1. The problem statement, all variables and given/known data
An arm with lenght AB, spins with [itex]\dot{θ}_1 [/itex] around a vertical axis, parallel to the [itex] \hat{Z}[/itex]-axis of a fixed set of axes. There is a horizontal hinge at a distance R from the axis of rotation. Around this hinge a homogenous rod, CD, with lenght L and mass m is able to turn.
The rod has a set of axis that move with it, its x-axis is parallel to the hinge. Its [itex] \hat{y}[/itex]-axis is parallel to CD. Its [itex] \hat{z}[/itex]-axis is perpendicular to the hinge, making a 90 degree angle with the [itex] \hat{y}[/itex]-axis. (See the picture for a clear overview)
The rotation of arm AB around the fixed [itex] \hat{Z}[/itex]-axis is measured by the angle [itex]θ_1[/itex] with respect to the fixed [itex] \hat{Y}[/itex]-axis. The rotation of rod CD is measured with angle [itex] θ_2 [/itex] with respect to the [itex] \hat{Z}[/itex]-axis. (The picture depicts the situation where [itex] θ_1 [/itex] = 0)
The spinvector of a random point along the rod is given by the components:
[tex]\omega_x = - \dot{θ}_2 ,\ \omega_y = \dot{θ}_1 cos(θ_2),\ \omega_z = \dot{θ}_1 sin(θ_2) [/tex]
Now determine the spin vector with respect to the fixed basis (Axes).
2. Relevant equations
[tex] Sin(θ) = \frac{O}{H} , Cos(theta) = \frac{A}{H} , Tan(θ) = \frac{O}{A} [/tex]
Knowledge of adding and substracting vectors.
3. The attempt at a solution
I managed to calculate the value of the new [itex] \omega_z [/itex]
1: Take the vertical component of all the three vecors, and add them together.
2: => [tex] \omega_z = \dot{θ}_1 sin^2(θ_2) + \dot{θ}_1 cos^2(θ_2) + 0 (\text{because x, hat does not have a Z component in its vector}) = \dot{θ}_1 [/tex]
Which matches the answer in my answer booklet.
But when calculating the x- and y components:
[tex] \omega_y = \sin(θ_1) (- sin(θ_2) \dot{θ}_1 \cos(θ_2) + sin(θ_2) \dot{θ}_1 \cos(θ_2)) + (\text{Appearently they cancel each other out, yet I still have to add the y-component of x,hat}) \frac{- \dot{θ}_2}{tan(θ_1)} [/tex]
[tex] = \frac{- \dot{θ}_2}{tan(θ_1)} [/tex]
Which does not equal the answer in the answer booklet, which is:
[tex] \omega_y = - \dot{θ}_2 sin(θ_1) [/tex]
Can someone point me what I am doing wrong? (It only seems like a minor error, but I can't find it)
You can see in the two attached pictures, what the setup looks like.
Ps. I'm sorry for my poor english, I tried translating the question from german as good as I could.
An arm with lenght AB, spins with [itex]\dot{θ}_1 [/itex] around a vertical axis, parallel to the [itex] \hat{Z}[/itex]-axis of a fixed set of axes. There is a horizontal hinge at a distance R from the axis of rotation. Around this hinge a homogenous rod, CD, with lenght L and mass m is able to turn.
The rod has a set of axis that move with it, its x-axis is parallel to the hinge. Its [itex] \hat{y}[/itex]-axis is parallel to CD. Its [itex] \hat{z}[/itex]-axis is perpendicular to the hinge, making a 90 degree angle with the [itex] \hat{y}[/itex]-axis. (See the picture for a clear overview)
The rotation of arm AB around the fixed [itex] \hat{Z}[/itex]-axis is measured by the angle [itex]θ_1[/itex] with respect to the fixed [itex] \hat{Y}[/itex]-axis. The rotation of rod CD is measured with angle [itex] θ_2 [/itex] with respect to the [itex] \hat{Z}[/itex]-axis. (The picture depicts the situation where [itex] θ_1 [/itex] = 0)
The spinvector of a random point along the rod is given by the components:
[tex]\omega_x = - \dot{θ}_2 ,\ \omega_y = \dot{θ}_1 cos(θ_2),\ \omega_z = \dot{θ}_1 sin(θ_2) [/tex]
Now determine the spin vector with respect to the fixed basis (Axes).
2. Relevant equations
[tex] Sin(θ) = \frac{O}{H} , Cos(theta) = \frac{A}{H} , Tan(θ) = \frac{O}{A} [/tex]
Knowledge of adding and substracting vectors.
3. The attempt at a solution
I managed to calculate the value of the new [itex] \omega_z [/itex]
1: Take the vertical component of all the three vecors, and add them together.
2: => [tex] \omega_z = \dot{θ}_1 sin^2(θ_2) + \dot{θ}_1 cos^2(θ_2) + 0 (\text{because x, hat does not have a Z component in its vector}) = \dot{θ}_1 [/tex]
Which matches the answer in my answer booklet.
But when calculating the x- and y components:
[tex] \omega_y = \sin(θ_1) (- sin(θ_2) \dot{θ}_1 \cos(θ_2) + sin(θ_2) \dot{θ}_1 \cos(θ_2)) + (\text{Appearently they cancel each other out, yet I still have to add the y-component of x,hat}) \frac{- \dot{θ}_2}{tan(θ_1)} [/tex]
[tex] = \frac{- \dot{θ}_2}{tan(θ_1)} [/tex]
Which does not equal the answer in the answer booklet, which is:
[tex] \omega_y = - \dot{θ}_2 sin(θ_1) [/tex]
Can someone point me what I am doing wrong? (It only seems like a minor error, but I can't find it)
You can see in the two attached pictures, what the setup looks like.
Ps. I'm sorry for my poor english, I tried translating the question from german as good as I could.
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