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Q30  Consider the decay of a free neutron at rest: Show that the two-body decay of this type must necessarily give an electron of fixed energy and, therefore, cannot account for the observed continuous energy distribution in the $\beta - decay$ of a neutron or a nucleus (Fig. 6.19).

[Note: The simple result of this exercise was one among the several arguments advanced by W.
Pauli to predict the existence of a third particle in the decay products of $\beta -decay$ This
particle is known as the neutrino. We now know that it is a particle of intrinsic spin ½ (like
e—, p or n), but is neutral, and either massless or having an extremely small mass
(compared to the mass of an electron) and which interacts very weakly with matter. The
correct decay process of the neutron is :$n \rightarrow p+ e ^-$ ]

By Einstein’s mass-energy relation we can write :                                                  Here  and C are constant thus two-body decay is unable to explain (or account for) the continuous energy distribution in the β-decay of a neutron.

Q29  Which of the following potential energy curves in Fig. 6.18 cannot possibly describe the elastic collision of two billiard balls? Here r is the distance between centres of the balls.

The potential energy of the system depends inversely on the separation between the balls. Thus the potential energy will decrease as the balls will come closer and will become zero as they touch each other. Thus elastic collision is best described only by the graph (v).

Q28  A trolley of mass 200 kg moves with a uniform speed of 36 km/h on a frictionless track. A child of mass 20 kg runs on the trolley from one end to the other (10 m away) with a speed of $4 ms ^{-1}$ relative to the trolley in a direction opposite to its motion, and jumps out of the trolley. What is the final speed of the trolley ? How much has the trolley moved from the time the child begins to run ?

The initial momentum of the system (boy + trolley) is given as :                                                                                                                                            Now assume v' is the final velocity of the trolley with respect to the ground. Then the final momentum will be :                                                         Conserving momentum :  ...

Q27 A bolt of mass 0.3 kg falls from the ceiling of an elevator moving down with an uniform speed of $7 m s ^{-1}$. It hits the floor of the elevator (length of the elevator = 3 m) and does not rebound. What is the heat produced by the impact? Would your answer be different if the elevator were stationary?

In this case, the heat produced is the loss in the potential energy.  Thus,            heat produced  =  mg h or                                            or                                            The heat produced (when the lift is stationary) will remain the same as the relative velocity of the bolt with respect lift still remains zero.

Q26  A 1 kg block situated on a rough incline is connected to a spring of spring constant $100 N m^{-1}$ as shown in Fig. 6.17. The block is released from rest with the spring in the unstretched position. The block moves 10 cm down the incline before coming to rest. Find the coefficient of friction between the block and the incline. Assume that the spring has a negligible mass and the pulley is frictionless.

Displacement (x) of the block is given as :   =  0.1 m. Using equilibrium conditions we can write :                                                 and                                                                           (  is the frictional force). We can write work done in terms of potential energy as :                                            or                                     ...

Q25  Two inclined frictionless tracks, one gradual and the other steep meet at A from where two stones are allowed to slide down from rest, one on each track (Fig. 6.16).  Will the stones reach the bottom at the same time? Will they reach there with the same speed? Explain. Given $\theta _1 = 30 \degree , \theta = 60 \degree ,$ and h = 10 m, what are the speeds and times taken by the two stones?

The FBD of the track is shown in the figure below :                                                Using the law of conservation of energy we have :                                             or                                             Hence both stones will reach the bottom with the same speed. For stone 1 we can write :                                     or                               ...

Q24 A bullet of mass 0.012 kg and horizontal speed $70 ms^{-1}$ strikes a block of wood of mass 0.4 kg and instantly comes to rest with respect to the block. The block is suspended from the ceiling by means of thin wires. Calculate the height to which the block rises. Also, estimate the amount of heat produced in the block.

We are given :                       Mass of the bullet m: 0.012 Kg      Mass of the block M: 0.4  Kg   The initial velocity of the bullet u: 70  m/s  The initial velocity of the block     :     0   The final velocity of the system (bullet + block): v For finding the final speed of system we will apply the law of conservation of momentum :                                     or                 ...

Q23  (b)  A family uses 8 kW of power. Compare this area to that of the roof of a typical house.

A typical has dimensions of  . The area of the roof of the house is . This is nearly equal to the area required for the production of the given amount of electricity.

Q23 (a)   A family uses 8 kW of power. Direct solar energy is incident on the horizontal surface at an average rate of 200 W per square meter. If 20% of this energy can be converted to useful electrical energy, how large an area is needed to supply 8 kW?

It is given that the efficiency of energy conversion is 20 per cent. According to question, we can write (equating power used by family) :                                                                        (Here A is the area required.) or                                                    or                                                           Thus required area is 200 m2.

