Ó 2000, W. E. Haisler Disk/Mass Kinetics Example 11

Ó 2000, W. E. Haisler Disk/Mass Kinetics Example 11

For the disk-mass system shown below, the system is initially at rest. The pinned supports can be considered frictionless. Ropes do not stretch, nor slip on disks. Determine the following:

a)  Angular accelration of each disk immediately after release of each suspended mass.

b)  Tension in each rope.

c)  Magnitude of the force on each pin supporting the disks.

Disk 1:

Disk 2:

Mass 3:

Mass 4:

Hence, we have 4 simultaneous equations to solve in terms of 6 unknowns: T1, T2, T3, a1, a2, and .

Oops, something is wrong! More unknowns then equations! What is wrong?

We neglected to say any thing about friction between the rope and the pulleys; or whether the rope stretches (kinematics). If the rope does not stretch and there is no slippage between the disks and the rope, then we can write other equations:

·  The tangential acceleration at the outer radius of disk 1 is: . Since the rope between the disk and mass 3 does not stretch and there is no slippage, then . Similarly, .

·  The rope between the two disks does not stretch nore is the slippage; hence: where q is the rotation of a disk. Thus, , and or . The last three equations (boxed) are called kinematic constraints.

Consider the following 5 cases:

1. 

2. 

3. 

4. 

5. 

Case 1:

Notice that since the mass of the suspended blocks are equal, the accelerations are zero.

Case 2:

Case 3:

Notice that even though the masses of the suspended blocks are different, the angular acceleration of each disk is equal (because they have same radius and rope does not slip on disks).

Case 4:

Notice that since the mass of the block on the left is larger then on the right, the angular acceleration of each disk is CCW (-).

Case 5:

Case 6: What happens in case #3 if the pin for each disk has a frictional torque of 50 Nm? Recall that friction opposes motion.

Notice that angular acceleration is now less than for case 3.

Case 7: Same as case 6, except radius of and .

Notice that the angular acceleration of disk 1 is now 1/2 that of disk 2.

c) Magnitude of the force on each pin supporting the disks?

Since each disk is spinning about its center of mass (and center of mass has no linear acceleration), there are no net forces on the pin support due to inertial forces. Hence, the only forces on free body 1 and 2 that must be reacted by the pin support are the rope tensions in x direction, and tension plus weight of suspended mass in y direction. Or, can apply COLM in x and y directions for each disk free-body to obtain: