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A 1 kg 1 \text{ kg} tablet with dimensions 30 cm × 30 cm 30 \text{ cm} \times 30 \text{ cm} sits charging on a smooth table. The charging cord is attached to one of its corners. Somebody trips on the cord, applying a force of 15 N 15 \text{ N} to the tablet in the direction shown.

What is the initial acceleration at point A A in m/s 2 ? \text{m/s}^2?

Assume that tablet has uniform mass.


The answer is 47.434.

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1 solution

Rohit Gupta
Mar 8, 2018

The force will accelerate the tablet as well as rotate it. To find the acceleration of center of mass, we can apply newton's second law , according to which, F net = m a cm a cm = F m F_{\text{net}} = m a_{\text{cm}} \\ a_{\text{cm}} = \frac{F}{m} The rotation of the tablet can be calculated by finding the torque about the center of mass and using rotational form of newton's second law , which states τ net = I α τ net = F × l 2 I = 1 6 m l 2 \tau_{\text{net}} = I \alpha \\ \tau_{\text{net}} = F \times \frac{l}{\sqrt {2}}\\ I= \frac{1}{6} ml^2 Therefore, F × l 2 = 1 6 m l 2 α α = 6 F 2 m l F \times \frac{l}{\sqrt {2}} = \frac{1}{6} ml^2 \alpha \\ \alpha = \frac{6F}{\sqrt{2} ml} \\

The motion of point A A can be seen as the combination of translation at an acceleration of the center of mass and rotation about the center of mass.

The combined motion will give the net acceleration of A A as

a A = a c m 2 + ( l 2 α ) 2 = ( F m ) 2 + ( l 2 6 F 2 m l ) 2 = F m 10 = 15 10 = 47.434 m/s 2 \begin{gathered} a_A = \sqrt {a_{cm}^2 + {{\left( {\frac{l}{{\sqrt 2 }}\alpha } \right)}^2}} \\ = \sqrt {{{\left( {\frac{F}{m}} \right)}^2} + {{\left( {\frac{l}{{\sqrt 2 }}\frac{{6F}}{{\sqrt 2 ml}}} \right)}^2}} \\ = \frac{F}{m}\sqrt {10} \\ = 15\sqrt {10} = \boxed{47.434 \text{ m/s}^2} \\ \end{gathered}

Yes...why didn't you insider weight.??.....its incorrect.

Jahnavi Pande - 3 years, 3 months ago

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If the tablet is sitting on a table-top (presumed horizontal), gravity is opposed by the normal reaction from the table, and so has no effect. Since the table is smooth, there are no friction forces, so the only relevant force acting on the tablet comes from the cable.

Mark Hennings - 3 years, 3 months ago

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Oh... Thanks

Jahnavi Pande - 3 years, 3 months ago

Would this mean that the vertical acceleration drops from Fy, the vertical component of F, to (Fy - 9.8) the instant the block lifts off of the table?

Justin Park - 3 years, 3 months ago

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@Justin Park Why should it lift off? It will slide. Remember the diagram is looking at the tablet from above.

Mark Hennings - 3 years, 3 months ago

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@Mark Hennings Ohhh I see now, I had misinterpreted the diagram. Thank you!

Justin Park - 3 years, 3 months ago

I don't know why you all are assuming that "normal reaction equals gravity and so will cancel out". Ofcourse they are equal when the tablet is sitting stationary but as soon as you apply the force which has a vertical component the normal reaction ceases to be equal to gravity, it decreases its magnitude accordingly. So your assumption that applied force is the only relevant force is incorrect.

Sarthak Agrawal - 3 years, 3 months ago

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@Sarthak Agrawal The 15 15 N force is acting horizontally, so has no vertical component. You should be looking at the diagram “from above”. The tablet is not resting on its edge on the tabletop, it is lying flat on it.

Mark Hennings - 3 years, 3 months ago

So "smooth" is supposed to mean "perfectly smooth" / "frictionless"? I have several smooth tables in my house, and none are frictionless.

Lars Huttar - 3 years, 2 months ago

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@Lars Huttar In the context of Mechanics questions, “smooth” means “no friction”, yes.

Mark Hennings - 3 years, 2 months ago

Why did you consider the rotation about the center of mass and not about the vertex opposite to A?

A Former Brilliant Member - 3 years, 3 months ago

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This is because the vertex opposite to A is accelerating and a non inertial frame of reference and hence going in its frame would make a pseudo force act at center of mass. This will make the equations why complex but you will eventually end up with the same answer.

