Classical Mechanics(linear acceleration)

[iontail]

Homework Statement

A reel of mass M and of radius a has tape wound round its axle
which has radius b . The tape has negligible mass. The reel is
initially at rest on a rough horizontal table when the free end of the
tape is pulled horizontally by a constant force T , as shown below.
The pull T is sufficiently small that the reel does not slide.
Show that the resulting acceleration is given by

a=dv/dt = a(a-b)T/M(k^2+a^2)
where V is the velocity of the centre of the reel and k is the radius
of gyration of the reel.

Homework Equations

not given

The Attempt at a Solution

working so far,

dv/dt= r*angular acceleration(al)
dv/dv=t*r/I i=moment of inertia

unable to complete the proof.
please help

 

[Dick]

Write the total kinetic energy (both rotational and translational) of the wheel as a function of v. Now the rate of change of KE is equal to T times the rate at which the tape is being reeled it. There are two things to consider for the tape rate, the wheel is rolling with velocity v and the tape is unwinding from that axis. You have to add them. BTW your given solution is clearly wrong. If a=b that formula would say the acceleration would be zero. I don't think that's right.

 

[Dick]

Hah! I'll bet I know what it is. Your figure must show the tape being pulled off of the bottom of the axle. So the v of wheel is AWAY from you. That would mean the v and unwinding are acting in opposite directions and need to be subtracted to get the net tape rate.