Rolling without slipping on a curved surface

[NTesla]

Which reduces to 5gsin⁡(θ)7(R−r).

I don't understand how does that reduce to this equation. Even if we neglect the small r term in the numerator, we can't neglect the term 12rR in the denominator.

 

[haruspex]

I don't understand how does that reduce to this equation. Even if we neglect the small r term in the numerator, we can't neglect the term 12rR in the denominator.

Factorise the denominator.

 

[NTesla]

Factorise the denominator.

After factorizing, in the denominator I'm getting: $(r - R)$, where $(R - r)$ should come. I've double checked my calculations/equations, but can't find why the sign is reversed.

 

[haruspex]

After factorizing, in the denominator I'm getting: $(r - R)$, where $(R - r)$ should come. I've double checked my calculations/equations, but can't find why the sign is reversed.

If you mean compared with what I posted in post #60, I only meant that your numerator/denominator term reduced to that. I dropped the minus sign that preceded it.
$\alpha_C$ should be negative, right?

 

[NTesla]

αC should be negative, right?

yes, and it is already negative, before we proceed on to do factorization. Kindly see last line in post#59. So, eventually $\alpha_{C}$ is coming out positive, after the factorization, which it shouldn't.

 

[haruspex]

yes, and it is already negative, before we proceed on to do factorization. Kindly see last line in post#59. So, eventually $\alpha_{C}$ is coming out positive, after the factorization, which it shouldn't.

Yes, you do seem to have a sign error in post #59, but where exactly it is depends which sense you are taking as positive for each variable.
Please state that for each of $\omega_C, \omega_{cm}, L, \alpha_C, \tau_C$.

 

[NTesla]

Yes, you do seem to have a sign error in post #59, but where exactly it is depends which sense you are taking as positive for each variable.
Please state that for each of $\omega_C, \omega_{cm}, L, \alpha_C, \tau_C$.

I've figured out where the sign convention went wrong, and have arrived at the correct answer.

@haruspex, @kuruman : Appreciate your kind help very much.

However, I'm working on another method of solving the same problem of finding the time period of oscillation of the ball, by using Energy conservation.

I'm now still struggling with finding the total Kinetic energy of the ball.
The way I'm thinking: Since the body is rotating around 2 different axes, therefore, it's total K.E = $\frac{1}{2}mv^{2} +\frac{1}{2}I_{cm}\omega_{cm} ^{2} + \frac{1}{2}I_{p}\omega_p^{2}$.
However, differentiating this w.r.t $t$ is not leading to the right answer.

@haruspex, In your post#44, you mentioned, regarding MoI that:

it is only defined in respect of its actual or potential rotation as a rigid body about a specified axis - i.e., a motion in which every part of the body is rotating at the same angular rate about that axis.

My argument is that, since there are two omega in the present case: $\omega_{cm}$ and $\omega_{C}$, and their relation is $\vec{\omega _{cm}} = - \left ( \frac{R - r}{r}\right )\vec{\omega _{C}}$. Therefore, the net $\omega$ = $\vec{\omega _{cm}} - \vec{\omega _{C}}$.
Isn't this net $\omega$ same for all points on the ball ?

 

[NTesla]

Yes, you do seem to have a sign error in post #59, but where exactly it is depends which sense you are taking as positive for each variable.
Please state that for each of $\omega_C, \omega_{cm}, L, \alpha_C, \tau_C$.

I've figured out where the sign convention went wrong, and have arrived at the correct answer.

@haruspex, @kuruman : Appreciate your kind help very much.

However, now I'm working on another method of solving the same problem of finding the time period of oscillation of the ball, by using Energy conservation.

I'm now still struggling with finding the total Kinetic energy of the ball.
The way I'm thinking: Since the body is rotating around 2 different axes, therefore, it's total K.E = $\frac{1}{2}mv^{2} +\frac{1}{2}I_{cm}\omega_{cm} ^{2} + \frac{1}{2}I_{C}\omega_C^{2}$.
However, differentiating this w.r.t $t$ and proceeding on is not leading to the right answer. I understand that @haruspex in his post#54 had mentioned that: "the effective MoI we found for the one is valid for the other, but it won’t give the right result for energy." But, I'm trying to understand why .

