Work Done by String Connecting to Uniform Vertical Disk | Solution and Equations

[Amar.alchemy]

Work done by String??

Homework Statement

You connect a light string to a point on the edge of a uniform vertical disk with radius R and mass M. The disk is free to rotate without friction about a stationary horizontal axis through its cente. Initially, the disk is at rest with the string connection at the
highest point on the disk. You pull the string with a constant horizontal force F until the wheel has made exactly one-quarter revolution about a horizontal axis through its center, and then you let go. (a) Find the work done by the string

Homework Equations

The Attempt at a Solution

<br /> \[\begin{array}{l}<br /> w = \tau \int\limits_0^{\frac{\pi }{4}} {d\theta } \\ <br /> w = FR\pi /4 \\ <br /> \end{array}\]<br />

Is this correct??Kindly help me

 

[kuruman]

It is correct with one exception. What fraction of π is one-quarter of a revolution?

 

[Amar.alchemy]

Sorry, it shuld be \[\frac{\pi }{2}\]...

However in the solution manual it is given as:
The torque is \[\begin{array}{l}<br /> \tau = FR\cos \theta \\ <br /> w = \int\limits_0^{\frac{\pi }{2}} {FR\cos \theta d\theta } \\ <br /> w = FR \\ <br /> \end{array}\]

So i am little bit confused...

 

[kuruman]

Is there a picture that goes with this? Your answer would be correct if the tension pulling on the wheel remained tangent to the wheel at all times as in a pulley. This is what I thought initially to be the case. Here it seems that the string is firmly attached to the rim of the wheel and the point of application of the force rotates with the wheel.

Draw yourself a wheel and put in a horizontal tension F at some angle, say the "1 o'clock" position. Call θ the angle between 12 o'clock and 1 o'clock. What is the torque? Note that the lever arm changes from R at 12 o'clock to zero at 3 o'clock.

 

[Amar.alchemy]

There is no picture given with this question and in the problem he clearly mentions that "The force applied is constant and horizontal"... so do you think is it appropriate to assume that point of application of the force rotates with the wheel??

 

[kuruman]

Yes, I think so. As you say, "clearly mentions it." I misread the problem initially. Also you get the answer in the solution manual that way.

 

[Amar.alchemy]

ok, now i understood the problem (the string is connected at a point instead of wrapping around the wheel)... Thanks :-)