2 Bon us question of Mathematics

Q.l

Q.2

If L = Lim x - > 0 /n(l + x) / n ( x + il + x 2 )

then find the value of L + 153

Two universities A and B write questions and their corresponding solutions for a high school mathematics tournament. University A writes 10 questions every hour but makes a mistake in their solutions 10% of the time. The university B writes 20 questions every hour and makes a mistake 20% of the time. Each university works for 10 hours and then sends all problems to a Miss 'C' for checking. However only 75% of the problems which she thinks are wrong are actually incorrect. Further she thinks that 20% of the questions from the university A have incorrect solutions, and that 10% of the questions from the university B have incorrect solutions. If the probability that a problem definitely written and solved correctly, randomly chosen by her,

Q.l

(a) (b)

Q.2

(a)

(b)

Q.3

(a) (b) Q.4

Q.5

1 UF 2Q —I i—VvW-10 V

was thought of as having incorrectly solved, is where p and q coprimes, then find the value of (p + q)-

PHYSICS QUESTION In the circuit shown, the switch S is in position-1 since a long time. At a certain moment t = 0, it is shifted to position-2. The 1 pF capacitor is initially uncharged. Find the current that flows through the 2 Q resistor as a function of time't' for t > 0. What percentage of the work done by the 10 V cell is lost as heat from the 2Q resistor, from t = 0 till infinity?

,20 V 2 (iF

• W r 1 D

A beam consisting of two wavelengths 8100 A and 4500 A is used to obtain interference fringes in a Young's double slit experiment. The distance between the slits is 2 mm and that between the plane of the slits and the screen is 100 cm. Find the least distance in millimeters from the central maxima on the screen where the bright fringes due to both the wavelengths coincide. Find the least distance in millimeters from the central maxima on the screen where the dark fringes due to bothrthe wavelengths coincide: A cylinder contains a tight fitting piston of mass 2 kg and cross-sectional area 10 cm 2. Under the piston, there is 1 mole of a diatomic gas at 300 K initially. The walls of the cylinder are heat insulating and the piston is also thermally insulating. By means of an electrical heater, the gas is slowly given a heat = 1000 Joules. The upper end of cylinder is open to the atmosphere having atmospheric pressure = 105 Pascals. Neglect any frictional loss. By what distance does the piston shift up? What is the final temperature of the gas? A solid sphere with a hollow cavity (of radius R/2) having net mass m and radius R is resting in. equilibrium on a rough horizontal floor, as shown. The sphere is tilted slightly and released. Find the time period of subsequent oscillations assuming that the sphere's surface does not slip over the floor. wnunWrWfuuuuu Two monochromatic and coherent point sources oflight, S, and S 0 of wavelength 4000 A, are placed at a distance 4 mm from each other. The line joining the two sources is perpendicular to a screen. The distance of the mid-point of S,S 7 from the screen is D = A/2 m. Find the radius (non-zero) ofthe smallest bright ring on the screen, using valid assumptions.

:{H=

Bansal Classes PHYSICS [2]

Q.6 A glass sphere of radius R has a point isotropic source of monochromatic light of wavelength X. The thickness of the glass wall is't' ( « R). The inner surface of the sphere is painted black so that it absorbs all the radiation incident on it. Find the maximum power of the source such that the sphere does not rupture due to the radiation pressure. Rupture stress of glass = a.

Q.7 In the figure shown, the sonic source of frequency 200 Hz is moving with a speed = 10 m/s. Find the beat frequency as heard by the listener L, who is s

himself moving with speed = 5 m/s. The reflecting wall is moving with a speed = 15 m/s. A wind is also blowing to the right with a speed = 5 m/s. Speed of sound in still air = 340 m/s. wall

Q.8 A sphere of mass'm' collides elastically with another stationary sphere of mass 'm/2' obliquely. Both the spheres are smooth and there are no external forces acting on them. Solve the equations of collision and find the maximum angle through which the sphere of mass'm' can be deflected w.r.t. its original direction.

