Fall Speed Based on Angle: 0°, 45°, 90°

[adjacent]

Homework Statement

Let the angle of the block(slope) to be 30°.
height=0.6m, length=1m. Breadth=1m
Smallest height on diagram is 0m

Let the red line to divide the block by 2(when viewed from the top)
The two wheels are of equal diameter and is connected by a rod.
The total mass of two wheels + rod is 1kg.
Let the angle be measured from the red line.(At the image,the wheel system is 90° to the line)
1-Rotate the wheel to 0°.
It will not move.
2-Rotate to 90°
it will fall at maximum speed.
3- Rotate to 45°
will be slower than 90°.

So I am thinking of the relation between the angle and the speed of fall.

Homework Equations

$F=ma$
...

The Attempt at a Solution

Sorry I am at Gr.10. So I haven't studied vectors,forces or speeds at angles.
-Just curious-

 

[voko]

A useful property here is the conservation of energy. When the system moves on the incline, its centre of mass gets lower. That means its potential energy is reduced. Which in turn means its kinetic energy is increased. Clearly when it goes "sideways" it needs to go a longer distance to gain the same amount of energy as compared with the case when it goes "straight".

Alternatively, you can look at the system so that your line of sight is always parallel to the axis of the system. Then you will see that it effectively rolls on an incline with some angle that depends on the "real" incline angle and "sideways" angle.

 

[adjacent]

Thank you.
Any specific equations for these kinds of "incline" things?

 

[voko]

I think you could try figuring out the "effective" incline angle. This is pure geometry.

 

[adjacent]

I don't understand what you mean by the axis of the system.
Anyway,I want to find the relationship.
So I have to make an equation relating incline angle and wheel angle.

 

[voko]

Let the top point of the red line be A. The lowest point of the red line is B. The point on the ground under A is C, so ABC is a right triangle, and the angle ABC is $\alpha$.

The system starts at A and rolls down the incline at some angle with the red line and it touches the ground at D. The angle DAB is the "sideways" angle $\beta$. The "effective" incline angle is ADC. Express angle ADC in terms of $\alpha$ and $\beta$.

 

[adjacent]

Let the top point of the red line be A. The lowest point of the red line is B. The point on the ground under A is C, so ABC is a right triangle, and the angle ABC is $\alpha$.

The system starts at A and rolls down the incline at some angle with the red line and it touches the ground at D. The angle DAB is the "sideways" angle $\beta$. The "effective" incline angle is ADC. Express angle ADC in terms of $\alpha$ and $\beta$.

Isn't D the same as B?I said in the OP that lowest point of the red line and the ground has a distance of 0m

 

[voko]

Isn't D the same as B?

How? The system rolls away from the red line, so it cannot end up at B.

I said in the OP that lowest point of the red line and the ground has a distance of 0m

Do you think I indicated otherwise?

 

[adjacent]

How? The system rolls away from the red line, so it cannot end up at B.

Oh sorry.I thought B as the point of ground.
Ok.Its a point on the red line and D is a point on ground(Where it touches)

How's that going to relate wheel angle and speed?

 

[voko]

Oh sorry.I thought B as the point of ground.
Ok.Its a point on the red line and D is a point on ground(Where it touches)

I do not think you understood me. Points B, C, D are all on the ground. Point B in particular is where the red line touches the ground. Point D is where the rolling system touches the ground. These two points can also be said to lie on the edge of the incline.

How's that going to relate wheel angle and speed?

I do not know what the "wheel angle" is and I would like you to answer my question about angle ADC first, anyway.

 

[adjacent]

Let the top point of the red line be A. The lowest point of the red line is B. The point on the ground under A is C, so ABC is a right triangle, and the angle ABC is $\alpha$.

The system starts at A and rolls down the incline at some angle with the red line and it touches the ground at D. The angle DAB is the "sideways" angle $\beta$. The "effective" incline angle is ADC. Express angle ADC in terms of $\alpha$ and $\beta$.

I would like you to answer my question about angle ADC first, anyway.

That seems to be a 3d problem.How am I going to do that?I haven't studied that yet.Can you give me a hint?

 

[voko]

You solve a 3D problem by disassembling it into 2D subproblems.

In this case, you have three triangles: ABC, ABD, and ADC. These are right triangles and they are all related to each other in some way. Triangle ABD and triangle ADC share hypotenuse AD. Triangle ABC and triangle ADC share cathetus AC. This is enough to express angle ADC via angle ABC ($\alpha$) and angle DAB ($\beta$), it takes just a bit of trigonometry.

 

[adjacent]

Is it OK to include other variables like BD,CB and AC?

 

[voko]

There is an answer that only has $\alpha$ and $\beta$. But we can discuss your approach even if you do not have that answer yet. Show what you have so far.

 

[adjacent]

OK.

I've got
$\alpha=tan^{-1}(\frac{h}{b})$
$\beta=tan^{-1}(\frac{S}{\sqrt{h^2+b^2}})$
$ADC=tan^{-1}(\frac{h}{\sqrt{s^2+b^2}})$

Not sure how to link $\alpha$,$\beta$ and ADC in one equation

EDIT:Latex problem.How do I fix that?

