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Consider this problem:

A simple coupling for connecting two axes in a motor consists of two cylindrical plates (r = 0.6 m) that can be pressed together if necessary. Plate A with mass mA = 6 kg is accelerated in 2 s to an angular velocity of ω1 = 7.2 rad / s. Clutch disc B with mB = 9 kg is still at rest. If the plates are now coupled together, they both rotate with a reduced angular velocity ω2.

What is the value of the angular momentum and torque of plate A before the coupling took place? Which angular velocity ω2 is reached after coupling?

Now I had no problem calculating the angular moment and torque of plate A before the coupling.Should be

$$ L_a = 7,776 kgm^2/s $$ $$ M_a = 3,888 Nm $$

Now for the angular velocity after the coupling I though of that this way. We have the plate A with La angular momentum, and plate B with Lb angular momentum.Since Lb = 0 because the plate is standing still we can take our angular momentum just to be La. For the moment of inertia we have to take into account that the total weight has changed,from 6 to 15kg. So I calculated Itotal and got $$ I_t = 2,7 kgm^2 $$ And now we can easily calcualte the angular velocity $$ \omega = 2,88 rad/s $$.

Now what is bugging me is the fact that I have simply assumed that the angular momentum stays the same after coupling.Since angular momentum is dependent on the moment of inertia and angular velocity I am not sure if I can make this assumption. If plate B was also moving how would I calculate the angular momentum than? Would simply adding them do the trick?

Qmechanic
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codeisfun
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3 Answers3

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In general, the angular momentum of a system is conserved in the absence of an external torque. If your two disks are on friction-less axles which line up with each other then: ${I_1}{w_1} = ({I_1} + {I_2}){w_2}$.

R.W. Bird
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I assume that for your problem the plates are balanced; that is, the center of mass of each plate is along the axis of rotation and the axis of rotation is a principal axis. For a fixed axis (or point) of rotation, that axis (point) should be taken as the origin from which to evaluate torques. I assume the two plates have a common axis of rotation (the two separate axes "line up"). With respect to the fixed axis (call it in the z direction) there is no net external torque on the system consisting of both plates, so angular momentum about this axis is conserved as @R.W. Bird says in an earlier answer. (Friction force/torques between the two plates are internal to the system and do not affect the total angular momentum of the system.) Assuming the two plates rotate about a common axis, the total angular momentum with respect to that axis is conserved whether or not the second plate is initially rotating. If the plates do not have a common axis of rotation, see the discussion in the next to last paragraph of this answer.

For balanced plates, the center of mass (CM) of each plate is along the common axis of rotation; also, the axis of rotation is a principal axis and the only component of angular momentum is along the z axis. If a plate is unbalanced, the implicit assumption is that the common axis of rotation, the z axis, is constrained to remain fixed; that is $\vec \omega$ only has a component along the $z$ axis. However, the axis of rotation is no longer a principal axis for an unbalanced plate. If the fixed axis is not a principal axis, it is still true that $J_z = I_z \omega$ is constant, where $J_z$ is the angular momentum in the z direction, $I_z$ is the moment of inertia about the z axis, and $\omega$ is the angular speed of rotation (constrained to be fixed in the z direction), but $J_x$ and $J_y$ may both also not be zero (inertia in general is a tensor and the products of inertia are not necessarily zero.) The total angular momentum varies in direction, describing a cone around the z axis of rotation as an unbalanced plate rotates, and thus there is a net external torque on the system, at right angles to the z axis of rotation. Whatever forces/torques are required to maintain the axis of rotation fixed do not contribute to the constant angular momentum about the z axis.

A recent question on this exchange dealt with a case where angular momentum is conserved but the axis of rotation is not fixed; see Conservation of angular momentum of disk and block on this exchange. If for your problem the two axis of rotation for the plates although in the same direction do not align, then you should consider the angular momentum about the center of mass of the two-plate system.

The text Mechanics, by Symon derives the appropriate relationships and has a good physical discussion of these considerations. The text, Analytical Mechanics, by Fowles has a good discussion of rotation about a fixed axis that is not a principal axis.

John Darby
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you have this two situations:

bevor the clutch engaged

$$I_e\,\dot{\omega}_e=\tau_e+\tau_c$$ $$I_c\,\dot{\omega}_c=-\tau_c$$

and after the clutch engaged

with $~\omega_e=\omega_c$

$$(I_e+I_c)\dot\omega_c=\tau_e$$

where e stay for engine and c stay for clutch

so I don't see any conservation of the angular momentum

Eli
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  • If the two disks are on a common axle, then when they make contact the two torques are equal and opposite. – R.W. Bird Feb 28 '21 at 18:23
  • To expand on the above comment by @R.W. Bird. Considering the two plates together as the system, there is no net external torque on the system; therefore, the angular momentum is constant. Only external torques can change the angular momentum of a system, similar to only external forces can change the linear momentum of a system. – John Darby Feb 28 '21 at 20:57
  • @R.W.Bird i extend the equations – Eli Mar 01 '21 at 08:31