A top is a toy that is made to spin on its pointed end by pulling on a string wrapped around the body of the top. The string has a length of 54 cm and is wound around the top at a spot where its radius is 2.2 cm. The thickness of the string is negligible. The top is initially at rest. Someone pulls the free end of the string, thereby unwinding it and giving the top an angular acceleration of 13 rad/s2. What is the final angular velocity of the top when the string is completely unwound?

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To find the final angular velocity of the top when the string is completely unwound, we can use the principle of conservation of angular momentum.

The formula for angular momentum is L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.

First, let's find the moment of inertia (I) of the top. The moment of inertia of a solid cylinder rotating about its central axis is given by the formula I = 0.5 * m * r^2, where m is the mass of the top and r is the radius.

Since we don't have the mass of the top, we need to find it using the given information. The length of the string is 54 cm, and it is wound around the top at a spot where its radius is 2.2 cm. The length of the string represents the circumference (2πr) that the string covers, and since it is wound around at radius 2.2 cm, we can find the circumference.

Circumference = 2πr
54 cm = 2π * 2.2 cm
r ≈ 8.63 cm

Now that we know the radius, we can find the mass. The mass is proportional to the radius, assuming the density is constant. Let's assume a density of 1 g/cm³ for the top.

Volume = π * r^2 * h
Volume ≈ 3.14 * (8.63 cm)² * 1 cm ≈ 236.24 cm³

The mass is then the product of the volume and density:

Mass ≈ 236.24 cm³ * 1 g/cm³ ≈ 236.24 g

Now we have the mass, so we can calculate the moment of inertia:

I = 0.5 * m * r^2
I ≈ 0.5 * 236.24 g * (2.2 cm)² ≈ 558.52 g*cm²

Next, let's find the initial angular velocity (ω) of the top. Since the top is initially at rest, the initial angular velocity is zero.

Now we can use the principle of conservation of angular momentum: L_initial = L_final.

The initial angular momentum (L_initial) is zero because the initial angular velocity is zero.

The final angular momentum (L_final) is the product of the final angular velocity and the moment of inertia (L_final = I * ω_final).

L_initial = L_final
0 = I * ω_final

Therefore, the final angular velocity (ω_final) is zero.

So, when the string is completely unwound, the final angular velocity of the top is zero.