Question

A circular disc reaches from top to bottom of an inclined plane of length 'L'. When it slips down the plane, it takes time ' t₁'. when it rolls down the plane, it takes time t₂. The value of `t_2/t_1` is `sqrt(3/x)`. The value of x will be ______.

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Answer 1

A circular disc reaches from top to bottom of an inclined plane of length 'L'. When it slips down the plane, it takes time ' t₁'. when it rolls down the plane, it takes time t₂. The value of `t_2/t_1` is `sqrt(3/x)`. The value of x will be 2.

Explanation:

When disc rolls down

 When disc slips down

Case 1:

a₁ = gsinθ .............(i)

Case 2:

`a_2 = (gsintheta)/(1 + I/(MR^2))`

`a_2 = (gsintheta)/(1 + (MR^2)/(2MR^2)) = 2/3gsintheta` ............(ii)

Using the second equation of motion for cases 1 and 2,

Case 1:

`L = 1/2a_1t_1^2` ............(iii)

`t_1 = sqrt((2L)/a_1)`

Case 2:

`L = 1/2a_2t_2^2`

`t_2 = sqrt((2L)/a_2)` .......(iv)

Now, using equations (i), (ii), (iii), and (iv), (iv)

Now, `t_2/t_1 = sqrt(a_1/a_2) = sqrt(3/2)`

x = 2