The correct option is
B √5gl The point of suspension is moving with velocity
u as shown in the figure.
![](https://s3-us-west-2.amazonaws.com/infinitestudent-images/ckeditor_assets/pictures/623018/original_New_Bitmap_Image.bmp)
From the frame of reference of point of suspension, ball will seem to be moving with same velocity in opposite direction. Hence the given case changes to
![](https://s3-us-west-2.amazonaws.com/infinitestudent-images/ckeditor_assets/pictures/623019/original_New_Bitmap_Image.bmp)
To complete the circle, tension in the thread when the ball is at the topmost point should be greater than zero.
Let the velocity of the ball at topmost point be
v and tension in the thread be
T.
![](https://s3-us-west-2.amazonaws.com/infinitestudent-images/ckeditor_assets/pictures/623042/original_New_Bitmap_Image.bmp)
Centripetal force will also be acting in the downward direction.
Thus
T+mg=mv2l
⇒T=mv2l−mg
Condition for ball to complete the circle,
T>0
⇒mv2l−mg>0
⇒v2l>g
v>√gl(1)
Hence minimum speed ball should have at the topmost point is
√gl.
Now the change in kinetic energy when ball moves from bottom most point to the topmost point will be,
ΔK.E=12mv2−12mu2(2)
Work done by the gravity force when ball moves from bottom most point to the topmost point will be,
W=−mg×2l
(negative sign indicates that the direction of displacement and gravity force is opposite)
In the inertial frame of reference using work-energy theorem,
ΔK.E=W
12mv2−12mu2=−2mgl
Substituting the value of
v from (1),
12(gl−u2)=−2gl
Solving the above equation, we get
u=√5gl
For detailed solution watch next video.