Question

A spool of thread has rolled under the bed! Fortunately, you get the free end of it
thread. The moment of inertia of a homogeneous, full cylinder about
Its axis of symmetry is: ? = (1/2) ?? ^ 2. The moment of inertia of the whole spin about the axis of rotation is: ???? = 15?? ^ 2. The friction between the reel and the floor is  ?? = 0.25 (static) and ?? = 0.2 (dynamic).

You start pulling the thread from under the bed. In which you first give the thread a powerful bounce in the horizontal direction. Due to the large force ? the reel now slides on the floor, and the frictional force is now dynamic with
coefficient of friction ?? = 0.2. We look at the motion of the reel after it has got the initial velocities ?0
and ?0, and after the force ? has stopped working.

a) Find an expression for the linear velocity ? (?) and for the angular velocity ? (?) as a function of
time. Show that the reel starts rolling without slipping after a while ?1:
?1 = (15/4) * (?0 /?)
.
Remember that the radius of the discs on the outside of the reel is ? = 3? when checking the rolling condition.

b) Calculate the velocity ?⃗1 = ?⃗ (?1) and the angular velocity ?⃗1 = ?⃗ (?1).

c) Calculate the spin of the reel relative to the contact point on the floor at two different times:
at a time ?0 just after the cord has stopped working when the reel has initial velocities ?0 and
?0, and at a time ?1 when the reel starts to roll without slipping. Express the spin using ?, ?
and ?0. Explain the result.

We now start to investigate what happens when you pull a  thread in another direction.

We assume that you pull gently with a force ? at an angle, where
cosθ = 2? / 3? = 2/3 (R is radius)
.
l) Draw a free body diagram for this case and name the forces. Find the torque at
both the point of contact with the floor and about the mass center.

m) Describe the motion of the reel in this case. What happens to the reel when you use an even bigger angle?

Answer 1

Solution in the uploaded images