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show that the length of rod is invariant under galilean transformations.


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Answer 1

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suppose the ordinates of two points A and B in two inertial frames S and S' are (x1, y1, z1), (x2, y2, z2), (x1', y1', z1'),(x2' , y2' , z2') respectively.

If S' moves with velocity v relative to S along x' axis, then according to Galilean transformation.

X1'= X1 - Vt , Y1'= y1 z1'= z1

X1'= X1 - Vt , Y1'= y1 z1'= z1X2'= x2 - vt, y2 '= y2, z2'= z2

The distance between the points A and B in the frames S'

= [( X2' - X1')² + (Y2' - Y1')² + ( Z2' - Z1')²]½

[{(X2 - vt)-(X1 - vt)}² +(y2 - y1)² + (z2 - z1)²]½

= [(x2 - x1)² + (y2 - y1)² + (z2 - z1)²]½

= Distance between the points in the frame S. Consequently the length of rod in invariant under Galilean transformation

Answer 2

suppose the ordinates of two points A and B in two inertial frames S and S' are (x1, y1, z1), (x2, y2, z2), (x1', y1', z1'),(x2' , y2' , z2') respectively.

If S' moves with velocity v relative to S along x' axis, then according to Galilean transformation.

X1'= X1 - Vt , Y1'= y1 z1'= z1

X1'= X1 - Vt , Y1'= y1 z1'= z1X2'= x2 - vt, y2 '= y2, z2'= z2

The distance between the points A and B in the frames S'

= [( X2' - X1')² + (Y2' - Y1')² + ( Z2' - Z1')²]½

[{(X2 - vt)-(X1 - vt)}² +(y2 - y1)² + (z2 - z1)²]½

= [(x2 - x1)² + (y2 - y1)² + (z2 - z1)²]½

= Distance between the points in the frame S. Consequently the length of rod in invariant under Galilean transformation