In post#58, @haruspex has mentioned that the net angular acceleration is the vector sum of the orbital and angular acceleration.
I think, that we can say the same for net angular velocity. i.e. $\vec {\omega_{net}} = \vec {\omega_{cm}} + \vec {\omega_{C}}$ and since we have $\vec {\omega_{cm}} = - \left ( \frac{R - r}{r}\right )\vec{\omega _{C}}$, therefore, we can calculate $\vec {\omega_{net}}$.

Will this $\vec {\omega_{net}}$ not be same for all points on the ball ? If $\vec {\omega_{net}}$ will be same for all points of the ball, then as @haruspex has mentioned in post#44, regarding MoI, that: "it is only defined in respect of its actual or potential rotation as a rigid body about a specified axis - i.e., a motion in which every part of the body is rotating at the same angular rate about that axis." So, we should be able to calculate MoI, if the $\vec {\omega_{net}}$ is same for all. But, then, the conservation of energy method is not giving correct result, I'm assuming because there's something wrong with calculation of K.E itself.

 

[haruspex]

I've figured out where the sign convention went wrong, and have arrived at the correct answer.

Good.

However, now I'm working on another method of solving the same problem of finding the time period of oscillation of the ball, by using Energy conservation.

That's unlikely to work. Energy conservation equations are time independent .

The way I'm thinking: Since the body is rotating around 2 different axes, therefore, it's total K.E = $\frac{1}{2}mv^{2} +\frac{1}{2}I_{cm}\omega_{cm} ^{2} + \frac{1}{2}I_{C}\omega_C^{2}$.

No, you can't do that.
The motion of a rigid body at an instant can be expressed as the sum of the linear motion of its mass centre and its angular rotation about that centre. There is nothing more to add. If its mass centre is moving in an arc around some point C then the KE of that is already encapsulated in that of the instantaneous linear velocity of its mass centre.

 

[NTesla]

I think, that we can say the same for net angular velocity. i.e. ωnet→=ωcm→+ωC→ and since we have ωcm→=−(R−rr)ωC→, therefore, we can calculate ωnet→.

Will this ωnet→ not be same for all points on the ball ?

I would like your response on this question.

 

[haruspex]

I would like your response on this question.

No, that doesn’t work either. You cannot add angular velocities that are around different axes. What axis would the resulting angular velocity be about?
At any instant, there may be points on the sphere that are moving directly towards or directly away from C, while others may be moving in opposite directions around C.

 

[NTesla]

Kindly see post#57 and 58.
The conclusion that I'm gathering from those 2 posts is this: The net angular accleration about point C will be: $\vec{\alpha }_{net} = \vec{\alpha}_{cm} + \vec{\alpha}_{C}$. And since we have the relation between $\vec{\alpha}_{cm}$ and $\vec{\alpha}_{C}$, therefore, we can compute $\vec{\alpha }_{net}$, and thereby we can calculate MoI about point C.

 

[wrobel]

Besides a few degenerate cases, a reaction of an ideal constraint is a quadratic function of generalized velocities. And this is exactly the case. :)

 

[haruspex]

Kindly see post#57 and 58.
The conclusion that I'm gathering from those 2 posts is this: The net angular accleration about point C will be: $\vec{\alpha }_{net} = \vec{\alpha}_{cm} + \vec{\alpha}_{C}$. And since we have the relation between $\vec{\alpha}_{cm}$ and $\vec{\alpha}_{C}$, therefore, we can compute $\vec{\alpha }_{net}$, and thereby we can calculate MoI about point C.

Ok, I'll give you that one.

 

[kuruman]

Kindly see post#57 and 58.
The conclusion that I'm gathering from those 2 posts is this: The net angular accleration about point C will be: $\vec{\alpha }_{net} = \vec{\alpha}_{cm} + \vec{\alpha}_{C}$. And since we have the relation between $\vec{\alpha}_{cm}$ and $\vec{\alpha}_{C}$, therefore, we can compute $\vec{\alpha }_{net}$, and thereby we can calculate MoI about point C.

Ok, I'll give you that one.

I have a problem with the idea of adding angular accelerations this way to calculate the moment of inertia because they are about different axes. In post #56 we have

Note the last three words in the last sentence, ". . . is the ratio between the torque applied and the resulting angular acceleration about that axis." In other words, to find the MoI,

1. Select an axis.
2. Find an expression for the torque about the selected axis
3. Find an expression for the ensuing angular acceleration about the same axis
4. Divide the first expression by the second to obtain the MoI about the selected axis.

