mechanics test questions

Question 1
A ball is thrown vertically upwards with a velocity u from top of a tower. It strikes the ground with a velocity 3u. The time taken by the ball to reach the ground is given by
(a) u/g
(b) 2u/g
(c) 3u/g
(d) 4u/g

Question 2
A body slides down an inclined plane of inclination . The coefficient of friction down the plane varies in the direct proportion to the distance moved down the plane(=kx). The body will move down the plane with
(a) constant acceleration =gsin
(b) constant acceleration =(gsin-gcos)
(c) constant retardation =(gcos-gsin)
(d) variable acceleration that first decreases from gsin to zero and that becomes negative.

Question 3
A escalator is moving downwards with a uniform speed u. A man of mass m is running upwards on it at a uniform speed v. If the height of escalator is h, the work done by man in going up the escalator is

(a) zero

(b) mgh

(c) mghu/(v-u)

(d) mghv/(v-u)

Question 4

A particle moves in a circular orbit with a uniform angular speed. However , the plane of the circular orbit is itself rotating at a constant angular speed. We may then say that

(a) the angular velocity as well as angular acceleration of particle are both constant.

(b) neither the angular velocity nor the angular acceleration of the particle are constant.

(c) the angular velocity of the particle varies but its angular acceleration is constant.

(d) the angular velocity of the particle remains constant but the angular acceleration varies.

 

Question 5

Two objects of mass m and 4m are at rest at infinite separation. They move towards each other under mutual gravitational attraction. Then at separation r, which one of the following is true.

(a) the total energy of the system is not zero

(b) the force between them is not zero

(c) the center of mass of the system is at rest

(d) all the above are true

Answer
1. b
2. d

3. d

4. c

5. d

 

 

Problem on Bohr atom model

 

Question:
A particle of charge equal to that of electron and mass 208 times the mass of electron (-meson)moves in a circular orbit around a nucleus of charge +3e (assume mass of nucleus to be infinite). Assuming that the Bohr atom model is applicable to this system
(a) derive an expression for the radius of the nth Bohr orbit
(b) find the value of n for which the radius of the orbit is approximately the same as that of the first Bohr orbit for the hydrogen atom
(c) find the wavelength of the radiation emitted when  -meson jumps from the third orbit to the first orbit (Rydberg constant = 1.097 x 107 m-1)  

Question:

A particle of charge equal to that of electron and mass 208 times the mass of electron moves in a circular orbit around a nucleus of charge +3e (assume mass of nucleus to be infinite). Assuming that the Bohr atom model is applicable to this system

(a) derive an expression for the radius of the nth Bohr orbit

(b) find the value of n for which the radius of the orbit is approximately the same as that of the first Bohr orbit for the hydrogen atom

(c) find the wavelength of the radiation emitted when   jumps from the third orbit to the first orbit (Rydberg constant = 1.097 x 107 m-1)  

Solution

(a) Let M be the mass of the  and q be its charge. The centripetal force of the circular motion  of  is provided by the Coulomb attraction between the nucleus and  i.e..

                     (1)

where v is the speed of the in the n th Bohr orbit of radius .

                      (2)

According to the Bohr postulate, the angular momentum of is an integral multiple of , i.e.,

                                (3)

eliminating v from 2 and 3 we get

putting q=e , Q=3e , and M=208m where m is the mass of the electron then

                               (4)

(b) The radius of the first Bohr orbit is

On equating it equal to equation 4 we find

(c) The total energy of in the nth permitted orbit is

 using equation 1

Substituting the value of  from equation 4 we get

where is the Rydberg constant whose value is given in the  question.

When jumps from third orbit to the first orbit , the energy if emitted radiation is

The wavelength of the radiation is given by

calculating for wavelength we get