String wrapping around a cylinder

Classical Mechanics Level 3

There is a cylinder with radius $R$ that is stationary and its axis is perpendicular to the horizontal axis. A soft and massless string of length $L$ that cannot increase its length when stretched is wrapped around the cylinder, a small ball is attached to the string end, touching the surface of the cylinder.

Suddenly, I hit the ball, causing the ball to have an initial tangential velocity of $v$ parallel to the horizontal axis and perpendicular to the cylinder's axis, so, the wrapped string begins to loosen from the cylinder.

Assume that there is no friction between the string and the cylinder, weight can be neglected, and the ball moves horizontally throughout the whole process. The time taken for the string to be completely loosen from the cylinder can be expressed as $\frac{L^a}{bvR}$ , where $a,b$ are integers. Find $a+b$ .

The answer is 4.

1 solution

A Former Brilliant Member
Jul 10, 2020

Nice Problem.
Let us assume that the string opens $\theta$ angle.
$\large ds=vdt$ $\large \int_{0}^{\frac{L}{R}} R \theta d\theta = \int_{0}^{t} vdt$ $\large \boxed{\textcolor{#20A900}{t=\frac{L^{2}}{2vR}}}$
Velocity will be constant because work done by all the forces is zero.

String wrapping around a cylinder

The answer is 4.

1 solution

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