IMPORTANT NUMERICALS
CHAPTER 1
[1] A sphere of mass 4 kg collides with a wall, at an angle of 300 with the wall and rebounds in the direction making an angle of 600
with its original direction of motion. Find the force on the wall if the ball is in contact with the wall for 0.1s.
[2]A small sphere of radius a is cut from a big sphere of radius R, having centre at origin. Find the centre of mass of the remaining
part.[Unsolved question 7]
[3] A circular plate of uniform thickness has diameter 56cm. Its centre is at origin. A circular portion of diameter 42cm is cut from
one of its edge, symmetric to X axis. What will be the position of new centre of mass
[4] Distance between two particles of mass m 1 and m2 is r. If r1 and r2 are the distance of these particles from centre of mass
respectively, then show that
 m1 
 m2 
r1  r 

 and r2  r 
 m1  m 2 
 m1  m 2 
[5]Three particles of equal mass m are the vertices of an equilateral triangle, of side a. Find the centre of mass of this system
CHAPTER 2
[6] Find the moment of inertia of a uniform circular disc about an axis passing through its geometrical centre and perpendicular to its
plane and radius of gyration.
[7] Four spheres each of mass of M and radius R are placed at the corners of a square having side ‘a’ find the moment of inertia of the
system about an axis along one of the sides of square.
[8] A string is wound around a smooth circular disc of mass M and radius R. A mass m is suspended from the free end
of the string. The mass is allowed to descend. Prove that the angular acceleration of the disc is mg / R (m + M / 2)
[9] The initial angular speed of a wheel is 10 rad/ s. Its angular displacement in 10 s is 50 rad. How many rotation will it make starting
form the beginning to the time it stops ? Find its angular acceleration also.
[10] A turntable rotates in horizontal plane with angular speed 20rpm, about a vertical axis passing through its centre. A man of 60kg is
standing at the edge of the table. If the man goes from the edge of the table to the centre, what would be the new angular speed
now.The mass of the turntable is 200kg.
[11] A child of mass M is sitting on the board of a merry go round 1m from its centre, rotating about an axis perpendicular to its plane
and passing through its centre. With what angular velocity should the merry-go-round be rotated so that the child is on the verge of
sliding on the board.coeficient of friction between the child and the board is 0.25. [g =10SI]
[12] A truck is moving at a speed of 54 km/h. The radius of its wheels is 50 cm. On applying the brakes the wheels stop after 20
rotations. What will be the linear distance travelled by the truck during this? Also find the angular acceleration of the wheels.
CHAPTER 3
[13] A satellite, of mass 200 kg, revolves round the Earth at a height of 1000 km from the surface of the Earth. Calculate (i) the binding
energy and (ii)escape speed of the satellite. Take G = 6.67 x 10-11 MKS, the radius of the Earth = 6400 km and the mass of the Earth =
6 x 1024 kg.
[14] ] Two objects of masses 1 kg and 2 kg respectively are released from rest when their separation is 10m. assuming that only
mutual gravitational forces act on them, find the velocity of each of them when separation becomes 5 m
[15] Four particles, each of mass m, are placed on the vertices of a square of side l. Calculate the gravitational potential energy of this
system of four particles. Also, calculate the gravitational potential at the centre of the square.
[16] An object is thrown from earth with a speed thrice the escape speed. Determine its speed when it escapes from the gravitational
field of earth.
[17] Prove that the ratio of the rate of change of ‘g’ at a height R e above the surface of the Earth (Re = radius of the Earth) to the value
of ‘g’ at the surface of the Earth is equal to – 1/4Re.
[18] A space craft goes directly from earth to sun. How far from the centre of the earth the gravitational force on it due to earth and sun
would be equal in magnitude. The distance between earth and sun is 1.49 x 10 8km. Mass of earth is 6 x 1024kg and mass of sun is 2 x
1030kg
[19] If earth were completely made of gold what would have been the acceleration due to gravity on its surface. Density of gold 19.3 x
103kg/m3
CHAPTER 4
[20] A steel wire of length 5m and diameter 10-3, is hanging vertically from a height of 5.22m. Asphere of radius 0.1m and mass 8  kg
is tied to the free end of the wire. When this sphere is oscillated like a simple pendulum it touches the floor at the lower most position.
Find the velocity of the sphere at the lowermost position. Y = 1.994 x 10 11 N/m2
[21] A mass of 15 kg is tied at the end of a steel wire of 1m length. It is whirled in a vertical circle with angular velocity 1rad/s. Cross
sectional area of wire is 0.06 cm 2. Find the elongation of the wire when the mass is at the lowermost position. Y = 2 x 1011 N/m2
[22] The pressure at a certain depth in sea is 80 atm. If the density of water at the surface of the sea is 1.03 x 10 3kg/m3 and
compressibility of water is 45.8 x 10-11 Pa-1 find the density of water at the depth. 1 atm = 1.013 x 105 Pa
[23] A wire of length 5m and diameter 2mm is hanging from a ceiling. A mass of 5kg is suspended from the lower end. Find the
increase in the volume. Poission’s ratio = 0.2 and Y = 2 x 1011 Pa, g = 10m/s2. Also find the change in potential energy
[24] Length and cross sectional area of a wire is 5m and 2.5mm 2, find the work required to be done to increase its length by 1mm.
