**SAMPLE PAPER-5**

Physics

**Questions**

**Section A**

**Select and write the correct answers to the following questions:**

(i) Select correct statement about the formula (expression) of moment of inertia (M.L) in terms of mass

(a) Different objects must have different expressions for their M.L.

(b) When rotating about their central axis, a hollow right circular cone and a disc have the same expression for the M.I.

(c) Expression for the M.L for a parallelopiped rotating about the transverse axis passing through its centre includes its depth.

(d) Expression for M.I of a rod and that of a plane sheet is the same about a transverse axis.

(ii) When seen from below, the blades of a ceiling fan are seen to be revolving anticlockwise and their speed is decreasing. Select correct statement about the directions of its angular velocity and angular acceleration.

(a) Angular velocity upwards, angular acceleration downwards.

(b) Angular velocity downwards, angular acceleration upwards.

(c) Both, angular velocity and angular acceleration, upwards.

(d) Both, angular velocity and angular acceleration, downwards.

(iii) Which of the following is an example of the first law of thermodynamics?

(a) The specific heat of an object explains how easily it changes temperatures.

(b) While melting, an ice cube remains at the same temperature.

(c) When a refrigerator is unplugged, everything inside of it returns to room temperature after some time

(d) After falling down the hills, a ball’s kinetic energy plus heat energy equals the initial potential energy.

(iv) Which of the following equations represents a wave travelling along theY-axis? Add symbol 56/4

(a) _

(b)

(c)

(d) _

(v) When the balance point is obtained in the potentiometer, a current is drawn from: (a) Both the cells and auxiliary battery

(b) Cell only

(c) Auxiliary battery only

(d) Neither cell nor auxiliary battery

(vi) A charged particle is in motion having initial velocity when it enter into a region of uniform magnetic field perpendicular to because of the magnetic force the kinetic energy of the particle will (add symbol on ):

(a) Remain unchanged

(b) Get reduced

(c) Increased

(d) Be reduced to zero

(vii) What is the energy required to build up a current of in an indicator of ?

(a)

(b)

(c)

(d)

(viii) A current through a coil of self inductance increases from 0 to in . What is the induced emf in the coil?

(a)

(b)

(c)

(d)

(ix) For a metal having a work function , the threshold wavelength is . What is the threshold wavelength for the metal having work function ?

(a)

(b)

(c)

(d)

(x) In which of the following systems will the radius of the first orbit of the electron be smallest?

(a) Hydrogen

(b) Singly ionized helium

(c) Deuteron

(d) Tritium

**68. Answer the following questions:**

(i) What is wattless current?

(ii) A beaker of radius is filled with water. Calculate the force of surface tension on any diametrical line on its surface. Surface tension of water is .

(iii) What should be the diameter of a water drop so that the excess pressure inside it is ? (Surface tension of water )

(iv) A solar cooker and a pressure cooker both are used to cook food. Treating them as thermodynamic systems, discuss the similarities and differences between them.

(v) Define the diathermanous substance.

(vi) What do you understand by term wave-particle duality?

(vii) Define inductive reactance.

(viii) Radiation of wavelength is incident on a metal having work function . Due to the presence of a magnetic field , the most energetic photoelectrons emitted in a direction perpendicular to the field along a circular path of radius . What is the value of the magnetic field ?

**69. Section B**

**70. Attempt any Eight of the following questions:**

- Do we need a banked road for a two-wheeler? Explain.
- A resistor held in running water carries electric current. Treat the resistor as the system:

(i) Does heat flow into the resistor?

(ii) Is there a flow of heat into the water?

(iii) Is any work done?

(iv) Assuming the state of resistance to remain unchanged, apply the first low of thermodynamics to this process.

- Using differential equation of linear S.H.M., obtain the expression for:

(i) Velocity in S.H.M.

(ii) Acceleration in S.H.M.

