A ball of mass 0.70 kg is attached to a vertical post by strings of length L_(1)=1.1m and L_(2)=1.7 m, as shown in (Eigure 1). The ball is set moving in a horizontal circle. Determine the maximum speed for the lower string to be slack (tension in cord=0 Express your answer with the appropriate units.

To determine the maximum speed for the lower string to be slack, we need to consider the forces acting on the ball and the centripetal force required to keep it moving in a circular path.<br /><br />The forces acting on the ball are:<br />1. Gravitational force (weight) acting vertically downward.<br />2. Tension in the upper string acting along the string towards the post.<br />3. Centripetal force required to keep the ball moving in a circular path, which acts horizontally towards the center of the circle.<br /><br />The centripetal force is provided by the tension in the upper string and the gravitational force. We can set up an equation for the centripetal force:<br /><br />\[ F_{\text{centripetal}} = \frac{m v^2}{r} \]<br /><br />where \( m \) is the mass of the ball, \( v \) is the velocity of the ball, and \( r \) is the radius of the circular path.<br /><br />The radius of the circular path is given by the length of the upper string, \( L_2 \), since the ball is moving in a horizontal circle.<br /><br />The gravitational force acting on the ball is given by:<br /><br />\[ F_{\text{gravity}} = m g \]<br /><br />where \( g \) is the acceleration due to gravity.<br /><br />The tension in the upper string provides the centripetal force, so we have:<br /><br />\[ T = \frac{m v^2}{L_2} \]<br /><br />The lower string will be slack when the tension in the lower string is zero. This occurs when the gravitational force is balanced by the tension in the upper string. Therefore, we have:<br /><br />\[ m g = T \]<br /><br />Substituting the expression for \( T \) from the centripetal force equation, we get:<br /><br />\[ m g = \frac{m v^2}{L_2} \]<br /><br />Solving for \( v \), we get:<br /><br />\[ v^2 = g L_2 \]<br /><br />\[ v = \sqrt{g L_2} \]<br /><br />Substituting the given values, we have:<br /><br />\[ v = \sqrt{(9.8 \, \text{m/s}^2)(1.7 \, \text{m})} \]<br /><br />\[ v \approx 5.05 \, \text{m/s} \]<br /><br />Therefore, the maximum speed for the lower string to be slack is approximately \( 5.05 \, \text{m/s} \).