Y=2x1011 SI
[25] Masses 2kg and 4kg are tied to two ends of a wire passing over a pulley. Cross sectional area of wire is 2cm 2, find longitudinal
strain in wire.
[g = 10 SI and Y=2x1011 SI]
CHAPTER 5
[26] The radius of a pipe decreases according to r = r 0e–ax ; where a= 0.50 m-1 and x is the distance of a cross – section from the first
end ( x = 0). Find the ratio of Reynolds number for two cross – sections lying at the distance of 2 m from each other. (Here, e is an
irrational number like π. We take e = 2.718).
[27] The diameter of one end of a tube is 2 cm and that of another end is 3 cm. Velocity and pressure of water at narrow end are 2 ms -1
and 1.5 x 105 N m-2 respectively. If the height difference between narrow and broad ends is 25. m, find the velocity and pressure of
water at the broad end. (Density of water is 1 x 10 3 kg m-3). The narrow end is higher.
[28] Find the work done to increase the volume of bubble, of soap solution, having a radius of 1 cm, to 8 times (Surface tension of soap
solution is 30 dyn cm -1).
[29] A bubble of soap solution of radius 2.4 x 10-4 m is lying in a cylinder filled with air having pressure 105 N m-2. Now, on compressing
the air, the radius of the bubble reduces to half. Find the new pressure of air in the cylinder. (Surface tension of soap solution is 0.08 N
m-1).
[30] The diameter of one end of tube is 2cm and other end is 3cm. Velocity and pressure at narrow end are 2m/s and 1.5x10 5 , if the
height difference between the two ends is 2.5m, find the velocity and pressure at broad end.
[31] Two rain drops of equal volume falling with terminal velocity 10m/s, merge while falling forms a larger drop. Find the terminal
velocity of the bigger drop
[32] Two soap bubbles of radius R1 and R2 merge to form a bubble of radius R. If atmospheric pressure is P and the surface tension of
soap is T, prove that P[R13 + R23- R3] = 4T[R2 - R12 – R22]
[33] The piston and nozzle of a syringe kept horizontal have diameter 5mm and 1mm. The piston is pushed with constant velocity
0.2m/s. Find the horizontal distance travelled by the water jet before touching ground. [ g = 10SI] Height of the syringe from ground is
1m.
[34] Water is flowing through a horizontal pipe of irregular cross section. If pressure at a point where velocity is 0.2m/s is 30mm of Hg,
what will be the pressure at a point where velocity is 1,2m/s [density of Hg = 13.6 CGS, g = 1000 CGS, density of water = 1 CGS]
CHAPTER 6
[35]What should be the lengths of brass and aluminium at 0oC, if the difference between their lengths is to be maintained 5cm at any
temperature.
[For brass  = 18x10-6/ oC, for aluminium  = 24x10-6/ oC]
[36]Derive the expression for the work done in adiabatic process
[37] Calculate work required to be done to increase the temperature of 1 mol ideal gas by 30 oC . Expansion of the gas takes place
according to relation V proportional to T 2/3. R = 8.3 J mol-1 K-1.
[38] The temperature of heat sink in a Carnot engine is 280 K and its efficiency is 40%. How much should the temperature of heat
source be increased, at constant temperature of sink, so that efficiency of the engine becomes 50%.
[39] Pressure and temperature of 10 g O2 are 3 x 105 N m-2 and 100C respectively. On heating the gas at constant pressure its volume
becomes 10 L. Calculate heat absorbed by the gas [b] change in internal energy of the gas [c] work done by the gas during
expansion. Take R = 8.3 SI
[40] An ideal gas is enclosed in a closed container of 0.0083 m 3 at 300 K temperature and pressure of 1.6 x 106 Pa. Find the
temperature and pressure of the gas if 2.49 x 104 J heat is supplied to the gas. Neglect expansion of the container. R = 8.3 J mol -1 K-1.
[41] What will be the mass and temperature of water obtained by giving 210kJ heat to ice of 1kg at –10oC [Cice= 2220kJ/Kg-K]
[42] In a Carnot engine the temperature of source is 500K and that of sink is 375K. If engine absorbs 600kcal of heat from source per
cycle find [a] work done per cycle [b] heat released to sink per cycle [c] efficiency
CHAPTER 7
[43] An oscillator of mass 100 g is performing damped oscillations. Its amplitude becomes half of its initial amplitude after 100
oscillations If its period is 2 s find the damping co – efficient.