- In Young’s double slit experiment what will we observe on the screen when white light is incident on the slits but one slit is covered with a red filter and the other with a violet filter? Give reasons for your answer.
- What are eddy currents? State applications of eddy currents.
- A set of three coils having resistances and are connected in parallel. This combination is connected in series with combination of three coils of the same resistances. Calculate the total resistance and current through the circuit, if a battery of emf 4.1 volt is used for drawing current.
- An alpha particle (the nucleus of helium atom) (with charge +2 ) is accelerated and moves in a vacuum tube with kinetic energy . It passes through a uniform magnetic field of and traces a circular path of radius . Obtain the mass of the alpha particle. , charge of electron ]
- Calculate the gyromagnetic ratio of electron (Given ).
- Alternating emf of is applied to a circuit containing an inductance of henry. Write an equation for instantaneous current through the circuit. What will be the reading of the galvanometer connected in the circuit?
- Explain the inverse linear dependence of stopping potential on the incident wavelength in a photoelectric effect experiment.
- The frequency of a photon is . Find its wavelength in .
- The common-base DC current gain of a transistor is 0.967. If the emitter current is . What is the value of base current.

**71. Section C**

**72. Attempt any Eight of the following questions:**

- While driving along an unbanked circular road, a twowheeler rider has to lean with the vertical. Why is it so? With what angle the rider has to lean? Derive the relevant expression. Why such a learning is not necessary for a four-wheeler?
- Calculate the average molecular kinetic energy:

(i) Perkmol

(ii) Perkg

(iii) Per molecule of oxygen at , given that molecular weight of oxygen is is and Avogadro’s number is molecules .

- Explain the capillary action.
- The displacement of an oscillating particle is given by where and are constants. Prove that the particle performs a linear S.H.M. with amplitude .
- A pipe closed at one end can produce overtones at frequencies and . Calculate the fundamental frequency.
- Monochromatic electromagnetic radiation from a distant source passes through a slit. The diffraction pattern is observed on a screen from the slit. If the width of the central maximum is , what is the slit width if the wavelength is:

(i) (visible light);

(ii) -(infrared radiation):

(iii) (X-rays)

- State any two sources of errors in meter bridge experiment. Explain how they can be minimized.
- Draw a neat labelled diagram of a suspended coil type moving coil galvanometer. What is the advantages of a radial magnetic field in a MCG.
- Obtain the expression for orbital magnetic moment of an electron rotating about the nucleus in an atom.
- In a Faraday disc dynamo, a metal disc of radius rotates with an angular velocity about an axis perpendicular to the plane of the disc and passing through its centre. The disc is placed in a magnetic field acting perpendicular to the plane of the disc. Determine the induced emf between the rim and the axis of the disc.
- Find the shortest wavelength in Paschen series if, the longest wavelength in Blamer series is .
- In a common-base connection, the emitter current is and collector current is . Determine the common base DC current again.

**73. Section D**

**74. Attempt any Three of the following questions:**

- (i) Derive an expression of excess pressure inside a liquid drop.

(ii) A wave of frequency is travelling with a speed of .

(a) What is the phase difference between two displacement at a certain point at times apart?

(b) What will be the smallest distance between two points which are out of phase at an instant of time.

- (i) A mixture of fuel and oxygen is burned in a constantvolume chamber surrounded by a water bath. It was noticed that the temperature of water is increased during the process. Treating the mixture of fuel and oxygen as the system: (a) Has heat been transferred?

(b) Has work been done?

(c) What is the sign of ?

(ii) A parallel beam of green light of wavelength passes through a slit of width . The intensity pattern of the transmitted light is seen on a screen which is away. What is the distance between the two first order minima?

- (i) State Gauss’s law. What is Gaussian surface?

(ii) A capacitor, a inductor and a resistor are connected in series with an AC source whose emf is given by (volt). What is the frequency, reactance, impedance, current and phase angle of the circuit?

- A spherical shell of radius with charge is expanded to a radius . Find the work done by the electrical forces in the process.
- Explain spectral distribution of blackbody radiation.

**75. Answer Key**

**76. Section A**

- (i) (b) When rotating about their central axis, a hollow right circular cone and a disc have the same expression for the M.L.

