Q45 of 48 Page 48

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?


(a) ball 1 and 3 the total mechanical energy is conserved.


For ball 1, friction is so large that starts rolling without slipping and gains rotational kinetic energy.


For ball 3, friction is negligible so there is no external force responsible for energy loss ∴ the energy is conserved.


(b) ball 1 acquires some rotational kinetic energy ∴ it will not be


able to reach C and thus will not reach point D


ball 2 loses its energy due to friction, thus it cannot reach point D


ball 3 has exactly the amount of energy to reach C and D because it does not roll and does not lose energy due to friction.


∴ only ball 3 will reach point D


(c) ball 1 and 2 starts moving in the back direction before


reaching point C.


for ball 1, it loses its energy while slipping back to point B, thus it will not reach back to point A


for ball 2, it will have clockwise rotation as it moves backs to point B due to which there will be kinetic friction coming to play due to which it will lose energy and won’t be able to reach point A.


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43

On complete combustion a litre of petrol gives off heat equivalent to 3×107 J. In a test drive a car weighing 1200 kg. including the mass of driver, runs 15 km per litre while moving with a uniform speed on a straight track. Assuming that friction offered by the road surface and air to be uniform, calculate the force of friction acting on the car during the test drive, if the efficiency of the car engine were 0.5.

 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.



 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.