Ó 2000, W. E. Haisler Disk, Spring, Mass System by Energy Method 4

A 5-kg disk (A) with diameter of 100 cm is attached to the face of a 10-kg disk (B) with diameter of 200 cm, and this 15 kg unit is free to rotate about the pinned support. A mass of 3 kg (mass C) is attached to a massless cable that is wound around the outer diameter of disk B as shown. A massless cable is attached to a massless spring and wound around disk A. The spring has a spring constant of 500 N/m and is initially unstretched. Assume no stretching or slippage of the cables.

a) How much will the spring have stretched after the 3 kg mass (C) has moved 20 cm downward?

b) If the system is initially at rest and mass C is allowed to fall, determine the angular velocity of the disk unit (A&B) after the 3-kg mass has fallen 20 cm downward.

c) How far will mass C fall before coming to rest and then rebounding upwards?

Solution

a) This is a kinematics problem. Let the position of C and D be and , and the angular velocity of the unit be . Then and where is angular position of the disk. The rotation is the same for each disk, hence,

Thus,

For ,

b) The mass moment of inertia for a disk about its center is . Hence,

COE gives

Beginning:

Since the system is at rest,

Assume for mass C is zero when

Spring is initially unstretched, so

· No external forces do work on the system, so

COE becomes

End:

Note from kinematics that , so .

(negative since is down but positive PE=mgh is up)

Substituting above terms into COE gives:

Solving for gives

c) When mass C comes to rest, and , so that . Hence COE becomes:

But , so that

Solving for gives