First slide

Conservation of mechanical energy

Question

A simple pendulum of length l and mass m is initially at its lowest position . It is given the minimum horizontal speed necessary to move in a circular path about the point of suspension. The tension in the string at the lowest position of the bob is

Moderate

A

3mg
B

4mg
C

5mg
D

6mg

Solution

$\begin{array}{rcl} T - m g \cos θ & = & \frac{m u^{2}}{R}, h e r e T i s t e n s i o n i n s t r i n g, m i s m a s s a t t a c h e d, \\ g i s a c c e l e r a t i o n d u e t o g r a v i t y, θ i s a n g l e m a d e b y t h e s t r i n g w i t h v e r t i c a l \\ u i s v e l o c i t y o f m a s s, R i s r a d i u s o f t h e c i r c l e \\ a t l o w e s t p o s i t i o n θ & = & 0, \cos 0 = 1, t h e n \\ T - m g & = & \frac{m {(\sqrt{5 g R})}^{2}}{R} \\ \Rightarrow & T = 6 m g \end{array}$

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