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KNOW IIT-JEEBy Previous Exam QuestionsPHYSICS1. A spherical ball of mass m is kept at the highest pointin the space between two fixed, concentric spheres Aand B (see figure in solution). The smaller sphere Ahas a radius R and the space between the two sphereshas a width d. The ball has a diameter very slightly lessthan d. All surface are frictionless. The ball is given agentle push (towards the right see figure in solution)The angle made by the radius vector of the ball withthe upward vertical is denoted by θ [IIT-2002](a) Express the total normal reaction force exerted bythe sphere on the ball as a function of angle θ.(b) Let N A and N B denote the magnitudes of thenormal reaction forces on the ball exerted by thesphere A and B, respectively. Sketch the variations ofN A and N B as functions of cos θ in the range 0 ≤ θ ≤ πby drawing two separate graphs in your answer book,taking cos θ on the horizontal axes.Sol. The ball is moving in a circular motion. Thenecessary centripetal force is provided by(mg cos θ – N). ThereforeBARDCd/2θ θmgcosθmgN AVmgsinθmv 2mg cos θ – N A =…(i)⎛ d ⎞⎜R + ⎟⎝ 2 ⎠According to energy conservation1 mv 2 ⎛ d ⎞= mg ⎜R + ⎟ (1 – cos θ) …(ii)2 ⎝ 2 ⎠From (i) and (ii) N A = mg (3 cos θ – 2) …(iii)The above equation shows that as θ increases N Adecreases. At a particular value of θ, N A will becomezero and the ball will lose contact with sphere A. Thiscondition can be found by putting N A = 0 in eq. (iii)0 = mg (3 cos θ – 2)∴ θ = cos –1 ⎛ 2 ⎞⎜ ⎟⎝ 3 ⎠2mgN AmgThe graph between N A and cos θFrom eq (iii) when θ = 0, N A = mg.1cos θ –1When θ = cos –1 ⎛ 2 ⎞⎜ ⎟ ; NA = 0⎝ 3 ⎠The graph is a straight line as shown.When θ > cos –1 ⎛ 2 ⎞⎜ ⎟⎝ 3 ⎠mv 2N B – (– mg cos θ) =dR +2mv⇒ N B + mg cos θ =2⎛ d ⎞⎜R + ⎟⎝ 2 ⎠Using energy conservation1 2 ⎡⎛d ⎞ ⎛ d ⎞ ⎤mv = mg ⎢⎜R+ ⎟ − ⎜R+ ⎟ cos θ⎥ 2 ⎣⎝2 ⎠ ⎝ 2 ⎠ ⎦N B5mg2mgcos θ…(iv)mv 2 = 2 mg [1 – cos θ] …(v)⎛ d ⎞⎜R + ⎟⎝ 2 ⎠From (iv) and (v) we getN B + mg cos θ = 2 mg – 2mg cos θN B – mg (2 – 3 cos θ)When cos θ = 32 , NB = 0When cos θ = – 1, N B = 5 mgTherefore the graph is as shown.2. A cylindrical block of length 0.4 m and area of crosssection 0.04 m 2 is placed coaxially on a a thin metaldisc of mass 0.4 Kg and the same cross section. Theupper face of the cylinder is maintained at a constanttemperature of 400 K and the initial temperature ofthe disc is 300 K. If the thermal conductivity of thematerical of all cylinder is 10 watt/kg. K, how longwill it take for the temperature of the disc to increasesto 350 K? Assume, for purpose of calculation, thethermal conductivity of the disc to be very high andXtraEdge for IIT-JEE 7 APRIL 2010