How can there be work due to torque?

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Work done by torque is possible because the equation W = TΔΘ relates torque and angular displacement, where torque (T) acts along the axis of rotation and ΔΘ represents the angle through which an object rotates. Unlike linear motion, in rotational motion, the relationship between torque and angular displacement allows for work to be done even when they are perpendicular. The concept of work in rotational dynamics differs from linear dynamics, where force perpendicular to motion does not do work. Understanding the movement of force in the context of torque reveals that as a shaft spins, the force contributes to the work done. This highlights the unique nature of work in rotational systems compared to linear systems.
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Homework Statement


I'm doing rotational problems and I'm having a hard time understanding how work done by torque is possible.

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The Attempt at a Solution


The equation is W = TΔΘ. Torque points along the axis of rotation and ΔΘ always points along the rotational motion of the object. These two are perpendicular to each other so how can there be work done? I know in linear motion, if the force is perpendicular to the motion of the object, there is no work done. Do these rules not apply to work done on rotating object?
 
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Go back to basics, work is force times distance moved, now look at where the force is in torque and how far it moves when a shaft is spinning.
 
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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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