Question

Which has a greater ideal mechanical advantage, a ramp that is 12 m long and 2 m high or a ramp that is 6 m long and 2 m high?

Answer 1

More torque is generated by the 12m ramp

Explanation:

Torque, denoted by `vec tau` is:
`vec tau = vecF xx vecr` where `xx` is the cross product.

If you're not familiar with cross products, this means that the magnitude of the torque, `tau` ,is:

`tau = (F)(r) sin(theta)` where theta is the angle between the force, F and the radius, r, which is the ramp's length.

Force and height are the same in both cases, so we can see that the longer 12m ramp will create more torque (i.e. mechanical advantage). This is because both r and `sin(theta)` are larger for the longer ramp.
`sin(theta)` is bigger for the longer ramp as `theta` is bigger for this ramp because it's closer to horizontal than the shorter ramp, which has a more acute `theta`. You can see this by drawing it out or use trigonometry if you want to find the exact angles.