A 60-cm-long, 500 g bar rotates in a horizontal plane on an axle that passes through the center of the bar. Compressed air is fed in through the axle, passes through a small hole down the length of the bar, and escapes as air jets from holes at the ends of the bar. The jets are perpendicular to the bar's axis. Starting from rest, the bar spins up to an angular velocity of 150 rpm at the end of 10 s. How much force does each jet of escaping air exert on the bar? If the axle is moved to one end of the bar while the air jets are unchanged, what will be the bar's angular velocity at the end of 10 seconds?

The part about moving the axle is rather confusing. I assume they mean doing the experiment over again with the axis at the end of the bar. The angular speed at the end of the 10 s will be less than for the first case (also after 10 seconds) for two reasons:

(1) Only one jet produces a torque about the axle, and
2) The moment of inertia of the bar is greater with the axle at the end.

To get the jet force, use this relation for the axle-in-the-middle case:
Torque = 2*(force)*(lever arm)
= (Moment of inertia*(angular acceleration rate)

To find the force exerted by each jet of escaping air on the bar, we can use the principle of conservation of angular momentum. The initial angular momentum of the bar is zero since it starts from rest. The final angular momentum is given by:

Angular momentum = (moment of inertia) x (final angular velocity)

The moment of inertia of a uniform rod rotating about an axis passing through its center is given by:

Moment of inertia = (mass x length^2) / 12

Given that the length of the bar is 60 cm and the mass is 500 g, we can calculate the moment of inertia:

Moment of inertia = (0.5 kg x 0.6 m^2) / 12 = 0.025 kg·m^2

Converting the final angular velocity from rpm to radians per second:

Final angular velocity = 150 rpm x (2π rad/1 min) x (1 min/60 s) = 15π rad/s

Substituting the values into the angular momentum equation:

Angular momentum = (0.025 kg·m^2) x (15π rad/s) = 0.375π kg·m^2/s

To find the force exerted by each jet of escaping air, we use the equation:

Force = Change in angular momentum / Change in time

Since the initial angular momentum is zero, the change in angular momentum is equal to the final angular momentum:

Force = (0.375π kg·m^2/s) / 10 s = 0.0375π N

Therefore, each jet of escaping air exerts a force of approximately 0.0375π N on the bar.

If the axle is moved to one end of the bar while the air jets remain unchanged, the moment of inertia of the bar will change. The new moment of inertia can be calculated using the parallel axis theorem:

Moment of inertia about new axis = Moment of inertia about center of mass + (mass x distance^2)

In this case, the distance is 60 cm/2 = 30 cm = 0.3 m, since the axle is at one end of the bar.

New moment of inertia = (0.5 kg x 0.6 m^2) / 12 + (0.5 kg x (0.3 m)^2) = 0.025 kg·m^2 + 0.0225 kg·m^2 = 0.0475 kg·m^2

Now we can use the conservation of angular momentum principle again to determine the new angular velocity. The final angular momentum remains the same, but the new moment of inertia is different:

Final angular velocity = Angular momentum / Moment of inertia

Final angular velocity = (0.375π kg·m^2/s) / (0.0475 kg·m^2) = 7.89π rad/s

Therefore, if the axle is moved to one end of the bar while the air jets remain unchanged, the bar's angular velocity at the end of 10 seconds will be approximately 7.89π rad/s.