A ball rolling on a circular track, starting from rest, has angular acceleration $\alpha$. Find an expression, in terms of $\alpha$, for the time at which the ball's acceleration vector a is ${45}^{\circ }$ away from a radial line toward the center of the circle.

Verified step by step guidance

1. Understand the problem: The ball is rolling on a circular track with angular acceleration \( \alpha \). The goal is to find the time \( t \) when the total acceleration vector \( \mathbf{a} \) makes a 45° angle with the radial direction. The total acceleration has two components: radial (centripetal) acceleration \( a_r \) and tangential acceleration \( a_t \).
2. Express the tangential acceleration \( a_t \): The tangential acceleration is related to the angular acceleration \( \alpha \) by \( a_t = r \alpha \), where \( r \) is the radius of the circular track.
3. Express the radial (centripetal) acceleration \( a_r \): The radial acceleration is given by \( a_r = \frac{v^2}{r} \), where \( v \) is the tangential velocity. Since the ball starts from rest and has angular acceleration \( \alpha \), the angular velocity \( \omega \) at time \( t \) is \( \omega = \alpha t \), and the tangential velocity is \( v = r \omega = r \alpha t \). Substituting this into the formula for \( a_r \), we get \( a_r = \frac{(r \alpha t)^2}{r} = r \alpha^2 t^2 \).
4. Set up the condition for the angle between \( \mathbf{a} \) and the radial direction: The total acceleration vector \( \mathbf{a} \) is the vector sum of \( a_r \) and \( a_t \). The angle \( \theta \) between \( \mathbf{a} \) and the radial direction is given by \( \tan \theta = \frac{a_t}{a_r} \). For \( \theta = 45° \), \( \tan 45° = 1 \), so \( a_t = a_r \).
5. Solve for \( t \): Substitute \( a_t = r \alpha \) and \( a_r = r \alpha^2 t^2 \) into the equation \( a_t = a_r \). This gives \( r \alpha = r \alpha^2 t^2 \). Cancel \( r \) (assuming \( r \neq 0 \)) and solve for \( t \): \( t = \sqrt{\frac{1}{\alpha}} \).

Key Concepts

Angular Acceleration

Angular acceleration (α) is the rate of change of angular velocity over time. It is a vector quantity that indicates how quickly an object is rotating and in which direction. In this scenario, it determines how fast the ball's rotational speed increases as it rolls along the circular track.

Centripetal and Tangential Acceleration

In circular motion, an object's acceleration can be divided into two components: centripetal acceleration, which points toward the center of the circle, and tangential acceleration, which is directed along the path of motion. The angle of 45° between the acceleration vector and the radial line indicates that the magnitudes of these two components are equal at that moment.

Kinematic Equations for Rotational Motion

Kinematic equations for rotational motion relate angular displacement, angular velocity, angular acceleration, and time. These equations are analogous to linear motion equations but involve angular quantities. They are essential for deriving the time at which the ball's acceleration vector achieves the specified angle with respect to the radial line.