2+b^2&=c^2\\b&=\sqrt{c^2-a^2}\\b&=\sqrt{(14.7\;\text{m})^2-(3\;\text{m})^2}\\b&=14.4\;\text{m}\\$

### Explore More

1. A crowbar is a very handy tool to get out tough nails, demolition of a wall or breaking into or out of jail. It can be used from either end. The man shown is prying out a board. He is applying 30 N of force, study the picture to see how he is using it and which dimensions should be used.
1. What is the Mechanical Advantage of this crowbar, shown above with its dimensions?
2. How much force is being applied to the wood plank?
2. A mover loads a 100-kg box into the back of a moving truck by pushing it up a ramp. The ramp is 4 m long and the back of the truck is 1.5 m high.
1. Calculate the potential energy gained by the box when it’s loaded into the truck.
2. Calculate the mechanical advantage of the ramp.
3. Calculate the force required to push the box up the ramp in the absence of friction.

1. a. M.A. = 12 b. Force out = 360 N
2. a. 1500 J b. 2.7 c. 375 N