A Ferris wheel of radius R speeds up with angular acceleration starting from rest. Find expressions for the (a) velocity and (b) centripetal acceleration of a rider after the Ferris wheel has rotated through angle ∆θ.

Verified step by step guidance

1. Start by identifying the relationship between angular acceleration (α), angular displacement (Δθ), and angular velocity (ω). Use the kinematic equation for rotational motion: $\omega ^2\; =\; \omega \_0^2\; +\; 2\alpha \Delta \theta$. Since the Ferris wheel starts from rest, $\omega \_0\; =\; 0$, so the equation simplifies to $\omega \; =\; \backslash sqrt\{2\alpha \Delta \theta \}$. This gives the angular velocity after the wheel has rotated through angle Δθ.
2. To find the linear velocity (v) of the rider, use the relationship between linear velocity and angular velocity: $v\; =\; \omega R$, where R is the radius of the Ferris wheel. Substitute $\omega \; =\; \backslash sqrt\{2\alpha \Delta \theta \}$ into this equation to get $v\; =\; R\backslash sqrt\{2\alpha \Delta \theta \}$.
3. Next, calculate the centripetal acceleration (a_c), which is caused by the circular motion of the rider. The formula for centripetal acceleration is $a\_c\; =\; \backslash frac\{v^2\}\{R\}$. Substitute $v\; =\; R\backslash sqrt\{2\alpha \Delta \theta \}$ into this equation.
4. Simplify the expression for centripetal acceleration. Substituting $v^2\; =\; (R\backslash sqrt\{2\alpha \Delta \theta \})^2$, we get $a\_c\; =\; \backslash frac\{(R^2)(2\alpha \Delta \theta )\}\{R\}$. Simplify further to obtain $a\_c\; =\; 2\alpha \Delta \theta R$.
5. Summarize the results: (a) The linear velocity of the rider is $v\; =\; R\backslash sqrt\{2\alpha \Delta \theta \}$. (b) The centripetal acceleration of the rider is $a\_c\; =\; 2\alpha \Delta \theta R$. These expressions depend on the radius of the Ferris wheel (R), the angular acceleration (α), and the angular displacement (Δθ).

Key Concepts

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time. It is typically denoted by the symbol α and is measured in radians per second squared (rad/s²). In the context of the Ferris wheel, it describes how quickly the wheel is speeding up from rest, affecting the angular velocity as the wheel rotates through a certain angle.

Tangential Velocity

Tangential velocity refers to the linear speed of a point on the circumference of a rotating object. It is calculated as the product of the angular velocity (ω) and the radius (R) of the wheel, expressed as v = ωR. As the Ferris wheel accelerates, the tangential velocity increases, allowing us to determine the speed of a rider after the wheel has rotated through a specified angle.

Centripetal Acceleration

Centripetal acceleration is the acceleration directed towards the center of a circular path, necessary for an object to maintain circular motion. It is given by the formula a_c = v²/R, where v is the tangential velocity and R is the radius of the circular path. For a rider on the Ferris wheel, this acceleration ensures that they remain in circular motion as the wheel rotates.