- i •: vacuum 1 mole •

of -mbWMmu-diatomic

gas §

31°( 4 m 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7

0 7 7 7 7

Q.9 A thermally insulated cylinder is divided into two parts by a heat insulating tight piston, which can move freely in the cylinder without friction. The left part of the cylinder contains one mole of an ideal diatomic gas and the right part is evacuated. The piston is connected to the right wall of the cylinder through a spring whose natural length is equal to the total length of the cylinder. The electrical heater is switched on for some time so that the gas temperature increases and the piston shifts slowly to the right. What percentage of the heat supplied by the heater goes in compressing the spring? Neglect the heat capacity of the piston or the cylinder.

Q. 10 A ball is thrown from a point O with some speed v 0 at an angle of 37° with the horizontal, such that the ball bounces from the vertical wall and returns to O. For the bounce, the coefficient of restitution is 5/8. What must be the value of v 0? g = 10 m/s2.

Q.ll A spherical body of mass M and radius R has a spherical cavity of radius R/2 inside it, as shown. The center of the cavity O is displaced from the geometric center of the sphere C by a distance R/2. A tiny body of mass m ( « M) is placed at a distance 2R from the geometric center of the first body.

(a) Find the force of gravitational attraction on the tiny body. (b) If the tiny body is released from rest, with what velocity will it hit the surface of the spherical

body? Q.12 The circuit shown is fed by an a.c. source having emf = (15 V) sin

200t, where time t is in seconds. Coil-1 has a resistance = 3 fl and inductance 20 mH, while coil-2 has a resistance = 6 0 and inductance 40 mH. Find the voltages across the two coils, V, and V 2 , as functions of time, t.

Q.13 A certain radionuclide is getting formed in some reactor at a constant rate = q (number per second). It undergoes alpha decay with half life T. At the moment t = 0, there are (4qT//n 2) number of radionuclide in the reactor.

(a) Find the number of radionuclide 'N' in the reactor at any later time t > 0 and plot a graph of N versus t.

(b) Find the number of alpha particles emitted till t = 2T.

—T'̂ r—nrew^— coil-l coil-2

^Bansal Classes PHYSICS [426]

Q. 14 In a modified Young's double slit experiment, there are three identical parallel slits S,, S 2 and S 3 . A coherent monochromatic beam of wavelength 700 nm, having plane wavefronts, falls on the slits, as shown. The intensity of the central point O on the screen is found to be 7 x i (H W/m 2. The distance SjS 2 = S 2 S 3 = 0.7 mm.

(a) Find the intensity on the screen at O if S, and S 3 are covered. {b) Find the intensity on the screen at 0 if only S 3 is covered. (c) All three slits are now uncovered and a transparent plate of thickness 1.4 pm and refractive index

1.25 is placed in front of S 2 . Find the intensity at point O. Q. 15 A jeep is moving at a certain moment with velocity = 10 m/s. The acceleration of the jeep is 'a'. A

man sitting in the jeep throws a ball with initial velocity = 20 m/s, at an angle of 53° with the horizontal, both w.r.t. himself. The motion of the jeep is in the same direction and vertical plane as the motion of the ball. Given: sin 53° = 4/5, cos 53° = 3/5. Neglect air resistance.

(a) Find the actual initial speed of the ball relative to an earth observer. (b) What should be the acceleration 'a' of the jeep so that the man is able to catch the ball? (c) What is the farthest distance ofthe ball from the man, as perceived by him, in part (B)? Q.16 Two blocks, 1 & 2, of masses m and 4m, interconnected by a massless spring of spring constant k, and are

resting on a frictionless horizontal floor. Forces F and 2F start acting on the blocks, at t = 0, as shown. (a) Write the earth frame work-energy theorem for the system, in terms F

\uMuuuu\uuu\um 2F.

