Solution 19.2.11 from the collection (resolution book) of Kepe O.E. 1989

19.2.11. Let us assume that a thread with a mass of 2 kg and a radius of gyration ρ = 6 cm is wound on a coil of radius r = 8 cm, which is pulled with a force F = 0.5 N. It is necessary to determine the angular acceleration of the coil, taking into account the absence of slipping.

To solve this problem, it is necessary to use the law of conservation of energy, which allows us to express the kinetic energy of the coil through its angular velocity. You should also use the moment of inertia of the coil and Newton's second law for rotational motion.

From the law of conservation of energy it follows that the work of external forces is equal to the change in the kinetic energy of the coil. The work done by external forces is equal to the product of force and displacement, which in this case is equal to the product of force and 2πr (the length of the thread wound on the spool). Thus, we get the equation:

F * 2πr = Δ(1/2 * I * ω^2),

where I is the moment of inertia of the coil, ω is the angular velocity of the coil.

It is known that the moment of inertia of the coil is equal to:

I = m * ρ^2/2,

where m is the mass of the thread on the spool.

Substituting the expression for the moment of inertia into the equation and solving it for angular acceleration, we obtain:

α = F * r / (m * ρ^2/4 + m * r^2) = 0.5 * 0.08 / (2 * 0.06^2 / 4 + 2 * 0.08^2) ≈ 3.27 rad/s^2.

Thus, the angular acceleration of the coil is approximately 3.27 rad/s^2.

Solution tasks 19.2.11

We present to your attention the solution to problem 19.2.11 from the collection “Problems in General Physics” by author O.E. Kepe, released in 1989. In our solution, we used the law of conservation of energy and the moment of inertia to determine the angular acceleration of the coil. The solution is presented in a clear and accessible form and can be useful for both students and physics teachers.

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