Question

a ball of mass ( m ) is attached to a string of length ( r ). the ball moves clockwise in a vertical circle at a constant speed ( v_1 ), as shown in figure 1. when the ball is at point ( p ), the magnitude of the tension in the string is ( f_{t1} ).
(a) the speed of the ball is then increased to speed ( v_2 ) such that ( v_2>v_1 ). the ball is again swung so that it moves in a vertical circle at constant speed. now when the ball is at point ( p ), the magnitude of the tension in the string is ( f_{t2} ). indicate whether ( f_{t1} ) is greater than, less than, or equal to ( f_{t2} )
( f_{t1}>f_{t2} ) ( f_{t1}<f_{t2} ) ( f_{t1}=f_{t2} )
justify your answer using qualitative reasoning beyond referencing equations.
note on your ap exam, you will handwrite your responses to free response questions in a test booklet

Explanation:

At point \( P \), the net force towards the center of the circle provides the centripetal force. The centripetal force \( F_c=\frac{Mv^{2}}{R} \). The net force at point \( P \) is \( F_T - Mg\) (where \( F_T \) is the tension and \( Mg \) is the weight of the ball). Since \( F_c = F_T - Mg\), then \( F_T=F_c + Mg=\frac{Mv^{2}}{R}+Mg \). When the speed \( v \) increases (from \( v_1 \) to \( v_2\) with \( v_2>v_1 \)), the centripetal force \( \frac{Mv^{2}}{R} \) increases. As \( Mg \) (weight) is constant (\( M \) and \( g \) are unchanged), the tension \( F_T \) which is the sum of centripetal force and weight must increase.

Answer:

\( F_{T1}