B

of speeds v, and v 2 , and displacements x, and x 2 of the two blocks. \utMu\\uu\u\u\v,ffl\mrv» (b) Find the maximum elongation ofthe spring during the motion of the two blocks, if F = 5mg. (c) Find the maximum speed of block-1 in the center of mass frame, if F = 5mg, Q.17 A uniform and thin rod AB of mass 5m and length L is kept stationary on a frictionless horizontal

surface. At a certain moment, a tiny ball of mass m, moving with a horizontal velocity = v Q collides inelastically with the rod, at a point whose distance from end A of the rod is z. The direction of v 0

is perpendicular to the rod, as shown. The coefficient of restitution for collision is 3/4. Just after the collision, let v, = velocity of the center of rod (rightwards), v ( = velocity of the ball, assumed leftwards and co = angular velocity of the rod.

(a) Write the condition for coefficient of restitution = 3/4 in terms of relevant parameters

(b) It is found that the velocity of B just after the collision is zero. Find z. (c) Assuming the condition of part (B), calculate the percentage of

energy lost during the collision. Q.18 A gaseous mixture initially at 300 K and 2 x 105 N/m 2 pressure contains 6 g of hydrogen and 8 gm

of Helium. The m ixture is expanded to four times its ori ginal volume, through an isobaric heating process. Then, it is isochorically cooled until its temperature again becomes 300 K. After that, the gas mixture is isothermally compressed to its original volume.

(a) Find the ratio of molar specific heats = y ofthe mixture. (b) Plot the process in P-V and P-T indicator diagrams, showing all values of P & T. (c) Find the efficiency of the entire cycle (take in 2 = 0.7) Q.19 Two radio stations broadcast their programmes at the same amplitude A, but at slightly different

frequencies ro_ and co2, where o)3 - co2 = 1000 Hz. A detector receives the signals from the two stations simultaneously. It can only detect signals of intensity > 2A 2,

(a) Find the time interval between the successive maxima of the intensity of the signal received by the detector. (b) Find the time for which the detector remains idle in each cycle of the intensity of the signal, Q.20 A long wire PQR is made by joining two wires PQ and QR of equal radii. Their lengths and

masses are respectively: 4.8 m and 0.06 kg; 2.56 m & 0.2 kg. The tension is 80 N. A sinusoidal wave pulse of amplitude 3.5 cm is sent along the wire PQ from the end P. No power is dissipated during the propagation ofthe wave pulse. Calculate the time taken by the pulse to reach the end R and the amplitude of reflected and transmitted wave pulses at Q.

^Bansal Classes PHYSICS [4]

Q.21

Q.22

Q.23

(a) (b) (c) Q.24

(a) '(b)

Q.25

Q.26

Q.27

Q.28

In the circuit shown, the potentiometer wire AB has a length = 100 cm and total resistance 10 Q. What should be the distance of the jockey from point A so that the reading of the ammeter is 0,5 A? The coil resistance of the ammeter is 1 Q. The cell at the top has an emf = 15 Volts and internal resistance 1 O. 0 5 Q 1—\AVv A soap bubble of radius r is blown at the end of a capillary of length / and of internal radius R. Surface tension of soap solution is T and coefficient of viscosity of air is r\. The volume of air flowing per second through the capillary is given by , where P is the excess pressure on

8rj/ soap bubble. Find the lifetime of the soap bubble. Two small balls A and B are interconnected by an inextensible string of length L. Mass of ball A= m, mass of ball B = 2m. The balls are resting on a frictionless horizontal surface, with the distance between them = 3L/5. In this position, ball A is suddenly given a horizontal velocity v 0, perpendicular to the line joining the two balls. Find the speed of ball B just after the string becomes taut. Find the impulse of the tension in string when the string becomes taut Find the steady tension in string much after the string has become taut. A wooden log of mass m with a cross-section shaped like an equilateral right-angled triangle can slide on a horizontal surface without friction. Two point-like bodies of masses m and 2m, tied to each other using a thread, are placed onto the log as shown in the figure. The length of the base of the log is L-54 cm. Friction and the masses of the thread and the pulley are negligible The bodies are released at a certain moment. What distance does the wooden log cover until the body of mass 2m reaches its bottom? Determine the speed ofthe bodies and that ofthe wooden log when the body of mass 2m reaches the bottom of the log. In a tennis racket, the c.m. is 12 inches from the end ofthe handle. The radius of gyration about an axis through the c.m. as shown in the figure is 8 inches. If the tennis ball is hit at a distance of 20 inches from the end ofthe handle, where should the player hold his racket so as not to feel any translational force when hitting the ball? We have two liquids of different densities. A force of 1.36 N can hold the same piece of metal in one of them, and of 0.82 N in the other. In what volume proportion should they be mixed so that the holding force is exactly 1 N? A cart on an inclined plane of angle 9 - 30° is balanced as shown by a weight of mass 10 kg. The cord Ais wound on a drum of diameter d, d. = 3d,