Rohit Gupta - 3 years, 2 months ago

Indian nerds... 😂

At R - 3 years, 3 months ago

I used 0.15*sqrt(2) meters for the radius. Why did you divide by root 2? It seems like it should be half the diagonal for the effective radius.

egenriether . - 3 years, 2 months ago

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If side length of a square is l l , its diagonal is 2 l \sqrt 2 l and the radius is half of it 2 l 2 = l 2 \dfrac{\sqrt 2 l}{2} = \dfrac{l}{\sqrt 2} .

Rohit Gupta - 3 years, 2 months ago

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Oh ok. I see it now! We both have the same radius but I used half the side length in my calculation instead of writing (0.3 / 2). Thanks

egenriether . - 3 years, 2 months ago

Why are you neglecting gravity when calculating translation about c.o.m

BETTER CODER - 3 years, 2 months ago

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Because it is kept on a HORIZONTAL and SMOOTH surface, gravity would act perpendicular to the surface and would be exactly balanced by the normal reaction from the surface. So it would cause no net effect.

Arush Sharma - 3 years, 2 months ago

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but there is an y component of force applied

BETTER CODER - 3 years, 2 months ago

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@Better Coder Yes, but the force is only in the x-y plane, whereas the weight is in the - ve z-direction

Arush Sharma - 3 years, 2 months ago

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@Arush Sharma Please consider this: The drawing presented in the problem is a top view of the tablet.

Laszlo Mihaly - 3 years, 2 months ago

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@Laszlo Mihaly now i got it

BETTER CODER - 3 years, 2 months ago

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@Better Coder i think it should be mentioned in the question

BETTER CODER - 3 years, 2 months ago

@Better Coder Yes I also got confused.

Pau Cantos - 3 years, 2 months ago

For the x component yes (cos(90)=0). But not for the y, for which the gravity is opposing with cos(180deg).

Piotr W - 3 years, 2 months ago

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@Piotr W The tablet is flat on a horizontal surface. The illustration is a top-down view.

Laszlo Mihaly - 3 years, 2 months ago

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@Laszlo Mihaly Ok, now i get it... Thx

Piotr W - 3 years, 2 months ago

Would not a perfect right angle such as this one produce a pure rotational movement? Initially, at least. If not, which angle would the force need to have with respect to the diagonal to produce a purely initial rotational movement?

And Yag - 3 years, 2 months ago

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If there is only one force acting on a body, there is no way to produce a pure rotational movement. It is possible to produce a pure translational movement when the line of force goes through the center of mass.

Laszlo Mihaly - 3 years, 2 months ago

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That makes sense. Thank you for answering my question.

And Yag - 3 years, 2 months ago

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@And Yag Agree, HOWEVER, will the translational part of the acceleration be the FULL F/m? I doubt that, intuitively, and need to spend some time applying the proper formalism (like Lagrangian equations) to prove...

Peter Baumgart - 3 years, 2 months ago

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@Peter Baumgart That was my gripe with it as well, but then I realized that I was framing the problem as a collision. That is, I was imposing a limit to the energy available "to share" between rotation and translation. Clearly this problems is not a collision, and therefore there is not a limit to the maximum energy available.

Nevertheless, if you come up with a proper proof then share it with us, please, as I would very much like to understand intuitively the whole process.

And Yag - 3 years, 2 months ago

Your value of the acceleration implies that alpha is about 111.803 per second squared, which does not seem reasonable. Not only that, but your value of the acceleration of the point A is close to five times the acceleration of gravity, which is also not very plausible.

Juan Castro - 3 years, 2 months ago

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Nobody said the question was realistic now, did they? Also, though g is a relatively high value, you can accelerate things faster than it.

Aryaman Singh - 3 years, 2 months ago

Why is it that the torques are always taken about the centre of mass?

Amogh Rakesh - 3 years, 2 months ago

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Actually you could choose to calculate it around ANY point, but around the CoM it is easier. The result of the integration will be the same...

Peter Baumgart - 3 years, 2 months ago

Are we just supposed to know this

Ivy Bones - 3 years, 2 months ago

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No, but you can take a physics course where it is explained in great detail.

Laszlo Mihaly - 3 years, 2 months ago

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Or you can just derive everything for yourself like I did. I forgot all the moment of inertia formulas, so I just rederived it with a very simple integral. Are you supposed to "know it"? Probably not. Understand it? Most definitely.

David Weisberg - 3 years, 2 months ago

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@David Weisberg Sure, you are right about the moment of inertia. I was responding to Peter Baumgart's and Ivy Bones questions.

Laszlo Mihaly - 3 years, 2 months ago

@David Weisberg did u take the integral in the moment of inertia in the limits -L/2 and L/2??But I am not getting the answer!!! Whereas I am getting 1/12ml^2.

erica phillips - 3 years, 2 months ago

Why isn't the INITIAL acceleration only leading to rotation? What causes the concurrent translational acceleration (F/m) of the COM? The torque is applied at 90 deg to the "arm" between the COM and the corner where the force is applied.

