# Uniform Circular Motion A planet orbits counterclockwise in a circle of radius \( R \) about a star in the \( x\text{-}y \) plane. Its initial position at \( t = \) is at \( \vec{r}() = R\hat{y} \). The period of the orbit is \( T = 7' \) (i.e., 7 minutes). ## (i) Position Vector \( \vec{r}(t) \) Let the angular frequency be \[ \omega = \frac{2\pi}{T} \] The position vector as a function of time: \[ \vec{r}(t) = R \sin(\omega t)\,\hat{x} + R \cos(\omega t)\,\hat{y} \] This form ensures that at \( t= \), \( \vec{r}() = R\hat{y} \). ## (ii) Velocity Vector \( \vec{v}(t) \) Take the time derivative of \( \vec{r}(t) \): \[ \vec{v}(t) = \frac{d\vec{r}}{dt} = R\omega \cos(\omega t)\,\hat{x} - R\omega \sin(\omega t)\,\hat{y} \] ## (iii) Acceleration Vector \( \vec{a}(t) \) Take the time derivative of \( \vec{v}(t) \): \[ \vec{a}(t) = \frac{d\vec{v}}{dt} = -R\omega^2 \sin(\omega t)\,\hat{x} - R\omega^2 \cos(\omega t)\,\hat{y} \] Or, \[ \vec{a}(t) = -\omega^2 \vec{r}(t) \] ## (iv) Direction of Vectors and Physical Intuition - **Position**: Always points from the origin to the planet. - **Velocity**: Always tangent to the circle, perpendicular to \( \vec{r}(t) \), in the direction of motion (counterclockwise). - **Acceleration**: Always points toward the center (centripetal), opposite to \( \vec{r}(t) \). This matches the uniform circular motion intuition: velocity is perpendicular to radius, acceleration is always inward. --- # Elliptical Motion The position for a planet in an elliptical orbit is given by: \[ \vec{r}(t) = 2b\sin(\omega t)\,\hat{x} + b\cos(\omega t)\,\hat{y} \] where \( b \) is a constant. ## (v) Velocity \( \vec{v}(t) \) Differentiate position with respect to \( t \): \[ \vec{v}(t) = \frac{d\vec{r}}{dt} = 2b\omega \cos(\omega t)\,\hat{x} - b\omega \sin(\omega t)\,\hat{y} \] ## (vi) Acceleration \( \vec{a}(t) \) Differentiate velocity with respect to \( t \): \[ \vec{a}(t) = \frac{d\vec{v}}{dt} = -2b\omega^2 \sin(\omega t)\,\hat{x} - b\omega^2 \cos(\omega t)\,\hat{y} \] ## (vii) Speed \( v(t) \) Speed is the magnitude of the velocity vector: \[ v(t) = |\vec{v}(t)| = \sqrt{(2b\omega \cos(\omega t))^2 + (b\omega \sin(\omega t))^2} \] \[ = b\omega \sqrt{4\cos^2(\omega t) + \sin^2(\omega t)} \] ## (viii) Angle Between \( \vec{v} \) and \( \vec{a} \) at \( t = \frac{2}{\omega} \) First, calculate \( \omega t = 2 \). Let’s write: - \( \vec{v} = (2b\omega \cos 2)\,\hat{x} - (b\omega \sin 2)\,\hat{y} \) - \( \vec{a} = (-2b\omega^2 \sin 2)\,\hat{x} - (b\omega^2 \cos 2)\,\hat{y} \) The cosine of the angle \( \theta \) between \( \vec{v} \) and \( \vec{a} \) is: \[ \cos\theta = \frac{\vec{v} \cdot \vec{a}}{|\vec{v}|\,|\vec{a}|} \] Dot product: \[ \vec{v} \cdot \vec{a} = (2b\omega \cos 2)(-2b\omega^2 \sin 2) + (-b\omega \sin 2)(-b\omega^2 \cos 2) \] \[ = -4b^2\omega^3 \cos 2 \sin 2 + b^2\omega^3 \sin 2 \cos 2 \] \[ = (-4 + 1)b^2\omega^3 \cos 2 \sin 2 = -3b^2\omega^3 \cos 2 \sin 2 \] Magnitudes: \[ |\vec{v}| = b\omega \sqrt{4\cos^2 2 + \sin^2 2} \] \[ |\vec{a}| = b\omega^2 \sqrt{4\sin^2 2 + \cos^2 2} \] So, \[ \cos\theta = \frac{-3b^2\omega^3 \cos 2 \sin 2}{b^2\omega^3 \sqrt{4\cos^2 2 + \sin^2 2} \sqrt{4\sin^2 2 + \cos^2 2}} \] \[ = \frac{-3 \cos 2 \sin 2}{\sqrt{4\cos^2 2 + \sin^2 2} \sqrt{4\sin^2 2 + \cos^2 2}} \] --- ## Summary Table | Quantity | Circular Orbit | Elliptical Orbit | |----------|---------------|---------------------------| | \( \vec{r}(t) \) | \( R\sin(\omega t)\,\hat{x} + R\cos(\omega t)\,\hat{y} \) | \( 2b\sin(\omega t)\,\hat{x} + b\cos(\omega t)\,\hat{y} \) | | \( \vec{v}(t) \) | \( R\omega \cos(\omega t)\,\hat{x} - R\omega \sin(\omega t)\,\hat{y} \) | \( 2b\omega \cos(\omega t)\,\hat{x} - b\omega \sin(\omega t)\,\hat{y} \) | | \( \vec{a}(t) \) | \( -R\omega^2 \sin(\omega t)\,\hat{x} - R\omega^2 \cos(\omega t)\,\hat{y} \) | \( -2b\omega^2 \sin(\omega t)\,\hat{x} - b\omega^2 \cos(\omega t)\,\hat{y} \) | --- **Diagram suggestion:** You may wish to draw the velocity and acceleration vectors at a sample location on both orbits to visualize their directions. *Alt text: Circular and elliptical orbits with position, velocity (tangent), and acceleration (inward/variable) vectors shown.*