2 rn

V

.j, which is on the same shaft as a drum of diameter a, J U l , on which is wound cord B. What is the mass M of the cart?

Through the Looking Glass: A narrow beam oflight has entered a large thin lass plate. Each refraction is accompanied by reflection of k = 30% of the beam's energy. What fraction ofthe light energy is transmitted through the plate 9

^ Bansal Classes PHYSICS [428]

20 cm

Q.29 Lake Placid: A radio receiver is set up on a mast in the middle of a calm lake to track the radio signal from a satellite orbiting the Earth. As the satellite rises above the horizon, the intensity of the signal varies periodically, the intensity is at a maximum when the satellite is 8j= 3° above the horizon and then again at 9 2 = 6° above the horizon. What is the wavelength X of the satellite signal? The receiver is h = 4.0 m above the lake surface.

Q.30 In the figure, water of density 1000 kg/m 3 flows through the pipe. The cross-section area at stations 1, 2 and 3 are 1 cm 2, 2 cm 2 and A cm 2, respectively. The thin vertical tubes that are connected to the pipe at these stations have water levels as indicated. Find the mass flow rate of water through the pipe and v 3 . [Take g = 10 m/s 2]

Q.31 A metal ring having three metallic spokes of lengths r=0.2 m is in a vertical plane and can spin around a fixed horizontal axis in a homogeneous magnetic field of a magnetic induction of B=0.5 T. The lines of magnetic field are perpendicular to the plane of the metal ring. Between the axis of the metal ring and its perimeter we connect a consumer of a resistance of 0.15 with the help of two sliding contacts. We fix a thread of negligible mass to the rim of the ring and wind it several times around the ring and to its end we fix a body of a mass of 20 g. At a given moment we release the body of mass m. The friction is negligible everywhere, the resistance of the ring, the spokes and the connected wiring is also negligible.

(a) What is the torque exerted on the ring with the spokes by the magnetic forces when the body of m is moving with a constant velocity?

(b) What current is flowing through the consumer when the velocity of the body of mass m is 3 m/s?

(c) What is the highest velocity of the body of mass m?

\\\\\\\ www

© / B

\ / s® ®

JL •> ® ® ® ® / s

Q.32

Q.33

Q.34

Q.35

Q.36

Figure shows a hypothetical speed distribution for particles of a certain gas: P (v) = Cv 2 for 0 < v < v Q and P(v) = 0 for v > v„. (a) Show that C = 3 /vJ , dN/N Find (b) the average speed of the particles, and (c) their rms speed. dv =P(v)

A neutron moving with a kinetic energy = 65 eV collides head-on and inelastically with a singly ionized helium atom at rest (in its ground state). Take the ionization energy of hydrogen atom =13.6 eV, Also, mass of Helium atom is four times that of a neutron. If the helium ion gets de-excited subsequently by emitting radiation, calculate the possible energies of the emitted photon(s) in eV. A board of mass m is placed on a frictionless inclined plane that makes an angle 0 = 37° with the horizontal. A block of same mass is placed on the board and is given a quick push up the board with initial velocity v = 8 m/s. Find the distane d covered by the block by the time its velocity drops to v/2. The board does not move relative to the plane. A 20 mH inductor is connected in series with a charged capacitor of capacitance 2 pF, having initial charge = 10 mC. After how much minimum time will the energy in the capacitor become half of its initial value? Leave answer in terms of n. A uniform and slender rod of mass 2m and length L is lying on a frictionless horizontal surface. Two insects, of mass m each, moving horizontally with velocities v and 2v hit the rod simultaneously and symmetrically and stick to it.

