- #1
rakso
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- Homework Statement
- Show that the Coriolius Force can be associated with a tensor of rank 3.
- Relevant Equations
- $$ F_{C} = -2m \bar{\omega} \times \bar{v} $$
m = Particles mass, Omega = Systems angular frequency, v' = particles velocity.
Attempt at a Solution:
$$ F_{C} = -2m \bar{\omega} \times \bar{v}^{'} = -2 \bar{\omega} \times \bar{p} = 2 \bar{p} \times \bar{\omega} $$
Let
$$ \bar{\omega} = \frac {\bar{r} \times \bar{v}} {r^2}, \alpha = \frac {2} {r^2} $$
Then
$$ \frac {F_{C}} {\alpha} = \bar{p} \times ( \bar{r} \times \bar{v}) = \bar{e_{i}} \varepsilon_{ijk} p_{j} (\bar{r} \times \bar{v})_{k}
= \bar{e}_{i} \varepsilon_{ijk} p_{j} \varepsilon_{klm} r_{l} v_{m} = \bar{e}_{i} p_{j} r_{l} v_{m} (\delta_{il} \delta_{jm} - \delta_{im} \delta_{jl}) = \\ \bar{e}_{i} p_{j} r_{i} v_{j} - \bar{e}_{i} p_{j} r_{j} v_{i} = \bar{e}_{i} p_{j} ( r_{i} v_{j} - r_{j} v_{i}) = \bar{e}_{i} p_{j} T_{ij} $$
Where
$$ T_{ij} = r_{i} v_{j} - r_{j} v_{i} $$ is a tensor of rank 2. I don't understand how I am supposed to show it's a tensor of rank 3? Have the third rank something to do with what reference system we chose? Because I'm not quite sure the velocity of the particle is relative to the spinning system or the Newtonian one.
Attempt at a Solution:
$$ F_{C} = -2m \bar{\omega} \times \bar{v}^{'} = -2 \bar{\omega} \times \bar{p} = 2 \bar{p} \times \bar{\omega} $$
Let
$$ \bar{\omega} = \frac {\bar{r} \times \bar{v}} {r^2}, \alpha = \frac {2} {r^2} $$
Then
$$ \frac {F_{C}} {\alpha} = \bar{p} \times ( \bar{r} \times \bar{v}) = \bar{e_{i}} \varepsilon_{ijk} p_{j} (\bar{r} \times \bar{v})_{k}
= \bar{e}_{i} \varepsilon_{ijk} p_{j} \varepsilon_{klm} r_{l} v_{m} = \bar{e}_{i} p_{j} r_{l} v_{m} (\delta_{il} \delta_{jm} - \delta_{im} \delta_{jl}) = \\ \bar{e}_{i} p_{j} r_{i} v_{j} - \bar{e}_{i} p_{j} r_{j} v_{i} = \bar{e}_{i} p_{j} ( r_{i} v_{j} - r_{j} v_{i}) = \bar{e}_{i} p_{j} T_{ij} $$
Where
$$ T_{ij} = r_{i} v_{j} - r_{j} v_{i} $$ is a tensor of rank 2. I don't understand how I am supposed to show it's a tensor of rank 3? Have the third rank something to do with what reference system we chose? Because I'm not quite sure the velocity of the particle is relative to the spinning system or the Newtonian one.
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