[alancalverd]

So why does the car roll down the hill when you don't apply the parking brake?

If you draw the vector you will see that the offset weight applies a torque (force x distance) causing the wheels to rotate. Therefore in order to stop the wheels rotating we need to apply a counter-torque to stop them.

[hamdani yusuf (OP)]

So why does the car roll down the hill when you don't apply the parking brake?

If you draw the vector you will see that the offset weight applies a torque (force x distance) causing the wheels to rotate. Therefore in order to stop the wheels rotating we need to apply a counter-torque to stop them.

You can put a wedge under the tyres.

[hamdani yusuf (OP)]

Option 3: Change the unit of a rotational radius to meters per radian, while keeping the unit of a geometric radius in meters. A radius is geometric if it can still have a defined value while the object is stationary. Rotational radius is only defined when there's a rotation.
Pros:
- Solve the problem from option 0.
- Solve the problem from option 1.
Cons:
- Something needs to be done.
- Currently existing textbooks need to be revised.
- Requires more characters because the unit of rotational angle can not be omitted.
- People need to unlearn the old standard and relearn the new standard, which can be hard for some of them.

*All options except option 0 share these cons.

A visual description might be useful in understanding the difference between geometric radius and rotational radius.

Geometric radius of the wheel is more or less constant. It's half of the diameter.
On the other hand, each point in the wheel has their own rotational radius. Furthermore, if the axle shaft is inserted to a hole other than the central one, their rotational radius will be different.

[hamdani yusuf (OP)]

I think you (and maybe others) have been misled by the notion of "rotational analogs of linear quantities".

I think you (and maybe others) have been misled by the deceptively simple formula for torque, without considering that it's correlated to other rotational quantities. As I mentioned in my video, torque is only one third of rotational quantities with problematic standard units. The other two are angular momentum and rotational inertia.

These three proposed units for new standard of rotational quantities are simply a logical consequences from changing the standard unit of rotational radius to meter per radian.

Here's the updated table for proposed new standard units, now including kinetic and potential energy, as well as torsional stiffness.


If you can find inconsistency in this table, please let me know.

These proposed units address inconsistencies in current standard units as shown in the table below.

[hamdani yusuf (OP)]

With the new proposed standard units, conversion with non-standard units of rotational quantities can be done naturally and with flexibility.

[alancalverd]

So how do you calculate the brake pad force required to stop the car from rolling down the hill? Vehicle manufacturers don't seem to find this an insuperable problem.

[hamdani yusuf (OP)]

So how do you calculate the brake pad force required to stop the car from rolling down the hill? Vehicle manufacturers don't seem to find this an insuperable problem.

https://www.akebono-brake.com/english/product_technology/product/automotive/disc/
You need to know the mass of the car, slope of the hill, gravity acceleration, frictional coefficient between brake pad and disc, size of the brake. Why do you insist to complicate things before properly understanding the simplest fundamentals? Newton developed his laws of motion using simplest cases first.

[hamdani yusuf (OP)]

Confusion, ad infinitum!

I'm sorry that I couldn't describe the problem simple enough for you to understand it. Let me try again.
Current standard units create inconsistencies when compared to the results from equations relating one rotational quantities to the others.



Compare them with the new proposed standard units, which are consistent with the relating equations.

[hamdani yusuf (OP)]

The proposed new standard units essentially come from distinction between rotational radius and geometric radius. By expressing rotational radius as length per rotational angle, we get the following results :
Rotational inertia = mass times radius of rotation squared = mass times (distance per rotational angle) squared
Angular momentum = momentum times radius of rotation = momentum times (distance per rotational angle)
Torque = force times radius of rotation = force times (distance per rotational angle)

[alancalverd]

You need to know the mass of the car, slope of the hill, gravity acceleration, frictional coefficient between brake pad and disc, size of the brake. Why do you insist to complicate things before properly understanding the simplest fundamentals? Newton developed his laws of motion using simplest cases first.

The mass of the car is m
Slope is θ
Gravitational acceleration is g
Frictional coefficient is μ
Brake disc radius to the pad is r

Now please show us how your definition of torque allows us to calculate the required brake pad force to prevent the car from rolling, and why everyone else's definition doesn't.

