Q48 of 48 Page 48

A balloon filled with helium rises against gravity increasing its potential energy. The speed of the balloon also increases as it rises. How do you reconcile this with the law of conservation of mechanical energy? You can neglect viscous drag of air and assume that density of air is constant.

According to Archimedes principle when a body is partially or fully dipped into a fluid at rest, the fluid exerts an upward force of buoyancy equal to the weight of the displaced fluid.


The balloon similarly experiences a buoyant force which provides it an acceleration .


If


are the mass , density and volume of the balloon andis the density of


.


Then,




Now writing acceleration a, in its differential form,




Integrating both the sides, we obtain,



Now for the height by which it raises till time t, we use equation of motion



Putting the value of a and initial velocity to be zero,



now the kinetic energy of the balloon at time t,


K.E


Rearranging the terms, we get,





But are terms which represent potential energy of air and helium balloon respectively.


∴


So, we can say the balloon raises by pushing the air downward, thus its potential energy increases as the air’s decreases.


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44

A block of mass 1 kg is pushed up a surface inclined to horizontal at an angle of 30° by a force of 10 N parallel to the inclined surface (Fig. 6.15). The coefficient of friction between block and the incline is 0.1. If the block is pushed up by 10 m along the incline, calculate

(a) work done against gravity


(b) work done against force of friction


(c) increase in potential energy


(d) increase in kinetic energy


(e) work done by applied force.



 45

A curved surface is shown in Fig. 6.16. The portion BCD is free of friction. There are three spherical balls of identical radii and masses. Balls are released from rest one by one from A which is at a slightly greater height than C.


With the surface AB, ball 1 has large enough friction to cause rolling down without slipping; ball 2 has a small friction and ball 3 has a negligible friction.


(a) For which balls is total mechanical energy conserved?


(b) Which ball(s) can reach D?


(c) For balls which do not reach D, which of the balls can reach back A?


 46

A rocket accelerates straight up by ejecting gas downwards. In a small time interval ∆t, it ejects a gas of mass ∆m at a relative speed u. Calculate KE of the entire system at t + ∆t and t and show that the device that ejects gas does work = ∆m u2 in this time interval (neglect gravity).

 47

Two identical steel cubes (masses 50g, side 1cm) collide head-on face to face with a speed of 10cm/s each. Find the maximum compression of each. Young’s modulus for steel = Y= 2 × 1011 N/m2.