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EXERCISE-LEVEL I
 
 

LEVEL  I

1. Show that Newton’s equation d2x/dt2=a(t) is invariant under transformation:
                x®x-vt
                t®t

2. Gravitational acceleration defined by a=GM/r2 gives dynamics for motion of a body under free fall. Find the dependence of r on t in free fall. 

3. Potential of a body is defined as: Energy present in body at a point divided by the mass of the body or the charge of the body or any fundamental property. We define force to be F=-(dV/dr)r, where r is a unit vector towards radius vector.

Why so? What could be a fundamental assumption of physics which would have yield such an equation?

4. Schrodinger wave equation for a mass is defined by:
 

 

Derive the de-Broglie relation (l=h/mv) from this.

This is a proof in wrong direction; originally, the second relation was used to arrive at the first equation. You can establish de-Broglie relation from Einstein’s equation E=mc2 and Planck’s energy relation E=hc/l.

<You will find more general derivations in Level II>

5. Operators are very fundamental to Physics. They act on something and give a result. For instance, d2/dz2 acts on x to give acceleration.
In Schrodinger equation, energy operator is -(h2/8p2m)(d2/dx2), why? Note after your proof that this operator returns energy as well as the wave function itself. Such operator is called the Eigen value operator. 

6. Lorentz transformation for relativity is defined by:

                x ® g(x - vt)
                y ® y
                z ® z
                t ® g(t – vx/c2)

                where
                             g = 1/(1 - v2/c2)1/2 
                             is called the Lorentz factor.

There is another relation:

x2 + y2 + z2 – c2t2 = constant.

Another relation in relativity is:

Px2 + Py2 + Pz2 – E2/c2 = constant = -m02c2
                                                where P=momentum, m0=rest mass, E=energy.
So, derive the Lorentz transformation like relation for Px, Py, Pz, E. Does this mean energy is not conserved? <Look at Level II for more>

 7. Planck radiation law is defined by:

 

This is energy per unit volume per unit frequency.
Therefore, energy per unit volume = òu(n, T) dn
Find this integral integrating from 0 to ∞. Is this limit compatible with Einstein’s relativity theory. <Look at Level 2 for evaluation of this integral>

8. Can you derive Kepler’s 3rd law, square of radius and cube of time period law, from the equation d2r/dt2=GM/r2 (law for gravity) by transforming r®lr where l is a number which represents change in radius and t®mt, t resembleng time taken for small displacement of planet? [Hint: Remember that law has same form for all planets.]

9. Reduced mass: Two bodies M and m are rotating with w about an axis (passing through centre of mass).

where the system has to be replaced by as single mass M¢ at radius R from O such that total energy of rotation and the total angular momentum remains unchanged.
Find M¢ and R in terms of M, m, r1, r2. M¢ is called the reduced mass. It does not affect the syatem’s state since conservation laws are intact.

10. Consider the Minkowski space
 

Two lines are drawn. Show that DOAB will have same area even after coordinate rotation. If have an expertize in tensors, you can prove that volume of any dimensional region (area for 2-D region) is invariant under coordinate transformation.

 

Level II
 
 
 

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