Question

Mechanical advantage is the ratio of the force required without the use of a simple machine to that needed when using the simple machine. Compare the force to lift an object with that needed to slide the same object up a frictionless incline and show that the mechanical advantage of the inclined plane is the length of the incline divided by the height of the incline \((d / h\) in Fig. \(4.25).\)

Short answer

Answer: The mechanical advantage of an inclined plane is equal to the length of the incline divided by the height of the incline (MA = d/h). This relationship shows that the inclined plane allows us to use a smaller force to slide an object up the plane as opposed to lifting it directly, with the mechanical advantage increasing as the incline length increases or the height decreases.

Step-by-step solution

Identify the forces involved in lifting the object

When lifting an object, the force required (ignoring air resistance) is equal to the weight of the object, which can be represented as \(F_{lift} = mg\), where \(m\) is the mass of the object and \(g\) is the acceleration due to gravity.

Identify the forces involved in sliding the object up the inclined plane

When sliding the object up a frictionless incline, there is a gravitational force pulling the object downward along the incline. To find the force needed to slide the object up the incline, we must resolve this gravitational force into two components: one parallel to the inclined surface and one perpendicular to it. The force parallel to the inclined surface is given by the equation \(F_{slide} = mg\sin(\theta)\), where \(\theta\) is the angle between the inclined surface and the ground.

Calculate the mechanical advantage of the inclined plane

Now that we have the equations representing the force required to lift the object and the force required to slide the object up a frictionless inclined plane, we can find the ratio of these two forces to determine the mechanical advantage of the inclined plane. The mechanical advantage (MA) is defined as the ratio of the input force to the output force, or in this case, the force required to lift the object divided by the force required to slide the object up the inclined plane: MA \(= \frac{F_{lift}}{F_{slide}}\) Using our previously defined equations, we can substitute them into the MA equation: MA \(= \frac{mg}{mg\sin(\theta)}\) Simplifying, we see that the mass and acceleration due to gravity cancel out: MA \(= \frac{1}{\sin(\theta)}\)

Relate the mechanical advantage to the length and height of the incline

Now, we can relate the mechanical advantage calculated in step 3 to the length and height of the incline. From the given diagram, we can use the trigonometric identity: \(\sin(\theta) = \frac{h}{d}\) Substituting this into our mechanical advantage equation, we get: MA \(= \frac{1}{\frac{h}{d}}\) This finally simplifies to: MA \(= \frac{d}{h}\) Thus, we have shown that the mechanical advantage of the inclined plane is equal to the length of the incline divided by the height of the incline.