1. 23. Consider a system of three charges q/3, q/3 and –2q/3 placed at points A, B and C, respectively, as shown in the figure. Take O to be the centre of the circle of radius R and angle CAB = 60°. Figure :   The electric field at point O is q 8πε0 R2 directed along the negative x-axis   The potential energy of the system is zero   The magnitude of the force between the charges at C and B is q2 54πε0 R2   The potential at point O is q 12πε0 R2   Hint Half-n-half Clue   2. 24. A radio active sample S1 having an activity of 5µCi has twice the number of nuclei as another sample S2 which has an activity of 10µCi. The half lives of S1 and S2 can b   5 years each   20 years and 10 years, respectively   20 years and 5 years, respectively   10 years each   Half-n-half Clue   3. 25. A transverse sinusoidal wave moves along a string in the positive x –direction at a speed of 10 cm/s. The wavelength of the wave is 0.5 m and its amplitude is 10 cm. At a particular time t, the snap-shot of the wave is shown in figure. The velocity of point P when its displacement is 5 cm is- Figure :   − √3π 50 i ^ m/s   − √3π 50 j ^ m/s   √3π 50 j^ m/s   √3π 50 i^ m/s   Half-n-half Clue   4. 26. A block (B) is attached to two unstretched spring S1 and S2 with spring constants k and 4k, respectively (see figure I). The other ends are attached to identical supports M1 and M2 not attached to the walls. The springs and supports have negligible mass. There is no friction anywhere. The block B is displaced towards wall 1 by a small distance x (figure II) and released. The block returns and moves a maximum distance y towards wall 2. Displacements x and y are measured with respect to the equilibrium position of the block B is y/x Figure :   4   2   1 2   1 4   Half-n-half Clue   5. 27. A bob of mass M is suspended by a massless string of length L. The horizontal velocity V at position A is just sufficient to make it reach the point B. The angle θ at which the speed of the bob is half of that at A, satisfies Figure :   θ = π 4   π 4 < θ < π 2   π 2 < θ < 3π 4   3π 4 < θ < π   Half-n-half Clue   6. 28. A glass tube of uniform internal radius (r) has a valve separating the two identical ends. Initially, the valve is in a tightly closed position. End 1 has hemispherical soap bubble of radius r. End 2 has sub-hemispherical soap-bubble as shown in figure. Just after opening the valve, Figure :   air from end 1 flows towards end 2. No change in the volume of the soap bubbles   air from end 1 flows towards end 2. Volume of the soap bubble at end 1 decreases   no change occurs   air from end 2 flows towards end 1. Volume of the soap bubble at end 1 increases   Half-n-half Clue   7. 29. A vibrating string of certain length l under a tension T resonates with a mode corresponding to the first overtone (third harmonic) of an air column of length 75 cm inside a tube closed at one end. The string also generates 4 beats per second when excited along with a tuning fork of frequency n. Now when the tension of the string is slightly increased the number of beats reduces to 2 per second. Assuming the velocity of sound in air to be 340 m/s, the frequency n of the tuning fork in Hz is   344   336   117.3   109.3   Half-n-half Clue