A longer plank, a lighter push

The idea

For an inclined plane of length L raising load mg to height h with effort F′, ignoring friction, F′ × L = mgh, so mechanical advantage = mg/F′ = L/h, above 1; a longer, shallower plane needs less effort.

Up the plank

A box is too heavy to lift onto a platform. You lay a plank from the ground to the platform and push the box up the plank.

Do you think a longer, gentler plank makes the push smaller or larger?

• Smaller
• Larger
• The same

Change the plank

The platform is 1 m high and the box weighs 60 N (an example). Change the length of the plank.

Longer plank, smaller effort

The work done on the box is F' × L, and this equals the potential energy gained, mgh. So F' = mgh / L. A longer plank made L bigger, so the effort F' became smaller. The load divided by the effort came out as L divided by h.

For an inclined plane of length L raising a load mg to a height h, ignoring friction, F' × L = mgh, so mechanical advantage = mg/F' = L/h, which is above 1. A longer, shallower plane needs less effort.

Notes

For an inclined plane, ignoring friction: F' × L = mgh and mechanical advantage = L/h, above 1. A longer, shallower plane needs less effort.

Check yourself

A ramp 50 cm long raises an object over a step 30 cm high. What is the mechanical advantage L/h, to two decimal places?

Answer: 1.67

L/h = 50 cm / 30 cm = 1.67.

A plank 4 m long raises a load to a platform 1 m high. What is the mechanical advantage?

Answer: 4

L/h = 4 m / 1 m = 4.

A load of 60 N is raised to a platform 1 m high along a plank 5 m long, ignoring friction. What effort is needed, in N?

Answer: 12 N

F' = mgh / L = 60 N × 1 m / 5 m = 12 N.

The same load is raised to the same height, first by a short steep plank and then by a longer, shallower plank. The longer plank needs…

• a larger effort. F' = mgh / L, so a larger L gives a smaller effort.
• a smaller effort — correct. Yes!
• the same effort. F' × L = mgh. If L is larger and mgh is the same, F' is smaller.