Peter Baumgart - 3 years, 2 months ago

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This is because it is an unbalanced force and as a result will cause both rotation and translational acceleration, at least for some time. Pure rotation can only be achieved if the forces on the different parts of the body cancel out, but the torques due to each add up.

Amogh Rakesh - 3 years, 2 months ago

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Makes sense, but FULL F/m? I need to find some formalism to treat it properly, like Lagrange equations, or something. Intuitively it does not make sense that you get full F/m acceleration of the whole thing...

Peter Baumgart - 3 years, 2 months ago

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@Peter Baumgart Yes, you have the full F/m. This is what Newton's law states, and you would get exactly the same from the Lagrangian or Hamiltonian formalism.

What your intuition is telling you is that if the line of the force does not go through the center of mass, there will be a rotational motion of the body, in addition to the translation. As a result certain parts of the tablet (notably the part that is on the opposite side of the center of mass relative to the line of force) will have less than F/m acceleration. In fact, there are parts of the table that accelerate in the opposite direction relative to the center of mass.

Laszlo Mihaly - 3 years, 2 months ago

There is a conceptual mistake taking rotation pivot in the center of the square. It will be in the left down corner. Because the surface of table will prevent squeezing this corner into the table. If we are discussing a inertia of rotating square, we must take an equation of rotating square, but with pivot at its corner, which is different. This is also important to notice, that the vector of movement of A caused by rotation will be directly upwards (perpendicular to surface). Another missing thing is the earth gravity which will cancel out a good deal of the pulling cord vertical part. Calculation is much more complicated than this. A back of mind calculation tells me that the gravity will cancel out 5 out of 10.6N. And even if we ommit the fact, that table will prevent roation at the center of square mass, and follow original solution, the 10.6N pulling right the thing will not cause simply presented acceleration, as the rotation will cause a slip in the center of gravity seen by the pulling vector. In other words, from the standpoint of this vector, center of mass for the tablet will move (rotation). Unfortunately i dont have much time to analyse these staring conditions... And calculated almost 5G acceleration seems somewhat too much for these realistic physical dimensions...

Piotr W - 3 years, 2 months ago

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The picture is one from above, looking down on the tablet which is lying flat on a horizontal table. Consequently there is no gravitational effect, and the table does not prevent the rotational motion. (If the picture was from the side, as you suggest, the tablet would be resting balanced on its edge, which is most unlikely.)

Mark Hennings - 3 years, 2 months ago

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Oh, I get it. But even then... linear acceleration equation cant be applied directly as shown on first picture (or can it?). Because you are not pulling at the center of mass, but eccentrically. Thus the 15n would not pull a mass of 1kg, but would pull a dynamic lever on which 1kg is hunged... My rough guess is that the force pulling center of mass, doubles. The rotational part calculation however seems ok.

Piotr W - 3 years, 2 months ago

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@Piotr W The force will indeed induce linear acceleration as well as rotational acceleration. You don’t have to apply a force directly at the centre of mass to obtain linear motion. Think how you can drag a box along the floor by pulling at the top.

If you forces are directed through the centre of mass, there will be no rotational motion. The only way to have no linear motion is if the resultant force is zero.

Mark Hennings - 3 years, 2 months ago

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@Mark Hennings Yes, of course. You willhave two components, but the linear one cant be calculated as it is. Because the effective motion vector is not parallel to the force applied (by motion vector i mean linear vector from center of tablet mass). Thus the force will not pull 1kg but less. So the linear gain will be higher than it currently is calculated. Issue is, that the two effects: rotation and linear movement are coupled. And cant be superpositioned, as it currently is shown (by simple movement + rotation). I think it is second order equation, not a linear one.

Piotr W - 3 years, 2 months ago

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@Piotr W If forces F j \mathbf{F}_j act at points r j \mathbf{r}_j on a solid body of mass m m , then the equations of motion are m d 2 g d t 2 = j F j d H d t = j r j F j m\frac{d^2\mathbf{g}}{dt^2} \; = \; \sum_j \mathbf{F}_j \hspace{2cm} \frac{d\mathbf{H}}{dt} \; = \; \sum_j \mathbf{r}_j \wedge \mathbf{F}_j where g \mathbf{g} is the position vector of the centre of mass, and H \mathbf{H} is the angular momentum about the centre of mass. If the forces are all planar, the angular momentum simply becomes I ω ˙ k I\dot{\omega}\mathbf{k} , where I I is the moment of inertia about the centre of mass, ω \omega is the angular velocity, and k \mathbf{k} is a unit vector normal to the plane.

