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 59 cm and is wrapped around the top at a place where its radius is 1.9 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 +10 rad/s2. What is the final angular velocity of the top when the string is completely unwound?
To find the final angular velocity of the top when the string is completely unwound, we can apply the equations of rotational motion.
First, let's determine the moment of inertia of the top. The moment of inertia (I) depends on the mass distribution and shape of the object. Since we know the radius of the top (r), we can assume it to be a solid cylinder and use the formula for the moment of inertia of a cylinder rotated around its central axis, which is given by:
I = (1/2) * m * r^2
where m is the mass of the top. However, we are not provided with the mass directly. We can use the given information to find the mass by using the formula for the circumference of a circle:
C = 2 * π * r
Given that the length of the string is 59 cm, we can set up the equation:
59 cm = 2 * π * 1.9 cm
Simplifying this equation, we can calculate the radius:
1.9 cm = (59 cm) / (2 * π)
Now, we have the radius of the top, which allows us to calculate the moment of inertia.
Next, we can use the equation of rotational motion:
τ = I * α
where τ is the torque applied, I is the moment of inertia, and α is the angular acceleration.
The torque (τ) can be calculated as the product of the force applied and the radius at which it is applied:
τ = F * r
Since the force is pulling on the string, it becomes the tension force. Therefore:
τ = T * r
Now we can rearrange the equation to solve for the tension force (T):
T = τ / r
We can substitute the given values into this equation:
T = (I * α) / r
Finally, we can use the equation of rotational motion to calculate the final angular velocity (ω) of the top when the string is completely unwound:
ω = 0 + α * t
Since the top is initially at rest, the initial angular velocity (ω₀) is zero. As the string is unwound, the angular acceleration (α) is given as +10 rad/s².
To find the time it takes (t) for the string to completely unwind, we can use the equation:
θ = ω₀ * t + (1/2) * α * t²
However, we do not have the value of θ (the angle through which the top rotates when the string is unwound). To solve for it, we can use the formula for the circumference of a circle:
C = 2 * π * r
When the string is completely unwound, the distance traveled by the string is equal to the circumference of the top. We can equate this distance to θ and solve for θ:
θ = 2 * π * r
Once we have θ, we can solve for t using the equation mentioned earlier.
Finally, we can use the value of t in the equation ω = α * t to calculate the final angular velocity (ω) of the top when the string is completely unwound.
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