Do we need to use any of the info about x_P? The tube should move independently of whatever happens to P
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I don't believe that Xp nor X dot p is used. I think they reuse problems and in a past problem you might've needed to find the velocity or acceleration of the particle, but for this problem, they just took that part out.
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I am also wondering what Eric Lee is asking. x_p and x_dot_p do not factor into the solution. I was able to solve for omega and alpha without them.
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Is it a valid analysis to take the derivative of omega found in the fixed axis, or should we convert it to the i,j,k system and then take the derivative?
In order to determine angular acceleration, you must first write the angular velocity in terms of components which are valid for all time.
* That is, if an angular velocity component is about a fixed axis, then it must be written as such. A time derivative of that fixed axis unit vector is then zero.
* If it is about a moving axis, it must be written as such. A time derivative of that moving axis unit vector is not zero, but is found using the relations derived in class. For example, di/dt = omega x i.
Once you have taken the derivative, you can then transform to either the fixed frame components or moving frame components.
Based on the diagram, are i-dot and j-dot both non-zero values for this question, while k-dot is equal to zero?
Actually, all three (I-dot, j-dot and k-dot) are non-zero in this case. But, in order to solve this problem, you need only k-dot.
For those looking for a similar lecture book example to look for a good start to the problem, I would suggest 3.B.4.
The rail in which P is on is there to throw you off. Assume P doesn't move along the rail in which it is constrained as x dot p is 0
As I was solving this problem, x_p was never used in my calculations. I was able to get answers that make sense, but I just did not know if I was missing something regarding the particle. Is there a component of angular velocity for the tube that is caused in some way by the particle? I figured that because x_p_dot is zero the particle probably does not impact the calculations in this case. Am I wrong to make this assumption?
I don't really like this problem - since you're asking for omega and alpha for the tube in terms of the little components (which are attached to the tube) it seems like it would all just be zero.
How I see it, it's like asking what the velocity of a car is relative to itself. That would be zero because the car is the car.
The tube is not rotating about any of its own axes because the axes move with it.
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The tube has two components of rotation: one component about the fixed I-axis (Omega), and the other about the moving k-axis (theta_dot).
The omega vector is NOT a vector that describes what the observer sees - instead it is a vector that describes how the observer moves. It is not relevant to your above analogy about what the observer sees.
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Should using the omega (angular velocity) before the cos and sin adjustments impact the answer for angular acceleration compared to if you differentiate the angular velocity with the sin and cos?
The problem statement says that theta is increasing however theta_dot is negative. Which sign should I use for theta_dot?
Do we need to use any of the info about x_P? The tube should move independently of whatever happens to P
I don't believe that Xp nor X dot p is used. I think they reuse problems and in a past problem you might've needed to find the velocity or acceleration of the particle, but for this problem, they just took that part out.
I am also wondering what Eric Lee is asking. x_p and x_dot_p do not factor into the solution. I was able to solve for omega and alpha without them.
Is it a valid analysis to take the derivative of omega found in the fixed axis, or should we convert it to the i,j,k system and then take the derivative?
In order to determine angular acceleration, you must first write the angular velocity in terms of components which are valid for all time.
* That is, if an angular velocity component is about a fixed axis, then it must be written as such. A time derivative of that fixed axis unit vector is then zero.
* If it is about a moving axis, it must be written as such. A time derivative of that moving axis unit vector is not zero, but is found using the relations derived in class. For example, di/dt = omega x i.
Once you have taken the derivative, you can then transform to either the fixed frame components or moving frame components.
Based on the diagram, are i-dot and j-dot both non-zero values for this question, while k-dot is equal to zero?
Actually, all three (I-dot, j-dot and k-dot) are non-zero in this case. But, in order to solve this problem, you need only k-dot.
For those looking for a similar lecture book example to look for a good start to the problem, I would suggest 3.B.4.
The rail in which P is on is there to throw you off. Assume P doesn't move along the rail in which it is constrained as x dot p is 0
As I was solving this problem, x_p was never used in my calculations. I was able to get answers that make sense, but I just did not know if I was missing something regarding the particle. Is there a component of angular velocity for the tube that is caused in some way by the particle? I figured that because x_p_dot is zero the particle probably does not impact the calculations in this case. Am I wrong to make this assumption?
I don't really like this problem - since you're asking for omega and alpha for the tube in terms of the little components (which are attached to the tube) it seems like it would all just be zero.
How I see it, it's like asking what the velocity of a car is relative to itself. That would be zero because the car is the car.
The tube is not rotating about any of its own axes because the axes move with it.
The tube has two components of rotation: one component about the fixed I-axis (Omega), and the other about the moving k-axis (theta_dot).
The omega vector is NOT a vector that describes what the observer sees - instead it is a vector that describes how the observer moves. It is not relevant to your above analogy about what the observer sees.
Should using the omega (angular velocity) before the cos and sin adjustments impact the answer for angular acceleration compared to if you differentiate the angular velocity with the sin and cos?
The problem statement says that theta is increasing however theta_dot is negative. Which sign should I use for theta_dot?