Problem set 89

1. A particle is at rest in rotating frame of reference. The pseudoforce(s) acting on the particle is(are)
1. None of these
2. Only the Coriolis force
3. only the centrifugal force
4. both Coriolis force and centrifugal force

Coriolis force is given by $F_{Co}=-2m(\vec\omega\times\vec v')$

Centrifugal force is given by $F_{c}=-2m\vec\omega\times(\vec\omega\times\vec r)$

Hence, in rotating frame, if particle is at rest only centrifugal force act on the particle. If particle is not at rest both forces act on the particle.

Hence, answer is (C)

2. A thin rigid rod of length $l$ is moving inside a sphere of radius $R(R>l)$ such that both of its ends are in contact with inner surface of the sphere. The degrees of freedom of the rod are:
1. Four
2. Three
3. Two
4. One

As both ends of the rigid rod are in contact with inner surface, the position of any one end can be determined with $\theta$ and $\phi$ coordinates with respect to coordinate system fixed at the center of the sphere. To determine position of other end we will require one more coordinate i.e. orientation of rod with respect to some fixed direction. Hence, degrees of freedom is three.

Hence, answer is (B)

3. For a system shown in figure given below, the Lagrangian function is given by : ($V=0$ at $y=0$)
   
1. $L=M\dot y^2+Mgy$
2. $L=\frac{1}{2}M\dot y^2+Mgy$
3. $L=\frac{1}{2}M\dot y^2-Mg(y-x)$
4. $L=M\dot y^2+Mg(y-x)$

$$L=T-V$$ $$T=\frac{1}{2}M\dot x^2+\frac{1}{2}M\dot y^2$$ But $\dot x=\dot y$ $$T=M\dot y^2$$ $$V=-Mgy$$ $$L=M\dot y^2+Mgy$$

Hence, answer is (A)

4. "Hamiltonian $H$ is not equal to the total energy $E$ (sum of kinetic and potential energies)", hold true for a system characterized with:
1. conservative forces and time-independent constraints
2. conservative forces and time-dependent constraints
3. dissipative forces and time-independent constraints
4. for every system irrespective of nature of forces and constraints

dissipative forces and time-independent constraints

Hence, answer is (C)

5. A particle moves under the action of force $\vec F=-\frac{1}{r^n}\hat r$. The particle moves in a closed orbit if:
1. n=-1 or n=2
2. n=1 or n=-2
3. n=-1 or n=-2
4. n=1 or n=2

According to Bertrand's theorem the central force of the form $F=\frac{-k}{r^{3-\beta^2}}$ can produce stable, closed, non-circular orbits if it satisfies the condition $\beta^2(\beta^2-4)(\beta^2-1)=0$. Hence, this condition is satisfied for $\beta=0,1,2$ i.e. $n=3-\beta^2=3,2,-1$

Hence, answer is (A)