Ratios of work to height with same mass

[michaelw]

This is a question from the MCAT, test 3R, and I am stumped as to why the answer is what it is.

"A mass is lifted from the ground to an altitude h1, requiring work W1. The work to lift an identical mass to height h2 is W2. If h2 is twice h1, what is the ratio of W2 to W1? Assume force due to gravity does not change betweenh1 and h2."

I thought this was a simple W = mgh problem, resulting in an answer of 2:1.. but the answer key states its root(2):1! Any idea why?

 

[geosonel]

This is a question from the MCAT, test 3R, and I am stumped as to why the answer is what it is.

"A mass is lifted from the ground to an altitude h1, requiring work W1. The work to lift an identical mass to height h2 is W2. If h2 is twice h1, what is the ratio of W2 to W1? Assume force due to gravity does not change betweenh1 and h2."

I thought this was a simple W = mgh problem, resulting in an answer of 2:1.. but the answer key states its root(2):1! Any idea why?

unless there's more to this problem than you indicated, it appears you're correct that the answer SHOULD be (W2/W1)=2.

 

[thepatientmental]

The reason why the ratio of W2 to W1 is not 2:1 is because the work done to lift an object is not directly proportional to the height it is lifted. In fact, the work done is directly proportional to the change in potential energy, which is given by the formula mgh.

In this scenario, when the mass is lifted to a height h1, the potential energy gained is mgh1. When the same mass is lifted to a height h2, the potential energy gained is mgh2. Since h2 is twice h1, the potential energy gained is 2mgh1.

Therefore, the ratio of W2 to W1 can be calculated as (2mgh1)/(mgh1) = 2. However, this is not the correct answer because we are assuming that the mass and the force due to gravity are constant. In reality, as the mass is lifted higher, the force due to gravity decreases slightly due to the inverse square law.

To account for this, we need to use the root(2) instead of 2. This is because the ratio of the force at h2 to the force at h1 is root(2):1, which means that the work done is also in the same ratio.

In conclusion, the correct ratio of W2 to W1 is root(2):1, taking into account the slight decrease in force due to gravity as the mass is lifted higher. I hope this helps clarify any confusion you had about the answer.