# Amusement Park Ride in 2: Step-by-Step Solution We have two rotating turntables: a large one (radius \( r_1 \), angle \( \theta_1 \)) and, attached to its edge, a smaller one (radius \( r_2 \), angle \( \theta_2 \)). Riders are strapped to the circumference of the small turntable. We need to find the acceleration (force per unit mass) experienced by a rider in both an inertial and a non-inertial frame. --- ## (a) Inertial Frame: Compute \( \vec{F}/m \) ### 1. **Position Vector of the Rider** Choose the center of the big turntable as the origin. - Let \( \vec{r}_1 \) be the vector from the center of the big turntable to the center of the small turntable. - Let \( \vec{r}_2 \) be the vector from the center of the small turntable to the rider. Express these vectors in terms of the fixed Cartesian unit vectors \( \hat{x} \), \( \hat{y} \): \[ \vec{r}_1 = r_1 (\cos \theta_1 \, \hat{x} + \sin \theta_1 \, \hat{y}) \] \[ \vec{r}_2 = r_2 (\cos \theta_2 \, \hat{x}' + \sin \theta_2 \, \hat{y}') \] where \( \hat{x}' \), \( \hat{y}' \) are axes rotated by \( \theta_1 \) with respect to the fixed axes. But in the fixed frame, \( \vec{r}_2 \) becomes: \[ \vec{r}_2 = r_2 [\cos(\theta_1 + \theta_2)\,\hat{x} + \sin(\theta_1 + \theta_2)\,\hat{y}] \] **Total position:** \[ \vec{R} = \vec{r}_1 + \vec{r}_2 = r_1 (\cos \theta_1 \, \hat{x} + \sin \theta_1 \, \hat{y}) + r_2 [\cos(\theta_1 + \theta_2)\,\hat{x} + \sin(\theta_1 + \theta_2)\,\hat{y}] \] \[ \boxed{ \vec{R} = [r_1 \cos \theta_1 + r_2 \cos(\theta_1 + \theta_2)]\,\hat{x} + [r_1 \sin \theta_1 + r_2 \sin(\theta_1 + \theta_2)]\,\hat{y} } \] --- ### 2. **Velocity and Acceleration** Assume \( \theta_1(t) \), \( \theta_2(t) \) are functions of time. - \(\dot{\theta}_1 = \omega_1\) - \(\dot{\theta}_2 = \omega_2\) - \(\ddot{\theta}_1 = \alpha_1\) - \(\ddot{\theta}_2 = \alpha_2\) #### Velocity: \[ \vec{V} = \frac{d\vec{R}}{dt} \] Take derivatives using the chain rule: \[ \frac{d}{dt} [\cos f(t)] = -\sin f(t) \cdot \dot{f}(t) \] \[ \frac{d}{dt} [\sin f(t)] = \cos f(t) \cdot \dot{f}(t) \] So, \[ V_x = -r_1 \sin\theta_1\, \dot{\theta}_1 - r_2 \sin(\theta_1 + \theta_2)(\dot{\theta}_1 + \dot{\theta}_2) \] \[ V_y = r_1 \cos\theta_1\, \dot{\theta}_1 + r_2 \cos(\theta_1 + \theta_2)(\dot{\theta}_1 + \dot{\theta}_2) \] #### Acceleration: Take the derivative of each term: \[ A_x = \frac{dV_x}{dt} = -r_1 [\cos\theta_1\, (\dot{\theta}_1)^2 + \sin\theta_1\, \ddot{\theta}_1] - r_2 [\cos(\theta_1+\theta_2)\, (\dot{\theta}_1+\dot{\theta}_2)^2 + \sin(\theta_1+\theta_2)(\ddot{\theta}_1 + \ddot{\theta}_2)] \] \[ A_y = \frac{dV_y}{dt} = -r_1 [\sin\theta_1\, (\dot{\theta}_1)^2 - \cos\theta_1\, \ddot{\theta}_1] - r_2 [\sin(\theta_1+\theta_2)\, (\dot{\theta}_1+\dot{\theta}_2)^2 - \cos(\theta_1+\theta_2)(\ddot{\theta}_1 + \ddot{\theta}_2)] \] **Or, after factoring:** \[ \boxed{ \begin{aligned} A_x = & -r_1 [\cos\theta_1\, (\dot{\theta}_1)^2 + \sin\theta_1\, \ddot{\theta}_1] \\ & - r_2 [\cos(\theta_1+\theta_2)\, (\dot{\theta}_1+\dot{\theta}_2)^2 + \sin(\theta_1+\theta_2)(\ddot{\theta}_1 + \ddot{\theta}_2)] \\ A_y = & -r_1 [\sin\theta_1\, (\dot{\theta}_1)^2 - \cos\theta_1\, \ddot{\theta}_1] \\ & - r_2 [\sin(\theta_1+\theta_2)\, (\dot{\theta}_1+\dot{\theta}_2)^2 - \cos(\theta_1+\theta_2)(\ddot{\theta}_1 + \ddot{\theta}_2)] \end{aligned} } \] So the force per unit mass in the inertial frame is this acceleration vector. --- ## (b) Noninertial Frame (Center of Small Turntable) Now, shift to a rotating frame whose origin is the center of the small turntable. ### 1. **Equation in Rotating Frame** Equation of motion in a rotating frame (center of small turntable): \[ \frac{\vec{F}}{m} = \vec{a}' + 2 \vec{\Omega} \times \vec{v}' + \vec{\Omega} \times (\vec{\Omega} \times \vec{r}') + \dot{\vec{\Omega}} \times \vec{r}' \] Where: - \( \vec{a}' \): acceleration of rider relative to small turntable (in this frame, just due to motion around small turntable) - \( \vec{v}' \): velocity relative to small turntable - \( \vec{\Omega} \): angular velocity of small turntable relative to inertial frame (this is \( \dot{\theta}_1 + \dot{\theta}_2 \)) - \( \dot{\vec{\Omega}} \): angular acceleration ### 2. **Terms to Compute** - **Relative position:** \( \vec{r}' = r_2 \hat{e}_r \) (in the small turntable's frame) - **Relative velocity:** \( \vec{v}' = r_2 \dot{\theta}_2 \hat{e}_\theta \) (motion around the small turntable) - **Relative acceleration:** \( \vec{a}' = r_2 \ddot{\theta}_2 \hat{e}_\theta - r_2 (\dot{\theta}_2)^2 \hat{e}_r \) (radial and tangential) But, as the small turntable itself is rotating due to the large turntable's motion, the **total angular velocity** of the small turntable is \( \omega_1 + \omega_2 \). #### Now, the terms: - **Coriolis acceleration:** \( 2\vec{\Omega} \times \vec{v}' \) - **Centrifugal acceleration:** \( \vec{\Omega} \times (\vec{\Omega} \times \vec{r}') \) - **Euler acceleration:** \( \dot{\vec{\Omega}} \times \vec{r}' \) Where \( \vec{\Omega} = (\dot{\theta}_1 + \dot{\theta}_2)\,\hat{z} \), \( \dot{\vec{\Omega}} = (\ddot{\theta}_1 + \ddot{\theta}_2)\,\hat{z} \). #### Each term in cylindrical coordinates (\( \hat{e}_r, \hat{e}_\theta \)): - **Coriolis:** \( 2 (\dot{\theta}_1 + \dot{\theta}_2) r_2 \dot{\theta}_2\, \hat{e}_r \) (since \( \hat{z} \times \hat{e}_\theta = \hat{e}_r \)) - **Centrifugal:** \( - r_2 (\dot{\theta}_1 + \dot{\theta}_2)^2\, \hat{e}_r \) - **Euler:** \( r_2 (\ddot{\theta}_1 + \ddot{\theta}_2)\, \hat{e}_\theta \) - **Relative acceleration:** \( - r_2 (\dot{\theta}_2)^2\, \hat{e}_r + r_2 \ddot{\theta}_2\, \hat{e}_\theta \) **So, sum all terms except \( \vec{F} \):** \[ \boxed{ \begin{aligned} \text{Total acceleration (all terms except } \vec{F}):\;\;\; & \Big[ -r_2 (\dot{\theta}_2)^2 + 2 r_2 (\dot{\theta}_1 + \dot{\theta}_2)\dot{\theta}_2 - r_2 (\dot{\theta}_1 + \dot{\theta}_2)^2 \Big]\, \hat{e}_r \\ & + \Big[ r_2 \ddot{\theta}_2 + r_2 (\ddot{\theta}_1 + \ddot{\theta}_2) \Big]\, \hat{e}_\theta \end{aligned} } \] Simplify: - Radial: \[ -r_2 (\dot{\theta}_2)^2 + 2 r_2 (\dot{\theta}_1 + \dot{\theta}_2)\dot{\theta}_2 - r_2 (\dot{\theta}_1 + \dot{\theta}_2)^2 \] \[ = -r_2 [ (\dot{\theta}_2)^2 - 2(\dot{\theta}_1 + \dot{\theta}_2)\dot{\theta}_2 + (\dot{\theta}_1 + \dot{\theta}_2)^2 ] \] Expand \( (\dot{\theta}_1 + \dot{\theta}_2)^2 = (\dot{\theta}_1)^2 + 2\dot{\theta}_1\dot{\theta}_2 + (\dot{\theta}_2)^2 \): \[ (\dot{\theta}_2)^2 - 2(\dot{\theta}_1 + \dot{\theta}_2)\dot{\theta}_2 + (\dot{\theta}_1)^2 + 2\dot{\theta}_1\dot{\theta}_2 + (\dot{\theta}_2)^2 \] \[ = (\dot{\theta}_2)^2 - 2\dot{\theta}_1\dot{\theta}_2 - 2(\dot{\theta}_2)^2 + (\dot{\theta}_1)^2 + 2\dot{\theta}_1\dot{\theta}_2 + (\dot{\theta}_2)^2 \] \[ = (\dot{\theta}_1)^2 \] So, the radial part is just \( -r_2 (\dot{\theta}_1)^2 \): \[ \boxed{ \text{Radial: } - r_2 (\dot{\theta}_1)^2 \hat{e}_r } \] - Tangential: \[ r_2 [\ddot{\theta}_2 + \ddot{\theta}_1 + \ddot{\theta}_2] = r_2 [\ddot{\theta}_1 + 2\ddot{\theta}_2] \] \[ \boxed{ \text{Tangential: } r_2 [\ddot{\theta}_1 + 2\ddot{\theta}_2] \hat{e}_\theta } \] --- ## (c) Physical Force in Both Frames and Comparison ### 1. **Noninertial Frame: \( \vec{F}/m \)** From above, in the noninertial (small turntable) frame: \[ \boxed{ \frac{\vec{F}}{m} = r_2 (\dot{\theta}_1)^2 \hat{e}_r - r_2 [\ddot{\theta}_1 + 2\ddot{\theta}_2] \hat{e}_\theta } \] ### 2. **Compare to Inertial Frame** In the inertial frame, \( \vec{F}/m \) was found by projecting the total acceleration onto Cartesian axes. ### 3. **Variable Transformation for Comparison

# Amusement Park Ride in 2D: Step-by-Step Solution We analyze a ride consisting of two rotating turntables: a large one with radius \( r_1 \) and a smaller one with radius \( r_2 \). The goal is to find the force per unit mass (\( F/m \)) experienced by a rider in both an inertial and a non-inertial frame. --- ## (a) Inertial Frame: Compute \( \frac{\vec{F}}{m} \) ### 1. **Position Vector of the Rider** Set the center of the larger turntable as the origin. - **Position of the large turntable**: \[ \vec{r}_1 = r_1 (\cos \theta_1 \, \hat{x} + \sin \theta_1 \, \hat{y}) \] - **Position of the smaller turntable** from the center of the large turntable: \[ \vec{r}_2 = r_2 [\cos(\theta_1 + \theta_2)\,\hat{x} + \sin(\theta_1 + \theta_2)\,\hat{y}] \] **Total position vector**: \[ \vec{R} = \vec{r}_1 + \vec{r}_2 = [r_1 \cos \theta_1 + r_2 \cos(\theta_1 + \theta_2)]\,\hat{x} + [r_1 \sin \theta_1 + r_2 \sin(\theta_1 + \theta_2)]\,\hat{y} \] ### 2. **Velocity and Acceleration** Assume \( \theta_1(t) \) and \( \theta_2(t) \) are time-dependent. #### Velocity: Using the chain rule: \[ \vec{V} = \frac{d\vec{R}}{dt} \] Calculating components: \[ V_x = -r_1 \sin\theta_1\, \dot{\theta}_1 - r_2 \sin(\theta_1 + \theta_2)(\dot{\theta}_1 + \dot{\theta}_2) \] \[ V_y = r_1 \cos\theta_1\, \dot{\theta}_1 + r_2 \cos(\theta_1 + \theta_2)(\dot{\theta}_1 + \dot{\theta}_2) \] #### Acceleration: Taking derivatives: \[ A_x = \frac{dV_x}{dt},\quad A_y = \frac{dV_y}{dt} \] Results in: \[ \boxed{ \begin{aligned} A_x = & -r_1 \left( \cos \theta_1 \, \dot{\theta}_1^2 + \sin \theta_1 \, \ddot{\theta}_1 \right) \\ & - r_2 \left( \cos(\theta_1+\theta_2) \, (\dot{\theta}_1+\dot{\theta}_2)^2 + \sin(\theta_1+\theta_2)(\ddot{\theta}_1 + \ddot{\theta}_2) \right) \\ A_y = & -r_1 \left( \sin \theta_1 \, \dot{\theta}_1^2 - \cos \theta_1 \, \ddot{\theta}_1 \right) \\ & - r_2 \left( \sin(\theta_1+\theta_2) \, (\dot{\theta}_1+\dot{\theta}_2)^2 - \cos(\theta_1+\theta_2)(\ddot{\theta}_1 + \ddot{\theta}_2) \right) \end{aligned} } \] --- ## (b) Noninertial Frame (Center of Small Turntable) ### 1. **Equation in Rotating Frame** In the frame of the small turntable, the equation of motion is: \[ \frac{\vec{F}}{m} = \vec{a}' + 2 \vec{\Omega} \times \vec{v}' + \vec{\Omega} \times (\vec{\Omega} \times \vec{r}') + \dot{\vec{\Omega}} \times \vec{r}' \] Where: - \( \vec{a}' \): acceleration of the rider relative to the small turntable - \( \vec{v}' \): velocity relative to the small turntable - \( \vec{\Omega} = (\dot{\theta}_1 + \dot{\theta}_2) \hat{z} \) ### 2. **Terms to Compute** - **Relative position**: \( \vec{r}' = r_2 \hat{e}_r \) - **Relative velocity**: \( \vec{v}' = r_2 \dot{\theta}_2 \hat{e}_\theta \) - **Relative acceleration**: \[ \vec{a}' = r_2 \ddot{\theta}_2 \hat{e}_\theta - r_2 (\dot{\theta}_2)^2 \hat{e}_r \] ### 3. **Adding Terms** 1. **Coriolis**: \[ 2 \vec{\Omega} \times \vec{v}' = 2 (\dot{\theta}_1 + \dot{\theta}_2) r_2 \dot{\theta}_2 \hat{e}_r \] 2. **Centrifugal**: \[ \vec{\Omega} \times (\vec{\Omega} \times \vec{r}') = - r_2 (\dot{\theta}_1 + \dot{\theta}_2)^2 \hat{e}_r \] 3. **Euler**: \[ \dot{\vec{\Omega}} \times \vec{r}' = r_2 (\ddot{\theta}_1 + \ddot{\theta}_2) \hat{e}_\theta \] ### Total Acceleration in Noninertial Frame: Combining all terms: \[ \boxed{ \begin{aligned} \text{Total Acceleration: }\;\;\; & \Big[ -r_2 (\dot{\theta}_2)^2 + 2 r_2 (\dot{\theta}_1 + \dot{\theta}_2) \dot{\theta}_2 - r_2 (\dot{\theta}_1 + \dot{\theta}_2)^2 \Big]\, \hat{e}_r \\ & + \Big[ r_2 \ddot{\theta}_2 + r_2 (\ddot{\theta}_1 + \ddot{\theta}_2) \Big]\, \hat{e}_\theta \end{aligned} } \] --- ## (c) Physical Force in Both Frames and Comparison ### 1. **Noninertial Frame: \( \frac{\vec{F}}{m} \)** From the noninertial frame: \[ \boxed{ \frac{\vec{F}}{m} = -r_2 (\dot{\theta}_1)^2 \hat{e}_r + r_2 [\ddot{\theta}_1 + 2\ddot{\theta}_2] \hat{e}_\theta } \] ### 2. **Comparison to Inertial Frame** To compare the two expressions for \( F/m \), we need to find a transformation relating the variables in the inertial frame to those in the non-inertial frame (e.g., transforming angular velocities and accelerations). ### 3. **Key Variable for Transformation** The variable needed for transformation to compare expressions is the relative angular position and the angular velocities \( \dot{\theta}_1 \) and \( \dot{\theta}_2 \). --- This structured solution outlines the calculations necessary to understand the forces acting on a rider in a complex amusement park ride setup.