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Motion in a Vertical Circle
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A stone tied to a string is rotated in a
Question
A stone tied to a string is rotated in a vertical circle. The minimum speed with which the stone has to be rotated in order to complete the circle :
A
decreases with increasing the mass of the stone
B
is independent of the mass of the stone
C
decreases with increasing the length of the string
D
is independent of the length of the string
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Solution
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Correct option is B)
Step 1: Energy Conservation [Ref. Fig.]
Dependence on m
Let
u
be the minimum speed required at
B
.
Apply energy conservation between point
A
and
B
we get
K
E
A
+
P
E
A
=
K
E
B
+
P
E
B
Taking point
A
as zero potential level
⇒
2
1
m
u
2
=
m
g
(
R
+
R
)
+
2
1
m
v
2
.
.
.
.
(
1
)
From equation
1
, we observe that mass
m
is cancelled on both sides.
⇒
u
is independent of
m
.
Steo 2: Solving Equation
Dependence on R
At highest point, Tension will be zero for minimum velocity
⇒
gravitational pull will provide necessary centripetal force
⇒
m
g
=
R
m
v
2
⇒
v
2
=
R
g
.... Putting in equation
(
1
)
We get,
2
1
m
u
2
=
m
g
(
R
+
R
)
+
2
1
m
(
R
g
)
⇒
u
=
5
R
g
⇒
u
is proportional to the square root of
R
.
Hence option
B
is correct.
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