Question

The sum of the x-components of unit vectors 𝑟̂ ̇ and 𝜃̂̇ for a particle moving with angular speed 2 rad s ^-1 at angle 𝜃 = 215° is__________ (rounded off to two decimal places)

Answer 1

Correct Answer: 2.7 - 2.85

Given:
Angular speed: \( \dot{\theta} = 2\ \text{rad/s} \) 
Angle: \( \theta = 215^\circ \)

Unit vector time derivatives:
\[ \dot{\hat{r}} = \dot{\theta}\,\hat{\theta}, \qquad \dot{\hat{\theta}} = -\dot{\theta}\,\hat{r} \] Unit vectors in Cartesian form:
\[ \hat{r} = (\cos\theta,\; \sin\theta), \qquad \hat{\theta} = (-\sin\theta,\; \cos\theta) \] Thus, \[ \dot{\hat{r}} = \dot{\theta}(-\sin\theta,\; \cos\theta) \] \[ \dot{\hat{\theta}} = -\dot{\theta}(\cos\theta,\; \sin\theta) \] x-components:
\[ (\dot{\hat{r}})_x = \dot{\theta}(-\sin\theta) \] \[ (\dot{\hat{\theta}})_x = -\dot{\theta}\cos\theta \] Sum: \[ (\dot{\hat{r}})_x + (\dot{\hat{\theta}})_x = \dot{\theta}(-\sin\theta - \cos\theta) \] Evaluate at \( \theta = 215^\circ \):
\[ \sin 215^\circ = -0.574,\qquad \cos 215^\circ = -0.819 \] Compute: \[ -\sin\theta - \cos\theta = -(-0.574) - (-0.819) = 0.574 + 0.819 = 1.393 \] Multiply by \( \dot{\theta} = 2 \): \[ 2 \times 1.393 = 2.786 \] Final Answer:
\[ \boxed{2.79} \]