V3-G3-

^ Bansal Classes PHYSICS [6]

Q.37

(a) (b) Q.38

(a) (b)

(c)

Q.41

4 m/s lm

Hvmummm" kg

1kg B rnrr

Q.39

Q.40

Q.42

The initial velocities of the insects are perpendicular to the rod, as shown. The distance of each insects's hit-point from the center of the rod is L/6. Just after hitting the rod, each insect starts walking along the rod, away from its center, with constant speed = v relative to the rod. As the rod rotates and moves, the insects finally reach the ends. Find the total angle rotated by the rod till this moment in radians. A thin uniform circular disc of radius R and mass m is hinged about its center point O, so that it is free to rotate about a fixed horizontal axis through O. The plane of the disc is vertical. A small body A, of mass m/2, is fixed at the rim of the disc, as shown. Initially, the line OA makes an angle of 60° with the vertical. The disc is now released from rest, Find the acceleration of point A just after release. Find the magnitude of horizontal and vertical reaction forces: F h o r and F v

on the hinge, just after the disc is released. In the figure shown, the spring constant is I6n 2 N/m and its right end is fixed to a vertical wall. The floor is smooth. A block of mass 1 kg is initially at a distance of 1 m from the other 1 kg block. The left block, touching a vertical wall, is imparted a velocity = 4 m/s towards the other block. All collisions are elastic. Find the time period of this oscillatory system. A ring of radius r = 1 m is placed on the top of an inclined plane and released from rest. The inclined plane makes an angle of 30° with the horizontal. The coefficient of friction between the ring and the incline is 0.2. Find the distance travelled by the centre of the ring by the time it completes one revolution, as it rolls down the incline. In the figure shown, a constant horizontal force F = mg/2 starts acting on the block of mass m, from the position shown. The spring is undeformed in the position shown and has a narual length L, while the blocks are initially stationary. The spring constant is unknown. The surface is frictionless. The mass of the hanging block is m/4, while the pulley is massless and frictionless. Find the initial acceleration of the block of mass. Write the work-energy equation for the system consisting of the two blocks, and the spring, for any general value of 9 = angle which the spring makes with the vertical. The maximum displacement of the bigger block is found to be LVJ . Based on this information, find the spring constant. A lift is moving up with a constant retardation = 2 m/s 2. When its upward velocity is 5 m/s, a boy in the lift tosses a coin, imparting it an upward initial velocity = 3 m/s, with respect to himself His fingers at the moment of toss are midway between the floor and ceiling, whose total height is 2.0 m. After how much time will the coin hit the floor or roof of the lift? Also find the distance travelled by the coin and its displacement in the earth frame till then. [Take g = 10 m/s 2] At a distance of 20 m from a point isotropic source of sound, the loudness level is 30 dB. Neglecting damping of sound, find the loudness level at a distance of 10 m from the source and the distance where the sound is not audible by humans.

[ \ w m m m s \ m

Q.43 In the figure shown, find the relative speed of approach/ separation of the two final images formed after the light rays pass through the lens on the far right, at the moment when u = 30 cm. The speed of object = 4 cm/s. The two lens halves are placed symmetrically w.r.t the moving object.

f=40cm

otfcct, infer 1--

V

f=60cm

40 cm

^ Bansal Classes PHYSICS [430]

Q . 4 4 H E A T C A P A C I T Y D E T E R M I N A T I O N O F A L I Q U I D U S I N G C A L O R I M E T E R :

Figure shows the Regnault's appratus to determine the specific heat capacity of a unknown liquid. A solid sphere of known specific heat capacity s, having mass m, and initial temperature 0,, is mixed with the unknown liquid filled in a calorimeter. Let masses of liquid and calorimeter are m 2 and m 3 respectively, specific heat capacities are s 2 and s 3 and initially they were at room temperature 0 2 . When the hot sphere is dropped in it, the sphere looses heat and the liquid calorimeter system takes heat. This process continues till the temperature of all the elements becomes same (say 0). Heat lost by hot sphere = mjS, (Qj—0) Heat taken by liquid & calorimeter = m 2 s 2 (0-0 2 ) + m 3 s 3 (0-0 2 ) If there were no external heat loss Heat given by sphere = Heat taken by liquid - calorimeter system

m,Sj (0,-0) = m 2 s 9 (0-0 2 ) + m 3 s 3 (0-0,) mjS j (0 j -0 ) m 3 s 3

m2(0-02) m2 Get s 2 =

steam Chamber

" 0 "

steam

Disk D -Water

(a)

