02 Galilean Cart

02 Galilean Cart#

To show and discuss an example of Galilean transformations

Two carts, easy rolling and able to carry a human being; construction as shown in Diagram.
Large funnel (outlet reduced to \(4 \mathrm{~mm}\)), suspended by three cords.
Clamping material to support the funnel-pendulum.
\(1 \mathrm{~kg}\) of salt.
Broom.

One person sits on the cart and fills the funnel with salt, keeping the outlet closed with a finger and gives the funnel-pendulum a deflection into the \(x^{'}\)-direction. The demonstrator moves the cart with constant speed along the front of the lecture hall (\(y\)-direction). As soon as the speed is constant, the person on the cart makes the pendulum go. A salt-track is written on the floor of the lecture hall (see Figure {number}). This track shows the recording of the movement of the swinging funnel in the \(x\)-\(y\) plane.

The same demonstration is performed but now with the funnel-pendulum swinging into the \(y^{'}\)-direction. A second salt-track appears on the floor (see Figure 31).

Again the salt track shows the recording of the movement of the swinging funnel in the \(x\)-\(y\) plane.

The results are discussed.

The pendulum moves in the \(x^{'}\)-\(y^{'}\)-\(z^{'}\)-frame according to: \(x^{'}=A \sin (\omega t+\phi)\); \(y^{'}=0 ; z^{'}=0\). The writing on the ground in salt is in the \(x-y-z\)-frame. The \(x^{'}-y^{'}-z^{'}-\) frame moves with a speed \(\mathrm{V}\) into the \(\mathrm{y}\)-direction., so a point measured in the \(x^{'}-\) \(y^{'}-z^{'}\)-frame will have an \(y\)-coordinate: \(y=V t\). (see Figure2).
When the pendulum swings into the \(y^{'}\)-direction, the movements in the \(x\)-y-zframe will be: \(y=V t+A \sin (\omega t+\phi) ; x=0\) and \(z=0\) (see Figure {number}).

As Figure 31 makes clear, the difference between much - and little salt is more pronounced when \(y=V t\) is steeper; that is at higher speeds of the cart.