What is the Relationship Between Restoring Force and Angle in a Pendulum?

[mancity]

Homework Statement

Which of the following is true regarding the restoring force on a pendulum?

I. The force is constant and dependent on the mass of the pendulum bob.

II. The force is always in the same direction.

III. The force is proportional to the distance from the rest point.

IV. The force is always directed toward the rest point.

Relevant Equations

n/a (conceptual problem)

I put the answer as (IV) but that happens to be wrong (or maybe it was only one of the multiple correct answers). Here is my reasoning:
I. the force is dependent on mass, but isn't always constant.
II. It's not always in the same direction, it points towards the rest point. Consider a point at top/ almost near the bottom. they won't be in the same direction
III. I believe it is proportional to the angle, not the distance.
IV. I'm pretty sure this is true, as intuitively that's what a restoring force does
Can someone help me as to why this answer of (IV) is wrong, and help guide me towards the correct answer(s)? Thanks

 

[Lnewqban]

Please, see:
https://courses.lumenlearning.com/suny-physics/chapter/16-4-the-simple-pendulum/

https://en.wikipedia.org/wiki/Pendulum

 

[haruspex]

Homework Statement: Which of the following is true regarding the restoring force on a pendulum?

I. The force is constant and dependent on the mass of the pendulum bob.

II. The force is always in the same direction.

III. The force is proportional to the distance from the rest point.

IV. The force is always directed toward the rest point.
Relevant Equations: n/a (conceptual problem)

I put the answer as (IV) but that happens to be wrong (or maybe it was only one of the multiple correct answers). Here is my reasoning:
I. the force is dependent on mass, but isn't always constant.
II. It's not always in the same direction, it points towards the rest point. Consider a point at top/ almost near the bottom. they won't be in the same direction
III. I believe it is proportional to the angle, not the distance.
IV. I'm pretty sure this is true, as intuitively that's what a restoring force does
Can someone help me as to why this answer of (IV) is wrong, and help guide me towards the correct answer(s)? Thanks

It's a poorly thought out question.

First, which force, the net force, gravity, the tension, or some component of one of those? I would say "the restoring force" could mean the component of the net force directed along the path of travel.

Next, is one supposed to consider that the path of the bob is an arc?
To get true SHM, we need to consider only a vanishingly small arc of swing so that we can approximate it as linear. That makes III and IV both true.
Treating it as the arc it really is, there is some ambiguity in "towards the rest point". That could mean directly towards it, or in that direction along the path of travel which leads to to the rest point. Combining my interpretation of restoring force with that latter interpretation of direction makes your answer defensible; III is not.

 

[mancity]

So, per my understanding, would the correct answer then be III & IV? (because that is indeed an answer choice. all other answer selections only have one, i.e. I, or II, or III, or IV).

Because I believe III only holds for small angles (in which we can approximate theta=sin(theta)), which would make that statement true.

 

[haruspex]

So, per my understanding, would the correct answer then be III & IV? (because that is indeed an answer choice. all other answer selections only have one, i.e. I, or II, or III, or IV).

Because I believe III only holds for small angles (in which we can approximate theta=sin(theta)), which would make that statement true.

As I wrote, depending on interpretation, IV also only holds for small angles, so III and IV is a reasonable answer.

 

[kuruman]

First, which force, the net force, gravity, the tension, or some component of one of those? I would say "the restoring force" could mean the component of the net force directed along the path of travel.

I think it's worse than that. The equation that one normally writes for the hanging mass involves a restoring torque, $\tau=-mgL\sin\theta$, not a restoring force. The small angle approximation appears not to be an issue because "small angles" is not mentioned as a premise to the choices.

 

[haruspex]

I think it's worse than that. The equation that one normally writes for the hanging mass involves a restoring torque, $\tau=-mgL\sin\theta$, not a restoring force.

True, but it can be expressed in terms of force on, and acceleration and displacement of, the bob. That could be either treating it as a straight horizontal line (small angle approximation) or displacement etc. around a curved path.

The small angle approximation appears not to be an issue because "small angles" is not mentioned as a premise to the choices.

We agree the question is shoddy, so small angles could have been the intent.

 

[mancity]

The correct answer as given by the system is III & IV. I believe small angles were indeed the intent for this problem.