Question

A length $L=15m$ if flexible tape is tightly wound. It is then allowed to unwinding as it rolls down a fixed incline that makes and angle $\theta ={30}^{\circ }$ with the horizontal, the upper end of the tape being fixed. Find the time taken (in second) by the tapes to unwind completely. Neglect radius at any time w.r.t. the length of the rope.

Answer 1

Given: A length L=15m if flexible tape is tightly wound. It is then allowed to unwinding as it rolls down a fixed incline that makes and angle $\theta ={30}^{\circ }$ with the horizontal, the upper end of the tape being fixed.

To find the time taken (in second) by the tapes to unwind completely. Neglect radius at any time w.r.t. the length of the rope.

Solution:

Here the tape can be taken as a disc as the it is very thin.

Here in this case,

$\rho (l-x)g\sin \theta -T=\rho (l-x)a........\left(i\right)$

$R\left(x\right)\times T=\frac{1}{2}\times (l-x)R^2\times \alpha .......\left(ii\right)$

$dx=R\left(x\right)d\sigma ........\left(iii\right)$

From eqn(ii) we get

$T=\frac{1}{2}\rho (l-x)a$

Substituting it in eqn(i), we get

$a=\frac{2}{3}g\sin \theta =a_0$

Hence,

$v=\sqrt{2a_{0}x}$

Hence the time needed is

$t=\sqrt{\frac{2s}{a}}=\sqrt{\frac{3l}{g\sin \theta }}$

substituting the values we get

$t=\sqrt{\frac{3\left(15\right)}{10\sin \left(30\right)}}⟹t=\sqrt{\frac{45}{5}}⟹t=\sqrt{9}⟹t=3sec$

is the time taken (in second) by the tapes to unwind completely. Neglect radius at any time w.r.t. the length of the rope.