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

# If the length of a simple pendulum is reduced to half and mass is doubled then the time period of simple is

1. Remains unchanged
2. Halved
3. Doubled
4. Decreased by a factor of $\sqrt{2}$

# 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. ML2
2. 12ML2
3. 1/12ML2
4. 1/4ML2
5. 1/2ML2

1. 150 meters
2. 100 meters
3. 75 meters
4. 50 meters
5. 25 meters

1. R
2. 1/R
3. R2
4. R3/2

# The product of generalized coordinate and its conjugate momentum has the dimension of

1. force
2. energy
3. linear momentum
4. angular momentum

# The dimensions of generalized momentum

1. are always those of linear momentum
2. are always those of angular momentum
3. may be those of linear momentum
4. may be those of angular momentum
5. both C and D