(b)

Calorimeter By measuring the final (steady state) temperature of the mixture, we can estimate s 2 : specific heat capacity of the unknown liquid. To give initial temperature (0,) to the sphere, we keep it in steam chamber ("O"), hanged by thread. Within some time (say 15 min) it achieves a constant temperature 0,. Now the calorimeter, filled with water (part C) is taken below the steam chamber, the wooden removable disc D is removed, and the thread is cut. The sphere drops in the water calorimeter system and the mixing starts. If specific heat capacity of liquid (s 2) were known and that of the solid ball (s^ is unknow then

( m 7 s 2 + m 3 s 3 ) ( 0 - 0 2 ) we can find s, = — — — — 1 "1 , (0 , -9 )

In the exp. of finding specific heat capacity of an unknow sphere (s 2) mass of the sphere and calorimeter are 1000 gm and 200 gm respectively and specific heat capacity of calorimeter is equal to 1/2 cal/gm/°C. The mass of liquid (water) used is 900 gm. Initially both the water and the calorimeter were at room temperature 20.0°C while the sphere was at temperature 80.0°C initially. If the steady state temperature was found to be 40.0°C. estimate specific heat capacity of the unknow sphere (s 2). (Use s w a t e r = 1 cal/g/°C) Also find the maximum permissible error in specific heat capacity of tinkown solid. What should be final temperature so that the error in s should ne minimum?

^Bansal Classes PHYSICS [431]

Q.45 END CORRECTIONS IN METER BRIDGE In meter bridge circuit, some extra length of wire called end corrections should be included at ends for accurate result. Suppose null point is obtained at /;, then

Qi L _ l i + a

Q 2 100-ZJ+p When known resistances are interchanged then balancing length is at l 2 .

R 2 L 2 + A

T i = i o o - / 2 + p The end corrections calculated from above readings are used to modify observation If 100 fi & 200 D values of known resistance is used to give null deflection at / , = 33.0 cm & on interchanging the known resistances the null deflection is found at 67.0 cm. Find the value of end correction. INDEX ERROR IN OPTICAL BENCH In u-v method the distance between object or image from the pole of mirror or les is required. Practically the position of holder when read from scale do not exactly give object or image distance. This mismatch is constant for every observation. To determine index error a needle (usually usedfor knitting) of known length is placed horizontally between the pole & object needle. The length of knitting needle gives actual object distance while the separation between holder index is read from the scale. Which becomes observed distance so index error (or excess reading) is e = Observed distance - Actual Distance For index correction the e is subtracted from observed reading to get correct reading. When a knitting needle of length 20.0 cm is adjusted between pole and object needle, the separation between the indices of object needle and mirror was observed to be 20.2 cm. Find the index correction for u. When the same knitting needle is adjusting between the pole and the image needle, the separation between the indices of image needle and mirror was found to be 19.9 cm. Find the index error for v. In some observation, the observed object distance (Separation between indices of object needle and mirror) is 30.2 cm, and the observed image distance is 19.9 cm. Using index correction from previous two equations, estimate the focal length of the concave-mirror. A conducting sphere of radius a is surrounded by another spherical thin conducting shell of radius b The space between them is filled with dielectric material of conductivity a and dielectric constant k. The charge Q. and Q 2 are given to the inner and outer shell at time t = 0. Find charge on outer shell at time t.

Q.48 The amplitude of the electric field in an electromagnetic wave of frequency © = 2.0 x 101 6 s~x

changes with times as E(t) = k (1 + cos Ht), where k is a constant and fi= 1.8 x 101 5 s~'. Would such a wave cause ionization of hydrogen atoms? If yes, what is the energy of the ejected electrons E e? Assume that atoms absorb light as photons. The ionization energy of hydrogen gas is E = 13.6 eV. the Planck constant h - 1.05 J * s.

