
Three types of simple machines—ramps, wedges, and screws—are related to each other because each contains an inclined plane. Less force is required to move an object up a ramp than is needed to lift it vertically, but the object must be moved a further distance. A wedge is composed of two ramps back to back, and it is often used to split a material, such as an axe through wood. A screw consists of a spiraling ramp that wraps around a cylinder.

ramp, wedge, screw


$$M{A}_{ramp}=\frac{{L}_{ramp}}{{h}_{ramp}}$$
 
$$M{A}_{wedge}=\frac{L}{h}$$
 
$$M{A}_{screw}=\frac{2\pi L}{p}$$
 

Review problems and questions 

 Describe the measurements and calculations that you need to make to determine the mechanical advantage of the following:
 a ramp
 a wedge
 a screw

 For the ramp, you need to measure (i) the difference in vertical height between the bottom and top of the ramp and (ii) the length of the ramp. The mechanical advantage is the ratio of the length over the height.
 For the wedge, you need to measure (i) the maximum thickness at one end and (ii) the length of the longest side. For a twosided wedge, the mechanical advantage is the ratio of the length to the thickness.
 For the screw, you need to measure (i) the pitch (the distance between threads) and the radius of the screw’s body (not the screw’s head!). The mechanical advantage is 2π times the radius of the screw’s body divided by the pitch.

 Is it easier to use a screwdriver with a wide handle or a narrow handle?

Answer: Increasing the radius (or diameter) of the screwdriver increases the mechanical advantage—making it easier to turn.
For a screwdriver and screw combination, the mechanical advantage is$$MA=\frac{2\pi L}{p}$$The mechanical advantage is directly proportional to the radius L of the screwdriver, so increasing the radius (or diameter) of the screwdriver increases the mechanical advantage—making it easier to turn.

 The blade of an axe has a length of 15 cm and a mechanical advantage of 12. What is the maximum thickness of the blade?

Answer: The maximum thickness is 1.3 cm.
The mechanical advantage of the wedge is the length L of the blade divided by its thickness h. Rearrange the terms to solve for h:$$MA=\frac{L}{h}\text{}\Rightarrow \text{}h=\frac{L}{MA}=\frac{15\text{cm}}{12}=1.25\text{cm}$$

 A screwdriver with a handle that is 4.0 cm in diameter is used to drive in a metric screw with a pitch of 2.5 mm.
 What is the mechanical advantage of this combination?
 If you want to increase the mechanical advantage to 90, what pitch should you use?

 The mechanical advantage is 50.
 A screw with a pitch of 1.4 mm or less is needed.

 Billy Bob is trying to decide whether to push a 50 kg dresser up a 4.5 m ramp or lift it directly into the back of a truck. The floor of the truck is 1.5 m above the surface of the street.
 What is the ideal mechanical advantage of the ramp?
 How much force is required to lift the dresser vertically upward into the back of the truck?
 If the ramp were frictionless, how much force is needed to push the dresser up it at constant speed?
 Billy Bob realizes that the ramp has a fairly rough surface. When he pushes the dresser up the ramp, he has to apply 225 N of force. What is the mechanical advantage of the ramp?
 How much work is required to lift the dresser vertically into the back of the truck?
 How much work is required to move the dresser up the ramp?
 What is the efficiency of the ramp?

Answer:  3
 490 N
 163 N
 2.18
 735 J
 1,013 J
 73%
Solution: The ideal mechanical advantage of the ramp is the ratio of the length of the ramp to its height:$$M{A}_{ideal}=\frac{{L}_{ramp}}{{h}_{ramp}}=\frac{4.5\text{m}}{1.5\text{m}}=3$$
 The force required to lift the dresser vertically is just the gravitational force:$${F}_{w}=mg=(50\text{kg})(9.8{\text{m/s}}^{2})=490\text{N}$$
 The mechanical advantage is the ratio of the output force to the input force. Rearranging terms to solve for the input force gives $$\begin{array}{ccc}M{A}_{ideal}=\frac{{F}_{o}}{{F}_{i}}& \Rightarrow & {F}_{i}=\frac{{F}_{o}}{M{A}_{ideal}}\end{array}=\frac{490\text{N}}{3}=163\text{N}$$
 The output force is still the same (490 N), but now more input force is required. The actual mechanical advantage of the ramp, accounting for friction, is$$M{A}_{ramp}=\frac{{F}_{o}}{{F}_{i}}=\frac{490\text{N}}{225\text{N}}=2.18$$
 The output work is the force (of gravity) times the distance:$${W}_{o}=(490\text{N})(1.5\text{m)}=735\text{J}$$
 The input work is the force required to push the dresser up the ramp times the distance:$${W}_{i}=(225\text{N})(4.5\text{m})=\mathrm{1,013}\text{J}$$
 The efficiency is the ratio of the output work to the input work:$$\eta =\frac{{W}_{o}}{{W}_{i}}=\frac{735\text{J}}{\mathrm{1,013}\text{J}}=0.73=73\%$$

Take a Quiz 