Rigid Body Rotation Application

AI Thread Summary
To calculate the angular velocity (\omega) of a rigid body rotation involving two vertically oriented shafts connected by a cross member, one shaft must remain stationary while a constant tangential load (F) is applied to the other. This setup generates a constant torque (τ = R * F) about the fixed axis, leading to a constant angular acceleration (β = τ/I_z), where I_z is the moment of inertia. The relationship between angular velocity and time must be considered, as \omega will change over time due to this constant acceleration. The kinetic energy of rotation can be expressed as K_{rot} = (1/2)I_z\omega^2, which may help in further calculations. Understanding these principles is crucial for solving the problem effectively.
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Homework Statement


I'm trying to include rigid body rotation in a problem I'm working on but can't seem to figure it out.

Two shafts oriented vertically are connected by a thin cross member of length R. Holding one shaft stationary and applying a constant tangential load F to the other shaft will cause rotation at some speed \omega. Given the mass m and moment of inertia I_z. Is it possible to calculate the angular velocity?

Homework Equations


Not sure what we need, but I believe it's going to involve energy.
K_{rot}=\frac{1}{2}I_z\omega^2
Other than that I'm not sure.

The Attempt at a Solution


No idea. I've been thinking about the problem for the past couple days but can't figure out how to determine the angular velocity given only these variables. If needed I may be able to supply other variables (this is a overly simplified example to give you an idea of the problem).

Thanks.
 
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I have no idea about this problem without a picture... ehild
 
Applying that constant tangential force F means constant torque (τ=R*F) with respect to the fixed axis and constant angular acceleration: β=τ/I, where I is the moment of inertia, again with respect to he fixed axis. The angular velocity will change with time.

ehild
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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