How Long to Increase Spool's Angular Velocity from 11.0 to 35.0 rad/s?

[dana711]

Homework Statement

A spool of thin wire rotates without friction about its axis. A man pulls down on the wire with force of 14.9N. How long does it take to increase the angular velocity of the spool from 11.0rad/s to 35.0rad/s?
Icm = 0.490kg*m^2
Inner radius, r = 0.280m
Outer radius, R = 0.600m

Homework Equations

w=d(theta)/dt
w=v/r

The Attempt at a Solution

im not sure how to do this. i know the equation for angular velocity but i don't know what to do with the force and time

 

[Doc Al]

Figure out the angular acceleration using Newton's 2nd law for rotation.

 

[Moetasim]

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As a scientist, it is important to break down the given information and apply relevant equations to solve the problem. In this case, we are dealing with angular velocity and time, and we are given the force applied to the spool, as well as its moment of inertia and radii.

First, we can use the equation w=d(theta)/dt to relate the angular velocity to the time. Since the spool is rotating without friction, we can assume that the angular acceleration is constant and use the formula w2=w1+at, where w1 and w2 are the initial and final angular velocities, respectively, and a is the angular acceleration.

Next, we can use the equation w=v/r to relate the angular velocity to the linear velocity. Since the spool is rotating without friction, the linear velocity at any point on the spool will be the same as the tangential velocity at that point. We can use this to find the tangential acceleration, aT, using the formula aT=F/m, where F is the applied force and m is the mass of the spool.

Now, we can use the formula aT=r*alpha, where alpha is the angular acceleration, to find the angular acceleration. Once we have the angular acceleration, we can plug it into the equation w2=w1+at to solve for the time, t, it takes for the angular velocity to increase from 11.0rad/s to 35.0rad/s.

Finally, we can use the given moment of inertia and radii to find the mass of the spool, and then use the formula F=m*aT to find the applied force in terms of the mass, and solve for the time, t, in terms of the mass, moment of inertia, and radii.

In conclusion, by breaking down the problem and applying relevant equations, we can solve for the time it takes for the angular velocity of the spool to increase from 11.0rad/s to 35.0rad/s.