In this case, the applied force produces the linear acceleration and angular acceleration exactly as described in the solution. In general, angular motion gives radial ( r ω 2 r\omega^2 -like) and transverse ( r ω ˙ r\dot{\omega} -like) components of acceleration. Since the tablet is initially stationary, there is no radial component of acceleration from the angular motion, and so we have the linear acceleration (parallel to the applied force) plus the transverse angular acceleration (perpendicular to the applied force).

Mark Hennings - 3 years, 2 months ago

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@Mark Hennings I will try to put it in yet another words. In frictionless world, the 15n force would make all the thing a) rotate with center of rotation in middle of tablet (as in resolution and Your explanation) B) accelerate along the force (again, as in resolution and Your explanation) C) move like a pendulum with like movement critically dumped down right. The 3rd compnent (c) is not taken into account in the solution. This component is dependant on both a and b. And it is present just because the force was applied excentrically. And the momentum of inertia is non zero (of course it is) which causes the point of rope connection to have also a down right component... Edit: not point of rope connection, but center of square. Edit2: maybe i am wrong... This component will manifest itself later, probably not initially... Ita too long since I last time solved such excercises...

Piotr W - 3 years, 2 months ago

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@Piotr W There is no option C. There is linear motion, and there is rotation. Pendula display both types of motion, but as a combination of the first two types of motion, not through some extra third factor. If you wrote down a few equations to explain what you are trying to do, I could maybe point out the problem.

Mark Hennings - 3 years, 2 months ago

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@Mark Hennings Mark, Thanks for this. I had no time since our discussion, but almost immediately after sending my last mail I found out why I did err... The issue is that I applied in my imagination a 'magic' mass-like properties to the point of force attachment. I also incorrectly thought that i'm applying force by pulling a cord... It's almost 20 years since I resolved last time this kind of calculations (university), so I applied too much of common physics to the selctive concepts of theoretical physics :)

One theoretical experiment is for example to imagine the continuous application of the force (exactly as described in the excercise) from the edge of square. It would produce a periodical 90 degree turns, forth and back, with the center of rotation exactly in the middle of the square. Because the motion is exactly symmetrical after passing 45 degrees (first advancing the rotation after passing 45degs reducing it). Of course there would be additional linear acceleration component, but if I forget about it - it explains exactly the same thing as is described by You.

Just two components (linear + rotational with center of rotation exactly in the middle of square)... So - again - sorry for my being so stubborn in the incorrect understanding of the issue :)

Piotr

Piotr W - 3 years, 1 month ago

Where does I = 1/6*ml^2 come from?

Liss Davison - 3 years, 2 months ago

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This is the moment of inertia of a uniform square lamina of mass m m and side length \ell about an axis through its centre and perpendicular to the plane of the lamina.

Mark Hennings - 3 years, 2 months ago

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How do I calculate it?Do I need multiple integral for this?

সামিন সালেক - 3 years, 1 month ago

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@সামিন সালেক You can do it as a double integral / 2 / 2 d x / 2 / 2 d y m 2 ( x 2 + y 2 ) \int_{-\ell/2}^{\ell/2}\,dx \int_{-\ell/2}^{\ell/2}\,dy \, \tfrac{m}{\ell^2}(x^2 + y^2)

Mark Hennings - 3 years, 1 month ago

Could u pls explain why is the r taken in torque formula T=F*r, is l/√(2)!!!

erica phillips - 3 years, 2 months ago

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That is the distance between the center of mass and the point where the charging cord is attached to the tablet. More accurately, that is the distance between the line of force and the center of mass.

Laszlo Mihaly - 3 years, 2 months ago

I must have missed the part where it was stated that this takes place in a zero-gravity environment.

Ban Yan - 3 years, 2 months ago

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It is not zero gravity. Here is the source of the misunderstanding: The illustration shows the tablet from top to down view, flat on the surface of the table. The tablet is horizontal.

Laszlo Mihaly - 3 years, 2 months ago

Very nice! But one thing I don't understand: Why is the acceleration at A that is caused by the rotation equal to l * α / sqrt(2) ? (see second picture in the solution).

Max Tröger - 3 years, 2 months ago

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Coz according to the question it has been asked to find the initial acceleration at the point A!!!

erica phillips - 3 years, 2 months ago

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Yes, but that didn't answer my questen. Why is the acceleration in this direction (caused by the rotation) equal to l * α / sqrt(2). How do you come up with this term?

Max Tröger - 3 years, 2 months ago

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@Max Tröger See firstly,the rotation happens along an arc and there is a formula which is a= αR,here R as we know is l/√(2),which is distance from the center of mass!!!

erica phillips - 3 years, 2 months ago

Does rotation is possible on smooth surface by acting a single force at a point on rigid body.i am in doubt is there possibility of rotation in the above scenario not considering friction of the surface

Thothadri Krishna B - 3 years, 2 months ago

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