 

[voko]

You meant those:
$\alpha=tan^{-1}(\frac{h}{b})$
$\beta=tan^{-1}(\frac{S}{\sqrt{h^2+b^2}})$
$ADC=tan^{-1}(\frac{h}{\sqrt{s^2+b^2}})$

(one closing curly brace was missing)

 

[adjacent]

You meant those:
$\alpha=tan^{-1}(\frac{h}{b})$
$\beta=tan^{-1}(\frac{S}{\sqrt{h^2+b^2}})$
$ADC=tan^{-1}(\frac{h}{\sqrt{s^2+b^2}})$

(one closing curly brace was missing)

Yeah,it's that.
How do I link those three together?

 

[voko]

As I said:

Triangle ABD and triangle ADC share hypotenuse AD. Triangle ABC and triangle ADC share cathetus AC.

So, can you express AD via AB and $\beta$? Then, can you express AB via AC and $\alpha$? And, finally, find the ratio of AC with AD?

 

[adjacent]

So, can you express AD via AB and $\beta$?

$AD=\frac{AB}{cos\beta}$

Then, can you express AB via AC and $\alpha$?

$AB=\frac{AC}{sin\alpha}$

And, finally, find the ratio of AC with AD?

How do I do that?I don't understand what you said.

 

[voko]

You are required to find a certain angle, let's call it $\gamma$. Is it related in any way with $$AC \over AD$$ ?

 

[adjacent]

You are required to find a certain angle, let's call it $\gamma$. Is it related in any way with $$AC \over AD$$ ?

Now I get
$\gamma=\frac{AB\sin\alpha}{\frac{AB}{\cos\beta}}$
Now I need to cancel AB somehow.
So ans is :(I always have trouble when 3 divisions are involved.Can you explain that as well?)

$\frac{\sin\alpha}{\cos\beta}$ = effective incline?

 

[voko]

Now I get
$\gamma=\frac{AB\sin\alpha}{\frac{AB}{\cos\beta}}$

Close, but not quite. Is it the angle that is a ratio of sides, or some function of the angle?

Now I need to cancel AB somehow.
So ans is :(I always have trouble when 3 divisions are involved.Can you explain that as well?)

Oh, that is very simple. When you have $$ {\frac a b \over \frac c d}$$ you eliminate the inner denominators by pulling $d$ up and pulling $b$ down, so that results in $$ ad \over bc $$ Note in your case $b = 1$.

$\frac{\sin\alpha}{\cos\beta}$ = effective incline?

Is it?

 

[adjacent]

Now I get
$\gamma=\frac{AB\sin\alpha}{\frac{AB}{\cos\beta}}$

Close, but not quite. Is it the angle that is a ratio of sides, or some function of the angle?

AC=$AB\sin\alpha$
AD=$\frac{AB}{\cos\beta}$

So $AC \over AD$ = $\frac{AB\sin\alpha}{\frac{AB}{\cos\beta}}$


So,$\gamma$=$\sin\alpha\cos\beta$ ?

 

[voko]

So, is it $$ \gamma = {AC \over AD} $$ or $$ \mathrm{function}(\gamma) = {AC \over AD} $$ ? And what $\mathrm{function}$ is that?

And please simplify that ratio!

 

[adjacent]

So, is it $$ \gamma = {AC \over AD} $$ or $$ \mathrm{function}(\gamma) = {AC \over AD} $$ ? And what $\mathrm{function}$ is that?

And please simplify that ratio!

Ah.You said just a ratio.From where did this FUNCTION come.
Anyway,I know what you mean.
It's $\gamma=tan^{-1}(\sin\alpha\cos\beta)$

 

[voko]

Ah.You said just a ratio.From where did this FUNCTION come.

Ratios of sides in a right triangle are always trigonometric functions of its angles.

Anyway,I know what you mean.
It's $\gamma=tan^{-1}(\sin\alpha\cos\beta)$

Why is that a $\tan^{-1}$? Can the ratio of a cathetus with the hypotenuse be a $\tan$?

 

[adjacent]

Ratios of sides in a right triangle are always trigonometric functions of its angles.



Why is that a $\tan^{-1}$? Can the ratio of a cathetus with the hypotenuse be a $\tan$?

Oh.I'm sorry it's $\sin^{-1}$

So $\gamma=\sin^{-1}(\sin\alpha\cos\beta)$

Then what does this have to do with the speed of the wheel?

 

[voko]

Very well. So now the problem is reduced from 3D to 2D. The system rolls along a straight line at angle $\gamma$ with the horizon.

Now go back to the beginning of the thread. Can you apply conservation of energy?

 

[adjacent]

$E_p=mgh$
h=0.6m here.
so $E_p=6J$
$E_k=\frac{1}{2}mv^2$
$v=\sqrt{12}$
Ah how does this relate to AD?

 

[voko]

$E_p=mgh$
h=0.6m here.
so $E_p=6J$

That is potential energy at the top of the incline. We need it at other points, too.

Let $x$ be the distance from point D (in the direction of A) where the system is located. Can you find the potential energy of the system at $x$?

$E_k=\frac{1}{2}mv^2$

That is correct for a point mass. But we have a rod and wheels, which rotate. Their rotation also has kinetic energy. Are you familiar with that?