(ii) (a) Angular velocity upwards, angular acceleration downwards

(iii) (b) While melting, an ice cube remains at the same temperature.

(iv) (a) _

(v) (d) Neither cell nor auxiliary battery

(vi) (a) Remain unchanged

(vii) (a)

(viii) (a)

(ix) (b)

(x) (d) Tritium

- (i) The current that does not lead to energy consumption, hence zero power consumption, is called wattless current.

(ii) Given,

We have,

.

(iii) Given:

We have,

(iv) Solar cooker:

- Solar cooker was invented by Horace Benedict de saussure in 1767.
- Solar cooker is a device used to cook food by using no fuel, instead of sunlight.
- Solar cookers use a parabolic reflector to collect the rays of the sun and focus them at the cooker to heat it and cook the food in the cooker.
- Today the solar cookers are a little bit expensive than pressure cookers.
- Big solar cookers can be used to make food for people on a larger scale.

**77. Pressure cooker:**

- Pressure cooker was invented by Denis papin.
- Pressure cookers are the most common cookers used in our houses and can be found in every house.
- Pressure cookers require water to convert it into steam for raising the internal temperature and pressure that permits quick cooking.
- Pressure cooker are cheaper than solar cookers.
- Pressure cookers requires a fuel for heating the liquid inside them.

(v) A substance through which heat radiations can pass is known as a diathermanous substance.

(vi) Condition or the structure of matter, electromagnetic radiation and material particles exhibit wave nature or particle nature this is known as wave particle duality.

(vii) The opposing nature of inductor to the flow of alternating current is called inductive reactance .

(viii) Given

,

Now, centripetal force,

**78. Section B**

- When a two-wheeler takes a turn along an unbanked road, the force of friction provides the centripetal force. The two-wheeler leans inward to counteract a torque that tends to topple it outward. First, friction cannot be relied upon to provide the necessary centripetal force on all road conditions. Secondly, the friction results in the wear and tear of the tyres. On a banked road at a turn, any vehicle can negotiate the turn without depending on friction and without straining the tyres.
- (i) Heat is generated into the resistor due to the passage of electric current. In the usual notation, heat generated .

(ii) Yes, water receives heat from the resistor.

(iii)

(iv)

Here, current through the resistor, resistance of the resistor, time for which the current is passed through the resistor, mass of the water, specific heat of water, rise in the temperature of water, pressure against which the work is done by the water, increase in the volume of the water.

- The general expression for the displacement of a particle in S.H.M. at time is

where is the amplitude, is a constant in a particular case and is the initial phase. The velocity of the particle is

From equation (1),

Equation (2) gives the velocity as a function of . The acceleration of the particle is

Equation (3) gives the acceleration as a function of . The minus sign shows that the direction of the acceleration is opposite to that of the displacement.

- In Young’s double-slit experiment, when white light is incident on the slits and one of the slit is covered with a red filter, the light passing through this slit will emerge as the light having red colour. The other slit which is covered with a violet filter will give light having violet colour as emergent light. The interference fringes will involve mixing of red and violet colours. At some points, fringes will be red if constructive interference occurs for red colour and destructive interference occurs for violet colour. At some points, fringes will be violet if constructive interference occurs for violet colour and destructive interference occurs for red colour. The central fringe will be right with the mixing of red and violet colours.
- Whenever a conductor or a part of it is moved in a magnetic field “cutting” magnetic field lines, or placed in a changing magnetic field, the free electrons in the bulk of the metal start circulating in closed paths equivalent to current-carrying loops. These loop currents resemble eddies in a fluid stream and are hence called eddy or Foucault currents [after Jean Bernard Leon Foucault (1819-68), French physicist, who first detected them].

**79. Applications:**

(i) Dead-beat galvanometer: A pivoted moving-coil galvanometer used for measuring current has the coil wound on a light aluminium frame. The rotation of the metal frame in the magnetic field produces eddy currents in the frame which opposes the rotation and the coil is brought to rest quickly. This makes the galvanometer dead-beat.

