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Question

Figure shows a rod of length 20 cm pivoted near an end and which is made to rotate in a horizontal plane with a constant angular spped. A ball of mass m is suspended by a string also of length 20 cm from the other end of the rod . If the angle θ made by the string with the vertical is 30, find the angular speed of rotation. Take g=10 m/s2

A
4.4 rad/s
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B
3.2 rad/s
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C
2.4 rad/s
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D
1.2 rad/s
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Solution

The correct option is A 4.4 rad/s
Let the angular speed be ω.

As it is clear from the figure, the ball moves in a horizontal circle of radius L+Lsinθ where L=20 cm. Its acceleration is, therefore , ω2(L+Lsinθ) towards the centre.

The forces on the bob are :-
a) the tension T along the string and
b) the weight mg
Resolving the forces along the radius and applying Newton's second law:
Tsinθ=ω2L(1+sinθ) ............(i)
Applying Newton's second law in the vertical direction:
Tcosθ=mg ..........(ii)

Dividing (i) by (ii),
tanθ=ω2L(1+sinθ)g
or ω2=gtanθL(1+sinθ)
Putting the values, we get,
ω=4.4 rad/s

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