Questions: What is the ratio of the centripetal acceleration of a point on the end of the rod to that of a point a distance L / 2 from the end of the rod?

Solution Steps

Centripetal acceleration \(a_c\) for a point on a rotating object is given by: \[ a_c = \omega^2 r \] where \(\omega\) is the angular velocity and \(r\) is the radius (distance from the axis of rotation).

• Point at the end of the rod: \(r = L\)
• Point at a distance \(L/2\) from the end of the rod: \(r = L/2\)

• For the point at the end of the rod: \[ a_{c1} = \omega^2 L \]
• For the point at a distance \(L/2\) from the end of the rod: \[ a_{c2} = \omega^2 \left(\frac{L}{2}\right) = \frac{\omega^2 L}{2} \]

The ratio of the centripetal acceleration at the end of the rod to that at a distance \(L/2\) from the end is: \[ \text{Ratio} = \frac{a_{c1}}{a_{c2}} = \frac{\omega^2 L}{\frac{\omega^2 L}{2}} = \frac{\omega^2 L}{\omega^2 L / 2} = 2 \]

Final Answer