(ii) Electric brakes: When a conducting plate is pushed into a magnetic field or pulled out very quickly, the interaction between the eddy currents in the moving conductor and the field retards the motion. This property of eddy currents is used as a method of braking in vehicles.

- In the following figure shows the electrical network. For resistance, and connected in parallel to equivalent resistance is given by

For resistance and connected in series, the equivalent resistance,

Thus, the total resistance,

The total resistance and current through the circuit are and 0.1 A respectively.

- Given: ,

Charge of an -particle,

10.

- Given:

Comparing with , we get

The instantaneous current through the circuit

is the reading of the galvanometer connected in the circuit.

- We have , where is the stopping potential, is the magnitude of the charge on the electron, his Planck’s constant, is the speed of light in free space, is the wavelength of the electromagnetic radiation incident on a metal surface and is the work function for the metal, and are constants, is constant for a particular metal.

Hence, it follows that as increases, increases.

The plot of verses is linear. This is because the energy associated with a quantum of radiation (photon) is directly proportional to the frequency of the radiation and hence inversely proportional to the wavelength of radiation.

- Given,

- Given: Current gain

Emitter current

**80. To find:**

The value of base current of the transistor.

- The common gain DC current us is given by

- The base current of the transistor is given by the formula

The value of base current of the transistor is .

**81. Section C**

- (i) When a bicyclist takes a turn along an unbanked road, the force of friction provides the centripetal

force; the normal reaction of the road is vertically up. If the bicyclist does not lean inward, there will be an unbalanced outward torque about the centre of gravity, fs.h, due to the friction force that will topple the bicyclist outward. The bicyclist must lean inward to counteract this torque (and not to generate a centripetal force) such that the opposite inward torque of the couple formed by and the weight

(ii) Since the force of friction provides the centripetal force,

If the cyclist leans from the vertical by an angle , the angle between and

Hence, the cylist must lean by an angle

(iii) When a car takes a turn along a level road, apart from the risk of skidding off outward, it also has a tendency to roll outward due to an outward torque about the centre of gravity due to the friction force. But a car is an extended object with four wheels. So, when the inner wheels just get lifted above the ground, it can be counterbalanced by a restoring torque of the couple formed by the normal reaction (on the outer wheels) and the weight. Consider a car of mass taking a turn of radius along a level road. As seen from an inertial frame of reference, the forces acting on the car are:

(a) The lateral limiting force of static friction on the wheels – acting along the axis of the wheels and towards the centre of the circular path which provides the necessary centripetal force. (b) The weight acting vertically downwards at the centre of gravity (C.G.).

(c) The normal reaction of the road on the wheels, upwards effectively at the C.G. Since maximum centripetal force limiting force of static friction.

In a simplified rigid-body vehicle model, we consider only two parameters – the height of the C.G. above the ground and the average distance between the left and right wheels called the track width.

The friction force on the wheels produces a torque that tends to overturn/rollover the car about the outer wheel Rotation about the frontto-back axis is called roll.

When the inner wheel just gets lifted above the ground, the normal reaction of the road acts on the outer wheels but the weight continues to act at the C.G. Then, the couple formed by the normal reaction and the weight produces a opposite torque which tends to restore the car back on the all four wheels.

The car does not topple as long as the restoring torque counterbalances the toppling torque Thus, to avoid the risk of rollover, the maximum speed that the car can have is given by

Thus, vehicle tends to roll when the radial acceleration reaches a point where inner wheels of the four-wheeler are lifted off of the ground

and the vehicle is rotated outward. A rollover occurs when the gravitational force passes through the pivot point of the outer wheels, i.e. the C.G. is above the line of contact of the outer wheels. Equation (3) shows that this maximum speed is high for a car with larger track width and lower centre of gravity.

- Given:

Molecular weight

Molar mass

(i) The average molecular kinetic energy per mol of oxygen the average kinetic energy per of oxygen

(ii) The average molecular kinetic energy per of oxygen

(iii) The average molecular kinetic energy per molecule of oxygen

- When a capillary tube is partially immersed in a wetting liquid, there is a capillary rise and the liquid meniscus inside the tube is concave,

Consider four points A, B, C, D of which point A is just above the concave meniscus inside the capillary and point is just below it. Points and are just above and below the free liquid surface outside.

