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I was trying to solve the following exercise (apologies in advance for potential mistakes in terminology); illustration will be attached below:

An object with mass $M$ is placed upon a frictionless, fixed slope. The slope's angle with respect to $-\hat{x}$ is $\alpha$. On top of the object is a man with mass $m$. Everything is frictionless and idealized and the only forces acting are gravitation and normal forces. I was asked to calculate the acceleration $\vec{a}$ of the object in a reference frame of my choosing.
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The reference frame I chose is on in which the $x$ axis is parallel to the slope and pointing downwards, and the $y$ axis is perpendicular to it and pointing upwards:
With my axes

The normal force will negate the effects of the force of gravity in the $y$ direction and so the acceleration will all be in the (positive) $x$ direction.

The bit that confused me is this:
At first, I chose to regard the object and the man as one object since they'll be moving together. Their combined mass is then $M+m$. Writing down some simple Newton laws equations gives me this (the only force acting in this direction is weight):
$$ (M+m)\cdot\vec{a} = (M+m) \cdot g \cdot \sin(\alpha) \cdot \hat{x} $$ $$\Rightarrow \vec{a} = g\cdot\sin(\alpha)\cdot\hat{x}$$

Afterwards, to see how it would play out, I tried to analyze it without looking at the man and the object as one body. In this case, the forces acting on the object in the $x$ direction are its own weight, as well as the normal force of the man standing on top of it (corresponding to the man's own weight), for a total of $$M\cdot\vec{a} = m \cdot g \cdot \sin(\alpha) \cdot \hat{x} + M \cdot g \cdot \sin(\alpha) \cdot \hat{x}$$ which then gives $$\vec{a} = \frac{M+m}{M} \cdot g \cdot \sin(\alpha) \cdot \hat{x}$$

In other words, I get completely different results. I feel like everything I do here is sort of hand-wavy in that many things are neglected and/or taken as ideal and it makes it difficult for me to "debug" this. At the same time, it seems like the fact that I can't easily explain this to myself should point at some pretty serious gap in my understanding, so I'd really appreciate any assistance in solving this issue.

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    $\begingroup$ Mostly looks good. Note that if the man was sitting on a frictionless platform, he would accelerate sideways at all. You might check that out. $\endgroup$
    – mmesser314
    Aug 24 at 16:44
  • $\begingroup$ Hey @mmesser314, thanks for the heads up! I've added a static friction note between the man and the object. $\endgroup$
    – ShyGuy
    Aug 24 at 16:52
  • $\begingroup$ Update: your comment and ACB's answer were very helpful. Thank you! $\endgroup$
    – ShyGuy
    Aug 24 at 16:53
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You are missing the crucial fact.

Everything is frictionless and idealized…

In your first method, you have considered that the man is moving with the object. For simplicity, change your x-axis to horizontal plane. There are vertical forces acting on the man, weight and normal force. But there is no horizontal force acting on the man. That means the man cannot move horizontally with the object. So you can't apply your first method considering them as a system.

Consider this way. There is a ball moving on a table. And another ball is placed stationary on the table apart. Are you going to take them as system and apply equations as in your first method?

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  • $\begingroup$ Ohhhh; I see now. So what I was missing is the friction between the man and the object. I suppose I got very rusty, very fast... :') Thank you very much! $\endgroup$
    – ShyGuy
    Aug 24 at 16:53
  • $\begingroup$ In regards to your rollback of my edit: note that I edited it in response to @mmesser314's comment, before your answer was posted. Also note that your answer is still relevant as even though I may have noted static friction existed between the man and the object, I didn't take it into consideration in my calculations (i.e didn't consider it when listing the forces acting upon the object). I did "think" about that static friction but in my mind, it was enough to mention everything was "idealized", and so my edited version better fits my intentions. All of that said, I don't really mind :) $\endgroup$
    – ShyGuy
    Aug 24 at 17:04
  • $\begingroup$ My answer was in progress before your edit. If you think my answer is still relevant, then no problem : ) $\endgroup$
    – ACB
    Aug 24 at 17:24

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