Inclined plane problem

[Vidrinho]

TL;DR Summary: Inclined plane; simple pendulum; true or false question; Newton's laws; dynamics

(CESGRANRIO-RJ) In the figure, the cart moves along an inclined plane, subjected only to gravitational interactions and the surface of the inclined plane. Stuck to the cart's ceiling, there's a simple pendulum whose wire is perpendicular to the system's direction of movement.

The following statements are made:
I) The cart is descending the inclined plane.
II) The cart's movement is uniform.
III) There's no friction between the cart and the plane.

It is true only for the statements in:

Figure:

I and II are false.
III is true
I can't prove III is true in a way that makes sense.

 

[A.T.]

I and II are false.

Are you sure about I ?

III is true
I can't prove III is true in a way that makes sense.

The pendulum is like an accelerometer. It indicates the direction opposite to proper acceleration, or the acceleration relative to a free falling frame, where the contact force accounts for all acceleration. If the pendulum is perpendicular to the incline the contact force has only a normal component.

 

[Vidrinho]

Are you sure about I ?

The pendulum is like an accelerometer. It indicates the direction opposite to proper acceleration, or the acceleration relative to a free falling frame, where the contact force accounts for all acceleration. If the pendulum is perpendicular to the incline the contact force has only a normal component.

I'm pretty sure about statement I. Force and velocity are different things. If we give the cart an initial velocity perpendicular to the plane and in the opposite direction to the cart's horizontal component, it could move up the plane with no other forces acting other than those specified in the problem, and therefore that would create the exact same scenario but with the cart moving up.

If the cart had greater acceleration, would the pendulum still be perpendicular? Could we prove there's no friction using the same logic?

 

[Orodruin]

Are you sure about I ?

It can be true or not depending on the state of motion of the cart. The point is that it is not necessarily true.

 

[A.T.]

The point is that it is not necessarily true.

The statement that it necessarily must be decending is indeed false. I would just be worried that choosing false will be taken as meaning that it can't be decending.

That's why I'm no fan of such binary multiple choice questions. One can fail at the statement logic interpretation, rather than the thing that is actually supposed to be tested.

 

[A.T.]

If the cart had greater acceleration, would the pendulum still be perpendicular? Could we prove there's no friction using the same logic?

What do you mean by "greater acceleration"? How would the magnitude of proper acceleration affect my argument? Only its direction matters for the direction of the pendulum.

 

[haruspex]

It can be true or not depending on the state of motion of the cart. The point is that it is not necessarily true.

How do we know that is the point? It says these statements are made, so imagine somebody makes those statements. The issue becomes whether they are deducing them from the same information we have or may have further information. In the former case we may respond no, you don’t know that, and in the latter, yes, you could be right.
I find it entirely ambiguous.

It is also unclear whether the diagram (with the perpendicularity) represents an ongoing arrangement or just a snapshot. If ongoing, (I) is necessarily true; if a snapshot, (I) and (III) are indeterminate.

And what is "uniform motion"? Can that be uniform acceleration or only uniform velocity? What about uniform jerk?

 

[erobz]

What I'm finding is that (iii) is true if this behavior is observed throughout the motion.

I take (ii) to mean if the cart is moving at constant velocity down the hill. Moving up the hill under only the influence of gravity and a dissipative friction force at constant velocity makes no sense to me.

@Vidrinho, I apply Newtons 2^nd Law to determine the angle between the string and vertical(inertial frame), and made a plot in Desmos playing around with the values for friction to see how the angle responds.

 

[haruspex]

What I'm finding is that (iii) is true if this behavior is observed throughout the motion.

Agreed.

I take (ii) to mean if the cart is moving at constant velocity down the hill. Moving up the hill under only the influence of gravity and a dissipative friction force at constant velocity makes no sense to me.

If we interpret the question as asking whether the statement could be true under appropriate circumstances, there is no need to assume it means moving down the hill. That would simply be one of the necessary circumstances.

 

[Lindsaatar]

Tried to prove it from scratch, so I hope it makes sense. PS I guess the expression of 'no friction' shall be referring to 'no sliding/no sliding friction' between slope&wheels. Otherwise the wheels would be useless, though results won't be affected physically either way.

 

[A.T.]

I guess the expression of 'no friction' shall be referring to 'no sliding/no sliding friction' between slope&wheels. Otherwise the wheels would be useless, though results won't be affected physically either way.

Yes, it doesn't matter what kind of friction. In fact, any external force on the cart, that has a component parallel to the incline, and is not a mass-proportional body-force (like gravity), will make the pendulum deviate from the indicated normal direction.

 

[haruspex]

PS I guess the expression of 'no friction' shall be referring to 'no sliding/no sliding friction' between slope&wheels. Otherwise the wheels would be useless

Are you suggesting that static friction is ok, as long as there is no axle friction?
Not so. Even if the wheels are rolling without frictional loss, their moments of inertia would reduce the linear acceleration, deflecting the pendulum from the shown angle. So yes, the wheels need to be frictionless against the plane, and therefore useless.

 

[A.T.]

Even if the wheels are rolling without frictional loss, their moments of inertia would reduce the linear acceleration, deflecting the pendulum from the shown angle. So yes, the wheels need to be frictionless against the plane, and therefore useless.

Useless or massless (or at least have all mass concentrated at the centers and thus no rotational inertia).

 

[haruspex]

Useless or massless (or at least have all mass concentrated at the centers and thus no rotational inertia).

Right, thanks. I intended to mention that option but got distracted.