Let and be the pressure at points and respectively.

Now,

The pressure is the same on both sides of the free surface of a liquid, so that

The pressure on the concave side of a meniscus is always greater than that on the convex side, so that

The excess pressure outside presses, the liquid up the capillary until the pressure at and (at the same horizontal level) equalize, i.e., becomes equal to Thus, there is a capillary rise.

For a non-wetting liquid, there is capillary depression and the liquid meniscus in the capillary tube is convex.

Consider again four points A, B, C and D when the meniscus in the capillary tube is at the same level as the free surface of the liquid. Points and are just above and below the convex meniscus. Points and are just above and below the free liquid surface outside.

The pressure at is greater than that . The pressure at is the atmospheric pressure and at . Hence, the hydrostatic pressure at the same levels at and are equal, . Hence, the liquid flows from to and the level of the liquid in the capillary falls. This continues till the pressure at is the same as that D, that is till the pressure at the same level are equal.

- Position of particles is given by

The velocity of particle is given by

Acceleration of particle is given by

Since the acceleration of particle is directly proportional to displacement and directed towards mean position, therefore, the motion is simple harmonic motion.

Now, let amplitude

Substituting and in (1), we get

Squaring (2) and (3) and adding, we have

Dividing (2), (3), we have

- The difference between the given frequencies of the overtones is . This implies that they are consecutive overtones. Let be the fundamental frequency of the closed pipe and the frequencies of the and consecutive overtones, where is an integer.

Given: , since only odd harmonics are present as overtones,

Therefore, the three given frequencies correspond to the second, third and fourth overtones, i.e, the fifth, seventh and ninth harmonics, respectively.

- Given:

Let be the slit wdith.

(i)

(ii)

(iii)

- The chief sources of error in the meter bridge experiment are as follows:

(i) The bridge wire may not be uniform in cross-section. Then the wire will not have a uniform resistance per unit length and hence its resistance will not be proportional to its length.

(ii) End resistances at the two ends of the wire may be introduced due to:

(a) The resistance of the metal strips

(b) The contact resistance of the bridge wire with the metal strips.

(c) Unmeasured lengths of the wire at the ends because the contact points of the wire with the metal strips do not coincide with the two ends of the metre scale attached.

Such errors are almost unavoidable but can be minimized considerably as follows:

(i) Readings must be taken by adjusting the standard known resistance such that the null point is obtained close to the centre of the wire. When several readings are to be taken, the null points should lie in the middle one-third of the wire.

(ii) The measurements must be repeated with the standard resistance (resistance box) and the unknown resistance interchanged in the gaps of the bridge, obtaining the averages of the two results.

Magnet

Moving Coil Galvanometer

**82. Advantages of radial magnetic field:**

(i) As the coil rotates, its plane is always parallel to the field. That way. the deflecting torque is always a maximum depending only on the current in the coil, but not on the position of the coil.

(ii) The restoring torque is proportional to the deflection so that a radial field makes the deflection proportional to the current. The instrument then has a linear scale ie, the divisions of the scale are evenly spaced. This maker it particularly straight forward to calibrate and to read.

- In the Bohr model of a hydrogen atom, the electron of charge – e performs a uniform circular motion around the positively charged nucleus. Let and be the orbital radius, speed and period of motion of the electron. Then,

Therefore, the orbital magnetic moment associated with this orbital current loop has a magnitude,

Therefore, the magnetic dipole moment associated with this electronic current loop has a magnitude

Multiplying and dividing the right-hand side of the above expression by the electron mass

where is the magnitude of the orbital angular momentum of the electron is opposite to .

which is the required expression.

According to Bohr’s second postulate of stationary orbits in his theory of hydrogen atom, the angular momentum of the electron in the th stationary orbit is equal to , where is the Planck constant and is a positive integer. Thus, for an orbital electron,

Substituting for in equation (4),

For

The quantity is a fundamental constant called the Bohr magneton, (or ) .

