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Question

A rod of length L is pivoted at one end and is rotated with a uniform angular velocity in a horizontal plane. Let T1​ and T2​ be the tensions at the points L4 and 3L4 away from the pivoted ends. Then,


A

T1=T2

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B

T1>T2

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C

T1<T2

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D

None of these.

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Solution

The correct option is B

T1>T2


Step 1: Given data

  1. The length of the rod is L.
  2. The tension of the rod at a distance L4is T1.
  3. The tension of the rod at a distance 3L4 is T2.

Step 2: Centripetal force due to a body

  1. The centripetal force due to rotation of a body of mass m is defined by the form, Fc=mω2r, where, r is the distance of the body from the center of rotation and ω is the angular velocity.

Step 3: Diagram

Step 4: Finding the tension on 3L4due to L4

Now, considering the tension act on the length 3L4 due to the length L4 is T1. And from the concept of rotation, we know that this tension is equivalent to the centrifugal force acting on the length 3L4.

Let the mass of the road is m. So mass per unit length is mL and dx be the small portion at a distance x from the point L4 on the road. And the mass of the portion is mLdx.

So, the tension on the mass length dx is dT=mLω2xdx (Using Fc=mω2r formula)

So, the tension on the length 3L4 is

T1=L4LmLω2xdxorT1=mω2Lx22L4L=mω2LL22-L232=mω2L.15L232orT1=mω2L.15L232.......................(1)

Step 4: Finding the tension on L4 due to 3L4

Again considering the tension act on the length L4 due to the length 3L4is T2. And from the concept of rotation, we know that this tension is equivalent to the centrifugal force acting on the length L4.

Let the mass of the road is m. So mass per unit length is mL and dx be the small portion at a distance x from the point 3L4 on the road. And the mass of the portion is mLdx.

Now, the tension on the mass length dx is dT=mLω2xdx.

So, the tension on the length L4 is

T2=3L4Lmω2LxdxorT2=mω2Lx223L4L=mω2LL22-9L232=mω2L.7L232orT2=mω2L.7L232.......................(2)

From equations 1 and 2 we get,

mω2L.15L232>mω2L7L232.

So, T1>T2

So. option (B) is correct.


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