Q.49 An air-filled parallel-plate capacitor with the plate area A is connected to a battery with an emf E and small internal resistance. One of the plates vibrates so that the distance between the plates varies as d = d 0 + a cos ©t (a « dQ). The capacitor break down when the instantaneous current in the circuit reaches the value of I. Find the maximum possible amplitude of vibrations a.

Q.50 Two simple pendulums of length L each are attached to the ceiling. The small balls attached to the strings have equal masses m. The weights are connected by a very light relaxed rubber band (not a spring) with the force constant k. At a certain moment, each ball is given a light quick push as shown, resulting in equal initial speeds. Find the period T of the ensuing motion.

^Bansal Classes PHYSICS [432]

(a)

(b) (c)

Q.47

Q.51

Q.52

Q.53

Q.54

Q.60

A proton (m, e) and an alpha particle (4m, 2e) approach each other from a large distance. Initially, their velocities are the same (v). Find the minimum separation r between the particles. A wooden cube with a side of d = 0.10 m is placed on a horizontal support. A bullet of mass m = 0.010 kg is shot vertically up through the support and through the cube. As the bullet passes through the cube, its speed decreases uniformly from v = 120 m/s to u = 115 m/s. Estimate the minimum mass M of the cube that would allow it not to lose contact with the support. In a strictly scientific experiment, a student athlete throws rocks out the window in all directions. All rocks have the same initial speed v. It turns out that all rocks' landing velocities make angles 0 or greater with the horizontal. Find the height h of the window above the ground. An insulated container is filled with a mixture ofwater and ice at t = 0°C. Another container is c filled with water that is continuously boiling at ^ = 100°C. In a series of experiments, the containers are connected by various thick rods that pass through the walls of the containers (refer diagram). The rod is insulated in such a way that there is no heat loss to surroundings. In experiment 1, a copper rod is used and the ice melts in T, = 20 min. In experiment 2, a steel rod of the same cross section is used and the ice melts in T 2 = 60 min. How long would it take to melt the ice if the two rods are used "in series"?

insulation

Q.55

Q.56

Q.57

Q.58

Q.59

How can you measure the resistance of an unknown resistor r with an ammeter and a voltmeter if you don't know the internal resistances of these devices? A voltage source is available. A dubmbell consists o f a light rod of length r and two small masses m attached to it. The dumbbell stands vertically in the corner formed by two frictionless planes. L After the bottom end is slightly moved to the right, the dumbbell begins to slide. Find the speed u of the bottom end at the moment the top end loses contact with the vertical plane. Bbi_ Find the maximum power of a heating element that can be constructed from a piece of wire that has a resistance of 536 Q. The element is to be powered by a constant voltage of V = 110 V. The current through the wire cannot exceed 2 A. A heavy block is attached to the ceiling by a spring that has a force constant k. A conducting rod is attached to the block. The combined mass of the block and the rod is m. The rod can slide without friction along two vertical parallel rails, which are a distance L apart. A capacitor of known capacitance C is attached to the rails by the wires. The entire system is placed in a uniform magnetic field B directed as shown. Find the period T of the vertical oscillations of the block. Neglect the electrical resistance of the rod and all wires. An electric circuit contains a battery with emf E and internal resistance r, two coils with inductances L, and L, and a resistor R, connected as shown. On the diagram, all shown parameters are given. Initially, both switches are open. Switch S, is then closed. After a while, switch S 2 is closed. What is the total charge Q that passes through the resistor after S- is closed? Figure shows three identical balls Mj, M 2 and M 3 each of radius 10 cm. The ball M 3 is given a certain velocity in the direction of AB such that after collision with M 2 , it (M 3) has a head-on collision with the ball Mj. Find the distance BC 2 (in cm) where B lies on the line joining the centres of M, and M 2 . The balls are assumed to be perfectly elastic. Given CjC 2 = 1 m.

M,

^ Bansal Classes PHYSICS [10]