- Suppose a thin conducting disc of radius is rotated anticlockwise, about its axis, in a plane perpendicular to a uniform magnetic field of induction points downwards. Let the constant angular speed of the disc be .

Consider an infinitesimal element of radial thickness at a distance from the rotation axis. In one rotation, the area traced by the element is . Therefore, the time rate at which the element traces out the area is

where is the frequency of rotation.

The total emf induced between the axle and the rim of the rotating disc is

For anticlockwise rotation in pointing down, the axle is at a higher potential.

- ?,

For shortest wavelength of Paschen series,

For longest wavelength in Balmer series

Divide (1) by (2), we get

- Given:

Common-emitter current gain,

Therefore, common-base current gain,

**83. Section D**

- (i) Consider a liquid drop of radius and surface tension T.

Due to surface tension, the molecules, on the surface film experience the net force in the inward direction normal to the surface.

Therefore, there is more pressure inside than outside. Let be the pressure inside the liquid drop and be the pressure outside the drop.

Therefore, excess of pressure inside the liquid drop is

Due to excess pressure inside the liquid drop the free surface of the drop will experience the net force in outward direction due to which the drop will expand. Let the free surface display by under isothermal conditions.

Therefore, excess of pressure does the work in displacing the surface and that work will be stored in the form of potential energy.

The work done by an excess of pressure in displacing the surface is

Force displacement

Excess of pressure surface area

displacement of the surface

Increase in the potential energy is

surface tension increase in area of the free surface

From equation (1) and (2).

The above expression gives us the pressure inside a liquid drop.

(ii) Given:

(a) , the path difference is the distance covered

Phase difference Path difference

(b) Phase difference

Path difference Phase difference

- (i) (a) Heat has been transferred from the chamber to the water both.

(b) No work is done by the system (the mixture of fuel and oxygen) as there is no change in its volume.

(c) There is an increase in the temperature of the water. Therefore, is positive for water.

For the system (the mixture of fuel and oxygen). is negative. (ii) Given:

- (i) Gauss’s law: The electric fuix over a closed surface of any size and any stage is equal to times total charge endorsed by that surface.

Gaussian surface: The surface over which Gauss’s theorem is applied is called as Gaussian surface,

(ii) Given: , (314) [volt]

Comparing with . we get

The frequency of the alternating emf as

Reactance

Peak current,

Phase angle,

- Consider a spherical conducting shell of radius placed in a medium of permittivity . The mechanical force per unit area on the charged conductor is

where is the surface charge density on the conductor. Given the charge on the spherical shell is , The force acts outward, normal to the surface.

Suppose the force displaces a charged area element through a small distance , then the work done by the force is

During the displacement, the area element sweeps out a volume

Therefore, the work done by the force in expanding the shell radius to is

This gives the required expression for the work done.

31. The spectral distribution of the thermal energy radiated by a blackbody (i.e., the pattern of the intensity of the radiation over a range of wavelengths or frequencies) depends only on its temperature.

The characteristics of blackbody radiation can be described in terms of several laws:

Planck’s law of blackbody radiation, a formula to determine the spectral energy density of the emission at each wavelength ( ) at a particular absolute temperature (T).

Wien’s displacement law, which states that the frequency of the peak of the emission increases linearly with absolute temperature (T). Conversely, as the temperature of the body increases, the wavelength at the emission peak decreases, .

Stefan-Boltzmann law, which relates the total energy emitted to the absolute temperature

Points to be noted: The blackbody radiation curves have quite a complex shape (describe by Planck’s law).

The spectral profile (or curve) at a specific temperature corresponds to a specific peak wavelength and vice versa.

As the temperature of the blackbody increases, the peak wavelength decreases (Wien’s law).

The intensity (or flux) at all wavelengths increases as the temperature of the blackbody increases. The total energy being radiated (the area under the curve) increases rapidly as the temperature increases (StefanBoltzmann law).

Although the intensity may be very low at very short or long wavelengths, at any temperature above absolute zero energy is theoretically emitted at all wavelengths (the blackbody radiation curves never reach zero).