- #1

JD_PM

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## Homework Statement

[/B]

A train stands in the middle of a rotating disk with an initial angular velocity of

$\omega_i$. The mass of the train is m and the moment of inertia of the train-disk is I. At one point the train departs on a straight track to a distance R from the disk's centre. (R smaller than the radius of the disk)

a) Find $\omega_f$,

**the angular velocity of the disk when the train is at a distance R from the centre.**

b) Show that if m, R and I are strictly positive,

**the total energy of the disk-train system is strictly smaller when the train has a distance R > 0 from the disk's centre compared with the total energy when**

**the train is in the middle.**

c) The train runs at a constant speed from the centre to

distance R.

**Show that the work performed by the braking force on the system is equal to**

:

:

$$W = - \int_0^R dr mr (\frac{I_i \omega_i}{I_i + mr^2})^2$$

d) Use the results from questions b) and c) to show that:

$$\int_0^R dr \frac {r}{(1 + r^2)^2} = \frac {1}{2(1 + r^2)}$$

## Homework Equations

- Conservation of angular momentum.

- Rotational kinetic energy.[/B]

## The Attempt at a Solution

[/B]

a) In this system the angular momentum is conserved due to the fact that the net external torque is zero. Therefore the spin angular momentum satisfies that:

$$I_i \omega_i = I_f \omega_f$$

$$\frac{I_i \omega_i}{I_i + mR^2} = \omega_f$$

Note that I applied the parallel axis theorem.

b) The energy of the disk-train-Earth system is conserved,

**but not if we do not take into consideration the Earth as part of the system.**

Actually, the total rotational kinetic energy is:

Actually, the total rotational kinetic energy is:

$$K = \frac{1}{2} I_{CM} \omega^2$$

Note that the higher is the angular velocity the higher is the energy.

**The angular velocity is higher when the train stands at the centre of the disk and lower when the train is placed at a distance R from the centre because of the conservation of angular momentum.**

**Therefore the energy is higher when the train stands at the centre of the disk and lower when the train is placed at a distance R from the centre.**

c)

**The friction force does a work on the system and the centripetal force is equal to the friction force so:**

$$F_c = \frac{m v^2}{r} = mr \omega^2$$

From here get the provided equation.

**HERE COMES MY BIG PROBLEM**

d)

**I do not know how to show the equation provided in this section**from sections b) and c):

$$\int_0^R dr \frac {r}{(1 + r^2)^2} = \frac {1}{2(1 + r^2)}$$

Any help is appreciated, thank you.

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