Simple? This is the sort of calculation we expect a 16-year-old schoolkid to do in his Applied Maths exams if he has any chance of becoming a scientist or engineer.

[hamdani yusuf (OP)]

This is my first video about rotational quantities.

I asked thestify.ai to review my essay in the video, formatted in pdf.

💪
What works well
The essay effectively identifies the inconsistencies in current standard units for rotational quantities and proposes a logical set of new units to address these issues.

⚔️
What can be improved
However, the argument would benefit from additional references to support the mathematical relationships and proposed units, as well as improved transitions between sections for better coherence.

❇️
Overall assessment
Overall, the document presents a solid foundation for its thesis but requires further elaboration and refinement to enhance clarity and persuasiveness.

Recommendations
High Impact
1.
Clearly articulate a strong and specific thesis statement in the introduction.
The essay currently lacks a clear and assertive thesis statement that defines the main argument regarding the inconsistencies in current standard units for rotational quantities. A well-defined thesis is crucial for guiding the essay's structure and argumentation, allowing readers to understand the central focus of the discussion.

High Impact
2.
Thoroughly discuss the implications of inconsistencies in standard units across various fields.
The essay does not adequately explore how inconsistencies in standard units for rotational quantities impact different fields. A thorough discussion of these implications would enhance the essay's relevance and depth, making the argument more compelling and applicable to real-world scenarios.

High Impact
3.
Incorporate additional scholarly sources to support claims and enhance credibility.
The essay lacks sufficient scholarly references to substantiate its claims, particularly regarding the mathematical relationships and proposed units. Incorporating credible sources would strengthen the arguments, provide necessary context, and ensure that each claim is well-supported throughout the essay.

[hamdani yusuf (OP)]

It also evaluates my evidence.

Evidence
Score: Can be improved
Evidence and thesis statement
Partially met

The thesis statement appears to center around the distinction and relationship between torque and energy, specifically how they are measured in the same units yet represent different concepts. However, the thesis lacks specificity and assertiveness, making it difficult to evaluate the evidence effectively. The evidence provided includes historical definitions, equations, and comparisons of rotational and linear quantities, which support the thesis but do not thoroughly strengthen it or address opposing views. There is insufficient evidence that challenges the thesis statement, which could enhance the argument's credibility.

Quality and types of evidence
Partially met

In the field of physics, particularly mechanics, evidence relevant to the discussion of torque and energy would include empirical data, historical definitions, and mathematical equations. For a given length of the essay and its field, the typical number of sources cited should be in the range of 5-10. The essay provides some relevant evidence, such as historical definitions and equations, but lacks a sufficient number of sources to substantiate the claims made. The integration of evidence is inconsistent, and there are instances of evidence that appear without proper referencing, which undermines the essay's academic rigor.

Evidence missing a reference
"Torque is a rotational force, while energy is the capacity to do work."
This statement lacks a reference, which is necessary to support the claim.

And gives recommendations.

Recommendations
1.
Strengthening the Thesis Statement
Instruction
Revise the thesis statement in the introduction to make it more specific and assertive, clearly outlining the argument being made about the relationship between torque and energy.

Where to find in the text
It started with two distinct physical quantities that have the same unit, even though they represent different concepts: Torque and Energy.
2.
Increasing the Quality and Number of Sources
Instruction
Incorporate additional scholarly sources that discuss the relationship between torque and energy, ensuring that each claim is supported by credible references throughout the essay.

Where to find in the text
Torque is a rotational force, while energy is the capacity to do work.

[hamdani yusuf (OP)]

The second video highlights some previous attempts to change standard units for some rotational quantities, especially rotational angle and rotational radius, and explain why they failed.

[hamdani yusuf (OP)]

You need to know the mass of the car, slope of the hill, gravity acceleration, frictional coefficient between brake pad and disc, size of the brake. Why do you insist to complicate things before properly understanding the simplest fundamentals? Newton developed his laws of motion using simplest cases first.