[44] Length of an elastic spring increases by 9 cm on suspending a body of mass 14.4 g its lower end. Now this body is pulled down by
3 cm and released so that its starts executing SHM. Find (i) amplitude and initial phase (2) angular frequency and period, (3) phase at t
= 3 s, (4) equation of displacement of this SHM and (5) displacement at t = 1.5 s, g = 100  cm s-2.
[45] Calculate the time in which the amplitude becomes A / 2 n in case of damped oscillations.
[46] A rectangular pipe having cross – sectional area A is closed at one end and at its other end a block having same cross – section is
placed, so that the system is airtight. In the equilibrium position of the block, the pressure and volume of air enclosed in the pipe are p
and V respectively. Prove that the block performs SHM when it is given a small displacement ‘x’ inwards released. Also find the period
of this SHM. Assume the walls to be frictionless and compression to be isothermal.
[47] Length of a second’s pendulum on the surface of the earth is l1 and it is l2 at a height ‘h’ from the surface of the earth. Prove that
2
h l2
the radius of the earth is given by Re =
l1  l2
[48] For an SHM, prove that a2T2 + 4  v2 = constant; where a and v are acceleration and velocity respectively at any distant and T is
the period.
[49] A simple pendulum is made by suspending small spear of brass at the end of a string. When it is oscillated in air its period is T .
2
Now this spear is immersed in a liquid of density ½ times that of brass and oscillated. Prove that its new period is
resitive damping oscillation.
2 T. Neglect all
[50] A body weighing 10kg has a velocity 6m/s after 1s of starting from mean position. If the time period of osscilation is 6s, find KE, PE
and total mechanical energy.
[51] A spring of force constant k and length l, is cut in to two parts of length l1 and l2, such that l1 = n l2 , obtain the values of force
constant k1 and k2 in terms of k
CHAPTER 8
[52] The length of a sonometer wire between its fixed ends is 110 cm. Where should two bridges S1 and S2 be placed in between the
ends so as to divide the wire into 3 segments whose fundamental frequencies are f 1 : f2 : f3 = 1 : 2 : 3
[53] The speed of sound in dry air at STP is 332m/s. Air is composed of 4 parts of nitrogen and 1 part of oxygen. Calculate speed of
sound in oxygen under similar condition. Density of oxygen and nitrogen are in the ratio 16:14
[54] The equation of a wave is y = Acos(ax +bt) [a] Determine the frequency and wavelength [b] If the intensity of relected wave is
0.64 times the incident wave the determine the equation of refected wave. [c] express the resultant displacement as a combination of
stationary and progressive wave.
[55] A SONAR system in submarine operates at 40kHz. An enemy submarine moves towards the SONAR with a speed of 360km/h.
What is the frequency of sound refelected by submarine. Speed of sound in water 1450m/s
[56] . The stationary waves produced in a 60 cm long string ties at both the ends with rigid supports are represented by y = 4sin (  x/
15) cos (96  t). Here, x and y are in cms and t is in seconds. Find out (1) position of nodes, (2) positions of unit – nodes (3) maximum
displacement of the particle at x = 5 cm (4) the equation of the component waves.
[57] A long wire PQR is made by joining two wires PQ and Or of equal radii. The wire PQ has length 4.8 m and mass 0.06 kg. The wire
QR has length 2.56 m and mass 0.2 kg. The wire PQR is under the tension of 80 N. Find the time taken by a wave produced at the end
P to reach the other end R
[58] Equation of a one dimensional propagating wave is y = 5 sin 30  (t – x / 240). Here y is in metre and t is in second. (1) Is the
particle of medium moving in +Y or – Y direction at the origin at time t = 0 ? i.e. what will be produced first – crest of trough ? (2) Find
the displacement, velocity of the particle and the slope of the wave at 480 m away from the origin at time t = 2 s. (3) Find the speed of
waves.
[59] Find the difference in the apparent frequencies of the sound of a car horn heard by a stationary listnerwhen the car is moving
towards and away from the listener with a speed of 54km/h. The frequency of sound from horn is 500Hz and speed of sound in air is
340m/s
[60] In case of progressive harmonic wave show that, the ratio of instantaneous velocity of particle to the wave speed is equal to the
negative of the slope.
[61] The amplitude of a progressive harmonic wave is 10m. The displacement of a particle at a distance 2m from the origin is 5m after
2s. Another particle which is at 16m from origin has displacement
5 3 m at time 8s. Find angular frequency and wave vector.
[62] At what temperature the hydrogen gas will have the speed of sound waves same as that in oxygen gas at 1200C. Density of
oxygen is 16 times that of hydrogen
[63] A wire having linear density 0.05g/cm is stretched between two rigid supports with a tension of 450N. The wire resonates with a
frequency of 420Hz. The next higher frequency at which the same wire resonates is 490Hz. Find the length of the wire.