Question

A circular table is rotating with an angular velocity of \( \omega \, \text{rad/s} \) about its axis (see figure). There is a smooth groove along a radial direction on the table. A steel ball is gently placed at a distance of \( 1 \, \text{m} \) on the groove. All the surfaces are smooth. If the radius of the table is \( 3 \, \text{m} \), the radial velocity of the ball with respect to the table at the time the ball leaves the table is \( x\sqrt{2}\omega \, \text{m/s} \), where the value of \( x \) is \(\dots\).

Answer 1

Correct Answer: 2

This problem concerns the motion of an object on a rotating surface. The objective is to find the coefficient \( x \) in the expression for the radial velocity of a steel ball as it exits the rotating surface.

The forces acting on the ball are considered. As the surface rotates at an angular velocity \( \omega \), a centrifugal force is exerted radially outward on the ball. This force is defined by the equation \( F = m \omega^2 r \), where \( r \) denotes the radial distance from the rotational center.

The ball originates at a radial distance \( r_0 = 1 \, \text{m} \) from the center. During its outward progression, it experiences no tangential force due to a smooth channel and accumulates radial kinetic energy solely from the centrifugal force. Upon reaching the edge of the rotating surface at \( r = 3 \, \text{m} \), its radial kinetic energy is quantified as:

\[ \frac{1}{2} m v_r^2 = \int_{r_0}^{3} m \omega^2 r \, dr \]

The integral calculation yields:

\[ \int_{1}^{3} \omega^2 r \, dr = \left. \frac{\omega^2 r^2}{2} \right|_{1}^{3} = \frac{\omega^2 (3^2 - 1^2)}{2} = \frac{\omega^2 (9 - 1)}{2} = 4\omega^2 \]

The kinetic energy is then equated:

\[ \frac{1}{2} m v_r^2 = m \cdot 4\omega^2 \]

Simplification leads to:

\[ v_r^2 = 8\omega^2 \]

\[ v_r = \sqrt{8}\omega = 2\sqrt{2}\omega \]

Consequently, the ball's final radial velocity is \( 2\sqrt{2}\omega \, \text{m/s} \). By comparing this with the expression \( x\sqrt{2}\omega \, \text{m/s} \), we deduce that \( x = 2 \).

This resultant value falls within the specified bounds of \( 2,2 \).