The mass of the car is m
Slope is θ
Gravitational acceleration is g
Frictional coefficient is μ
Brake disc radius to the pad is r

Now please show us how your definition of torque allows us to calculate the required brake pad force to prevent the car from rolling, and why everyone else's definition doesn't.

Simple? This is the sort of calculation we expect a 16-year-old schoolkid to do in his Applied Maths exams if he has any chance of becoming a scientist or engineer.

What's the initial velocity?
How far the car is allowed to move before it's stopped?

Current standard units of rotational quantities has following weaknesses:
- It leads inevitably to ghostly appearances and disappearances of the radian in the dimensional analysis of physical equations.
- A perennial problem in the teaching of mechanics, where radian appears on one side of an equation, but not on the other side.
- The typical advice of ignoring radians during dimensional analysis and adding or removing radians in units according to convention and contextual knowledge is "pedagogically unsatisfying".
- Inflexibility of conversion between radian and other units of angle in some equations.

They can still be used, if you are willing to ignore those weaknesses, and find a way around them.

[hamdani yusuf (OP)]

Now please show us how your definition of torque allows us to calculate the required brake pad force to prevent the car from rolling, and why everyone else's definition doesn't.

What's the initial velocity?
How far the car is allowed to move before it's stopped?

As I mentioned earlier, the weakness of current standard unit of torque becomes obvious when actual angular acceleration is not zero. How does the angle unit of radian suddenly appear, instead of the other units?
The case with levers used by Archimedes to introduce the concept of torque is only good when the quantitative value of angular acceleration is ignored.

[hamdani yusuf (OP)]

Can you do it without any change in position of the lever?

In principle, yes. In practice, any real lever will bend a bit (indeed some torque wrenches use the bending to measure the tporque), but the applied torque is independent of the elasticity of the lever: whether you use a rigid bar or a flexible one, torque is just the product of force x distance.

Here's a thought experiment in a workshop. We wanted to release a bolt from a corroded valve. It's clamped on a bench using a vise. A large wrench was used in an attempt to turn the bolt to release it. After a force was applied, the bolt didn't turn. It moved the whole bench instead.
It reminds you the definition of torque in terms of cross product between force and radius of rotation. It's not the radius of the object.

Here's the diagram for simplified version of the case.

The length of the wrench only represents an expected value for the radius of rotation. But the actual rotation in this case is around the bottom of left leg of the bench. The radius that should be used to calculate torque is the radius of actual rotation.
Let me remind you that unexpected results come from false assumptions.

Now, in other case where there is no actual rotation, even so slightly, what is the radius of rotation that you will use to calculate torque?

Imagine there's a crack at the middle of the lever of the wrench. When the force is applied at its end, the wrench breaks at the cracked point, and the actual rotational radius is half the expected radius, which makes the torque that you produce half of what you expected.
Every point in the system has a potential torque as the force is applied, according to the distance from the applied force and the angle between the distance vector and the direction of the force. But the actual torque is defined by the actual axis of rotation, which determines the radius of rotation.

[alancalverd]

What's the initial velocity?
How far the car is allowed to move before it's stopped?

Initial velocity = 0 (the question referred to the parking brake, which you only use once the car has stopped moving)
Permitted rolling distance = 0, obviously.

Now please answer the question: what force is required for the brake pads to prevent the car rolling down the hill? You may use the data I  gave in reply 509 above.

[alancalverd]

How does the angle unit of radian suddenly appear, instead of the other units?

Because, since the circumference of a circle is 2π r, the ratio of distance moved along a circular path s / radius of movement r is s/r radians if s and r are in the same units.

[hamdani yusuf (OP)]

How does the angle unit of radian suddenly appear, instead of the other units?

Because, since the circumference of a circle is 2π r, the ratio of distance moved along a circular path s / radius of movement r is s/r radians if s and r are in the same units.

Exactly. Then why it disappeared in the unit of torque?

[hamdani yusuf (OP)]

Initial velocity = 0 (the question referred to the parking brake, which you only use once the car has stopped moving)
Permitted rolling distance = 0, obviously.

This is the only condition your definition of torque can work, i.e. where angular acceleration is zero.