# Thermodynamics

# Friction and melting of an ice cube

Take 15 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and start searching for the solution.

A detailed solution is then proposed to you.

If you have more questions, feel free to ask them on the forum.

An ice cube at temperature , slides on the counter of a bar with an initial velocity and mass.

The coefficient of friction on the bar .

The energy expended by the friction is used to melt the ice.

The water formed leaves the ice with negligible relative speed.

Note the mass latent heat of the melting ice.

## Question

Determine, as function of time, the speed of the ice cube.

### Hint

Apply the second law of Newton in a closed system.

### Solution

Considering the ice at time , mass and the same mass at time , made of ice (of mass ) and liquid water of mass (here, ).

A momentum balance applied to this system gives :

Is, at the order :

## Question

What is the mass of the ice at the end of movement ?

Calculate the ratio for a speed , knowing that .

### Hint

### Solution

It reflects the energy expended by the friction used to melt the ice cube :

By integration :

The movement stops when the speed is zero, or at the time :

The mass of ice cube is then :

Numerical application :

The ice does not melt practically, the kinetic energy is negligible compared to the latent heat of fusion.