Q22 (b)  A person trying to lose weight (dieter) lifts a 10 kg mass, one thousand times, to a height of 0.5 m each time. Assume that the potential energy lost each time she lowers the mass is dissipated. Fat supplies $3.8 \times 10 ^ 2 J$ of energy per kilogram which is converted to mechanical energy with a 20% efficiency rate. How much fat will the dieter use up?

Efficiency is given to be 20 per cent. Thus energy supplied by the person :                                                      Thus the amount of fat lost is :                                                     or

Q22 (a)  A person trying to lose weight (dieter) lifts a 10 kg mass, one thousand times, to a height of 0.5 m each time. Assume that the potential energy lost each time she lowers the mass is dissipated.  How much work does she do against the gravitational force?

The work done against the gravitational force is given by :                                               =   Number of times the weight is lifted  work done in 1 time.                                               or                                           or

Q21 (c)  The blades of a windmill sweep out a circle of area A. Assume that the windmill converts 25% of the wind’s energy into electrical energy, and that$A = 30 m ^2 , v = 36 Km /h$ and the density of air is $1.2 Kg m^{-3}$ What is the electrical power produced?

It is given that 25 per cent of wind energy is converted into electrical energy. Thus electric energy produced is :                                                               or                                                 Now the electric power is given by  :                                                   or                                                                 or           ...

Q21 (b)   The blades of a windmill sweep out a circle of area A. What is the kinetic energy of the air?

The kinetic energy is given by :                                                      or                                                 or                                               Thus the kinetic energy of wind is        J.

Q21 (a) The blades of a windmill sweep out a circle of area A.  If the wind flows at a velocity v perpendicular to the circle, what is the mass of the air passing through it in time t?

The volume of wind   =    here   is the swept circle and  is the velocity.  Thus the mass of the wind is  : -      ,     is the density of the air. Hence mass of wind flowing through windmill in time t is     .

Q20  A body of mass 0.5 kg travels in a straight line with velocity $v = ax ^{3/2}$ where$a = 5 m ^{-1/2 }s ^{-1}$. What is the work done by the net force during its displacement from x = 0 to  x = 2 m ?

The relation between work done and the kinetic energy is given by :                                      Using the relation     we can write :                        Initial velocity  =   0    (at x  =  0 ) And          the final velocity    =      (at  x  = 2). Thus work done is :                                       or                                                  or                     ...

Q19  A trolley of mass 300 kg carrying a sandbag of 25 kg is moving uniformly with a speed of 27 km/h on a frictionless track. After a while, sand starts leaking out of a hole on the floor of the trolley at the rate of $0.005 Kg s^{-1}$. What is the speed of the trolley after the entire sandbag is empty?

Since the sand is falling in the trolley thus the force generated on the system (trolley and sandbag) is an internal force. There is no external force thus momentum of the system doesn't change. Hence speed remains the same i.e.,  27 Km/hr.

Q18  The bob of a pendulum is released from a horizontal position. If the length of the pendulum is 1.5 m, what is the speed with which the bob arrives at the lowermost point, given that it dissipated 5% of its initial energy against air resistance?

Consider the extreme position (horizintal) :-                        The kinetic energy at this position is zero as velocity is zero.                       Thus total energy is given by :     Now consider the mean position (lowermost point) :  Here the potential energy of bob is zero.                Whereas   kinetic energy is  :                                                                 ...

Q15  A pump on the ground floor of a building can pump up water to fill a tank of volume 30 min 15 min. If the tank is 40 m above the ground, and the efficiency of the pump is 30%, how much electric power is consumed by the pump ?

Mass of the water is :                                             or                                                 Thus the output power is given by :                                              or                                                          or                                                          or                                                          Also, we are given...

Q17  The bob A of a pendulum released from 30o to the vertical hits another bob B of the same mass at rest
on a table as shown in Fig. 6.15. How high does the bob A rise after the collision? Neglect the size of
the bobs and assume the collision to be elastic.

This is an elastic collision thus the transfer of momentum will take place. It is given that bob B is at rest and bob A has some velocity. So in momentum transfer, bob B will gain the velocity in the left direction whereas bob A will come to rest (complete momentum transfer takes place). Hence bob A will not rise.

Q16  Two identical ball bearings in contact with each other and resting on a frictionless table are hit head-on by another ball bearing of the same mass moving initially with a speed V. If the collision is elastic, which of the following (Fig. 6.14) is a possible result after collision?

The initial kinetic energy of the system is given by :                                                        or                                                    Case (i):-      The final kinetic energy is :                                                            Thus the kinetic energy is not conserved in this case. Case (ii):-      The final kinetic energy is :                          ...
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