# A particle of mass M moves in a plane under the influence of a force F = – kx directed towards the origin. What is the Lagrangian of the system in polar coordinate (r, θ)

1. $\mathbf{M\left&space;(&space;\frac{dr}{dt}&space;\right&space;)^{2}+&space;\frac{1}{2}kr^{2}}$
2. $\mathbf{M\left&space;(&space;\frac{dr}{dt}&space;\right&space;)^{2}&space;-&space;\frac{1}{2}kr^{2}}$
3. $\mathbf{\frac{1}{2}M\left&space;[&space;\left&space;(&space;\frac{dr}{dt}&space;\right&space;)^{2}&space;+&space;r^{2}\left&space;(&space;\frac{d\theta&space;}{dt}&space;\right&space;)^{2}&space;\right&space;]&space;+&space;\frac{1}{2}kr^{2}}$
4. $\mathbf{\frac{1}{2}M\left&space;[&space;\left&space;(&space;\frac{dr}{dt}&space;\right&space;)^{2}&space;+&space;r^{2}\left&space;(&space;\frac{d\theta&space;}{dt}&space;\right&space;)^{2}&space;\right&space;]&space;-&space;\frac{1}{2}kr^{2}}$

# The center of mass of velocity is defined by

1. $\mathbf{V_{cm}&space;=&space;\sum&space;m_{i}&space;v_{i}}$
2. $\mathbf{V_{cm}&space;=&space;\sum&space;\frac{\mathrm{d}&space;r_{i}}{\mathrm{d}&space;t}}$
3. $\mathbf{V_{cm}&space;=&space;m_{i}\sum&space;\frac{\mathrm{d}&space;r_{i}}{\mathrm{d}&space;t}}$
4. $\mathbf{V_{cm}&space;=&space;\frac{1}{M}\sum&space;m\frac{dr_{i}}{dt}&space;}$

# A bob of mass m is moving in XY-plane as a simple pendulum. Its Lagrangian function is:

1. $\mathbf{\frac{m}{2}(\dot{\theta&space;}L)&space;-&space;mgL&space;\\sin&space;\theta}$
2. $\mathbf{\frac{m}{2}(x)&space;-&space;mgy}$
3. $\mathbf{\frac{m}{2}(\dot{\theta&space;}L)^{2}&space;-&space;mgL(1&space;-&space;cos&space;\theta)}$
4. $\mathbf{\frac{m}{2}(\dot{\theta&space;}L)^{2}&space;-&space;mgL(1&space;-&space;\\sin&space;\theta)}$

1. 150.5
2. 13.6
3. 10.2
4. 1.227

# The nature of bonding in diamonds is

1. ionic
2. covalent
3. van der Waals
4. metallic

1. 0.866 m
2. 1.866 m
3. 2.866 m
4. 3.866 m

1. 10-2 Å
2. 1 Å
3. 10 Å
4. 102 Å

# The energy gap in a semiconductor is

1. Equal to insulator
2. Less than an insulator
3. More than an insulator
4. None of these

# A lens of focal length 20 cm has a power of

1. 0.20 diopters
2. 5 diopters
3. 10 diopters
4. 50 diopters