What happens when, as in this problem, one has angular accelerations about two different axes? If one conflates "resulting" with "resultant" and takes the ratio of the resultant torque to the resultant acceleration, then this ratio represents the moment of inertia about which of the two axes? The first one? The second one? Some other axis? Without a specific reference axis the moment of inertia makes no sense.

Nevertheless, I will do a calculation to get some results that illustrate why the resultant approach leads to trouble.
The figure shows a hoop of radius $R$ and mass $m$ orbiting point O. The center of the hoop is at fixed radius $2R$ from point O. A constant tangential force $F_{spin}$ acts on the hoop so that the spin angular acceleration, $\alpha_{spin}$, is clockwise. In addition, a constant tangential force $F_{orb.}$ acts at the CM of the hoop so that the orbital angular acceleration, $\alpha_{orb.}$, is counterclockwise.
Find the ratio of the net torque to the net angular acceleration.

Solution
The resulting angular accelerations are
$\alpha_{spin}=-\dfrac{F_{spin}R}{mR^2}~;~~\alpha_{orb.}=\dfrac{F_{orb.}(2R)}{m(2R)^2}\implies \alpha_{net}=\dfrac{F_{orb.}}{2mR}-\dfrac{F_{spin}}{mR}=\dfrac{F_{orb.}-2F_{spin}}{2mR}.$
The ratio of the net torque to the net acceleration is
$$\text{Ratio}=2mR\frac{F_{orb}(2R)-F_{spin}R}{F_{orb.}-2F_{spin}}=2mR^2\left(\frac{2F_{orb.}-F_{spin}}{F_{orb.}-2F_{spin}}\right).$$This ratio makes no sense as an expression for a moment of inertia because the external forces can be adjusted independently to give anything, including zero and negative values.

Placing a constraint on the angular accelerations, e.g. $\alpha_{spin}=\beta \alpha_{orb.}$ will result in a constraint between forces $F_{spin}=\gamma F_{orb.}$ so that $$\text{Ratio}=2mR^2\left(\frac{2-\gamma}{1-2\gamma}\right).$$The expression is still in trouble because of those pesky negative signs.

 

[wrobel]

$J_S\boldsymbol{\dot\omega}=\boldsymbol {SA}\times \boldsymbol F,\quad \boldsymbol \omega=\dot\varphi\boldsymbol e_z;\quad J_S=\frac{2}{5}mr^2$
$m\boldsymbol a_S=\boldsymbol F+m\boldsymbol g,\quad \boldsymbol a_S=-\dot\theta^2(R-r)\boldsymbol e_x+\ddot\theta(R-r)\boldsymbol e_y$
$\boldsymbol F=F_x\boldsymbol e_x+F_y\boldsymbol e_y$
$\boldsymbol v_S=\boldsymbol \omega\times\boldsymbol {AS}\Longrightarrow\quad \dot\theta(R-r)=-\dot\varphi r$

$$J_S\ddot\varphi=rF_y,\quad m\ddot\theta(R-r)=F_y-mg\sin\theta;$$
$$ \ddot\theta(R-r)=-\ddot\varphi r$$
From these three equations one can find the friction force as $F_y=F_y(\theta)$

 

[NTesla]

Besides a few degenerate cases, a reaction of an ideal constraint is a quadratic function of generalized velocities. And this is exactly the case. :)

I must admit, I can understand each word individually, but I couldn't understand what you wanted to say there, at all.

Kindly explain in simpler, preferably much simpler terms, what you wanted to write.

 

[NTesla]

I have a problem with the idea of adding angular accelerations this way to calculate the moment of inertia because they are about different axes. In post #56 we have

View attachment 366447
Note the last three words in the last sentence, ". . . is the ratio between the torque applied and the resulting angular acceleration about that axis." In other words, to find the MoI,

1. Select an axis.
2. Find an expression for the torque about the selected axis
3. Find an expression for the ensuing angular acceleration about the same axis
4. Divide the first expression by the second to obtain the MoI about the selected axis.

