Running around with paper

Classical Mechanics Level 3

Have you ever tried running around with a piece of paper stuck to your stomach, trying to accelerate so that the paper doesn't fall off (Look mom, no hands!)? No? Well I have, and I decided to create this problem based off this weirdness of mine...

Assume you have the situation described above. Calculate the value of acceleration needed to keep the paper on your stomach if

Air resistance is neglected
The mass of the paper is 0.0045 kilograms
Your mass is 75 kilograms
Gravitational acceleration is 9.8 m/s squared
Coefficient of friction between your shirt and paper is 0.07
Answers are all in $\text{m/s}^2$

Enjoy!

The answer is 140.

1 solution

William G.
Aug 14, 2015

To keep the paper on your stomach, the force of friction between the paper and your body must equal the weight of the paper. Therefore, set mg(weight) equal to normal force times coefficient of friction. However, through analysis, you realize that the force applied by you accelerating on to the paper is equal to the normal force. Therefore (again), set weight (of paper) equal to your mass times your acceleration times the coefficient of friction. Now finally you can plug in the variables for the equation and solve for acceleration. I hope you liked this problem!!

I believe you calculated normal force incorrectly. The normal force between the paper and your body is the mass of the paper times acceleration, not the your mass times acceleration (when you measure the normal force between you and the earth, do you use the mass of the earth?). This makes sense because the acceleration required should not depend of your mass - putting the paper on a car with the same coefficient of friction should also require the same acceleration. The correct answer should be 140 m/s^2. This is reasonable because .07 is a very low coefficient of friction (comparable to steel on ice).

Manvith Narahari - 5 years, 9 months ago

Yeah, normal force $\mathbb{N} = m_{paper} \cdot a$

Kishore S. Shenoy - 5 years, 9 months ago

But, in order to keep the paper on the air (I mean, not falling), you need to apply a force on it, while accelerating. Then, the normal force must be equal the force the paper makes on you (so that it will result on 0), because, considering Newton's Third Law, the force the paper applies on you is equal to the one you apply on it. Then, considering the force's modulo (which was calculated before, using the vertical forces, as Earth Attraction and friction force), you can calculate the person's acceleration using his/her mass. Remembering: the paper must be kept on your shirt. Then, I believe the problem's author is right. If I am wrong, do not hesitate in correcting me.

Bruno Oggioni Moura - 5 years, 9 months ago

@Bruno Oggioni Moura – I Think you concept is wrong.

Using Newton's Third Law won't create zero because it acts on two bodies.
It's you who is accelerating and in your frame, the paper has a psuedo force which acts on you and thus making the normal provided by you,

$\large \mathbb{N} = m_{paper}\cdot a_{provided~by~ you},$

and NOT your mass times acceleration.

Kishore S. Shenoy - 5 years, 9 months ago

@Kishore S. Shenoy – I get your point of view. Sorry for that misconception :/

Bruno Oggioni Moura - 5 years, 9 months ago

@Bruno Oggioni Moura – It's OK! We need someone to clear our doubts. It might be you or me or someone else in each case!

Kishore S. Shenoy - 5 years, 9 months ago

@Kishore S. Shenoy – Agreed. Thank you!

Bruno Oggioni Moura - 5 years, 9 months ago

Running around with paper

The answer is 140.

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