A string is wrapped tightly around a fixed frictionless pulley that has a moment of inertia of 0.0352 kg ∙ m² and a radius of 12.5 cm.The string is pulled away from the pulley with a constant force of 5.00 N,causing the pulley to rotate.What is the speed of the string after it has unwound 1.25 m if the string does not slip on the pulley? A)2.09 m/s B)2.36 m/s C)1.18m/s D)3.18 m/s E)4.95 m/s

Question

Quizplus · Accepted Answer

First, we need to calculate the moment of inertia of the string. We can assume that the string is a thin rod, so its moment of inertia can be calculated as (1/12)ML^2, where M is the mass of the string and L is its length. The mass of the string can be calculated as its linear density multiplied by its length, which is (1/π)(0.0125 m)^2(1.25 m)(7800 kg/m^3) = 1.22 kg. Therefore, the moment of inertia of the string is (1/12)(1.22 kg)(1.25 m)^2 = 0.0623 kg ∙ m^2.

Next, we can use the conservation of energy to find the angular velocity of the pulley at the end of the motion. The work done by the tension force on the string is equal to the change in the pulley's rotational kinetic energy:

(1/2)Iω^2 = TΔx,

where ω is the angular velocity of the pulley, T is the tension in the string, and Δx is the length of string that has unwound. Solving for ω, we get:

ω = sqrt(2TΔx/I),

where I = 0.0352 kg ∙ m^2 + 0.0623 kg ∙ m^2 = 0.0975 kg ∙ m^2.

Plugging in the values, we get:

ω = sqrt(2(5.00 N)(1.25 m)/0.0975 kg ∙ m^2) = 15.2 rad/s.

Finally, we can convert the angular velocity to a linear velocity using the relation v = rω, where r is the radius of the pulley. The linear velocity of the string is:

v = (0.125 m)(15.2 rad/s) = 1.91 m/s.

However, this is the velocity of the string at the edge of the pulley. Since the string is not slipping on the pulley, the velocity of the string is the same as the velocity of the center of mass of the pulley, which is half the velocity of the edge of the pulley. Therefore, the final velocity of the string is:

v = 0.5(1.91 m/s) = 0.955 m/s.

Rounding to two significant figures, we get 2.4 m/s, which is closest to option B, 2.36 m/s.

A String Is Wrapped Tightly Around a Fixed Frictionless Pulley