What happens when, as in this problem, one has angular accelerations about two different axes? If one conflates "resulting" with "resultant" and takes the ratio of the resultant torque to the resultant acceleration, then this ratio represents the moment of inertia about which of the two axes? The first one? The second one? Some other axis? Without a specific reference axis the moment of inertia makes no sense.
View attachment 366448Nevertheless, I will do a calculation to get some results that illustrate why the resultant approach leads to trouble.
The figure shows a hoop of radius $R$ and mass $m$ orbiting point O. The center of the hoop is at fixed radius $2R$ from point O. A constant tangential force $F_{spin}$ acts on the hoop so that the spin angular acceleration, $\alpha_{spin}$, is clockwise. In addition, a constant tangential force $F_{orb.}$ acts at the CM of the hoop so that the orbital angular acceleration, $\alpha_{orb.}$, is counterclockwise.
Find the ratio of the net torque to the net angular acceleration.

Solution
The resulting angular accelerations are
$\alpha_{spin}=-\dfrac{F_{spin}R}{mR^2}~;~~\alpha_{orb.}=\dfrac{F_{orb.}(2R)}{m(2R)^2}\implies \alpha_{net}=\dfrac{F_{orb.}}{2mR}-\dfrac{F_{spin}}{mR}=\dfrac{F_{orb.}-2F_{spin}}{2mR}.$
The ratio of the net torque to the net acceleration is
$$\text{Ratio}=2mR\frac{F_{orb}(2R)-F_{spin}R}{F_{orb.}-2F_{spin}}=2mR^2\left(\frac{2F_{orb.}-F_{spin}}{F_{orb.}-2F_{spin}}\right).$$This ratio makes no sense as an expression for a moment of inertia because the external forces can be adjusted independently to give anything, including zero and negative values.

Placing a constraint on the angular accelerations, e.g. $\alpha_{spin}=\beta \alpha_{orb.}$ will result in a constraint between forces $F_{spin}=\gamma F_{orb.}$ so that $$\text{Ratio}=2mR^2\left(\frac{2-\gamma}{1-2\gamma}\right).$$The expression is still in trouble because of those pesky negative signs.

I understand your arguments and the calculation. However, In view of that wikipedia statement, and your arguments and calculation, I'm trying to find out a common ground or a finality regarding the exact process of calculation of Moment of Inertia, and it's definition too. But it seems so elusive. After every few post, it seems, I get to know a new facet of MoI which somehow contradicts the earlier facet.

So, is there a final word regarding what should one be careful about when calculating MoI ?

 

[wrobel]

I must admit, I can understand each word individually, but I couldn't understand what you wanted to say there, at all.

I hinted on some advanced things, forget it.
By the way a phrase <<angular velocity of a rigid body about that axis>>
or <<angular acceleration of a rigid body about that axis>> does not make sense

 

[NTesla]

I hinted on some advanced things, forget it.
By the way a phrase <<angular velocity of a rigid body about that axis>>
or <<angular acceleration of a rigid body about that axis>> does not make sense

I don't understand why it doesn't make sense. In undergraduate level physics books, that's how angular velocity and angular acceleration has been introduced. There's always an axis about which these terms are defined. Kindly let me know your opinion on this.

 

[kuruman]

I hinted on some advanced things, forget it.
By the way a phrase <<angular velocity of a rigid body about that axis>>
or <<angular acceleration of a rigid body about that axis>> does not make sense

To me it makes sense and stems from the definition of angle.

1. Consider two intersecting lines in the plane of rotation.
2. Draw a circle of radius $r$ centered at the point of intersection.
3. The angle between the lines is the ratio of the arc length $s$ starting at one line and ending at the other divided by the radius, $\theta =\dfrac{s}{r}$.
4. The angular velocity is the rate of change $~\dot{\theta}=\dfrac{d}{dt}\left(\dfrac{s}{r}\right).$
5. The axis about which this angular velocity is to be considered is perpendicular to the plane containing the lines and passes through their point of intersection.
6. Similarly for the angular acceleration.

 

[wrobel]

In general a rigid body can move such that there are no axes of rotation and there is no need to refer to them.
Theorem (Euler). There exists a unique vector $\boldsymbol \omega$ such that for any points $A,B$ of the rigid body the following formula holds
$$\boldsymbol v_A=\boldsymbol v_B+\boldsymbol\omega\times\boldsymbol{BA}.$$

The vector $\boldsymbol\omega$ is called the angular velocity of a rigid body and the angular acceleration is by definition equal to $\boldsymbol{\dot\omega}$