Question

A pendulum consists of a small object hanging from the ceiling at the end of a string of negligible mass. The string has a length of 0.78 m. With the string hanging vertically, the object is given an initial velocity of 2.9 m/s parallel to the ground and swings upward in a circular arc. Eventually, the object comes to a momentary halt at a point where the string makes an angle θ with its initial vertical orientation and then swings back downward. Find the angle θ.

Answer 1

Given length of string l = 0.78 m

inital velocity = 2.9m/s

When the pendulum is at rest we are giving a velocity parallel to horizontal and it rises to a height h

that is Kinetic energy at bottom = 1/2mv²

potential energy when velocity becomes ZERO =mgh

the entire KE is coverted into PE

1/mv² =mgh

h = v²/2g

= 2.9²/2*9.81

= 0.4286 m

h =0.4286 m

So the vertical height climbed by mass before stopping is 0.4286 m

From figure

h =l- lcos

1- cos = h /l

cos = 1-0.4286/0.78

= Cos^-1 (0.4505) = 63.223 deg

Given is the angle string makes with inital velocity orientation(horizontal

Finally = 90+ (alternate interior angles)

= 153.223 deg

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