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Question 1Physics

Two spheres, one solid and one thin-walled hollow, have equal mass and equal radius. Both are released from rest at the same height on a rough incline steep enough to prevent slipping. Which sphere reaches the bottom first, and why?

  1. The solid sphere, because rolling friction does less work on a smaller moment of inertia.
  2. The solid sphere, because its smaller moment of inertia about the centre gives it a larger linear acceleration. Correct answer.
  3. The hollow sphere, because its mass is distributed farther from the centre, increasing the gravitational torque.
  4. Both reach together, because gravitational acceleration is independent of mass distribution for rolling.

Explanation

For rolling without slipping down an incline, the acceleration is a = g sin θ / (1 + I/(mR²)). A solid sphere has I = (2/5)mR², while a thin hollow sphere has I = (2/3)mR², so the solid sphere has the smaller denominator and hence the larger acceleration. Therefore the solid sphere reaches the bottom first.

Question 2Physics

An ideal gas is compressed slowly and adiabatically in a thermally insulated cylinder. Which statement about the gas is correct?

  1. Its temperature falls because the gas does positive work.
  2. Its temperature remains constant because no heat is exchanged.
  3. Its temperature change depends only on the initial pressure, not on the compression ratio.
  4. Its temperature rises because work is done on the gas while Q = 0. Correct answer.

Explanation

For an adiabatic process Q = 0, and the first law gives ΔU = W_on gas. During compression work is done on the gas, so ΔU is positive and the temperature rises. The relation TV^(γ−1) = constant shows that reducing the volume raises T.

Question 3Physics

A thin biconvex lens made of glass of refractive index 1.5 is immersed in a liquid of refractive index 1.5. What happens to its focal length?

  1. It becomes infinite; the lens ceases to converge or diverge. Correct answer.
  2. It becomes twice the value in air.
  3. It becomes half the value in air.
  4. It remains unchanged, because focal length depends only on the lens geometry.

Explanation

The lens maker's formula is 1/f = (n_lens/n_medium − 1)(1/R1 − 1/R2). When the refractive indices are equal, the factor in brackets is zero, so 1/f = 0 and the focal length becomes infinite. The lens then produces no net convergence or divergence.

Question 4Physics

A point charge +Q is placed at the centre of an initially neutral, thick conducting spherical shell. The shell is then grounded. Which statement describes the final charge distribution?

  1. Both surfaces have zero charge because the shell is grounded.
  2. The inner surface has −Q, the outer surface has zero charge, and the net shell charge is −Q. Correct answer.
  3. The inner surface has +Q, the outer surface has −Q, and the net shell charge is zero.
  4. The inner surface has −Q, the outer surface has +Q, and the net shell charge is zero.

Explanation

The electric field inside the conducting material must be zero, so a Gaussian surface within the metal encloses zero net charge; this requires −Q on the inner surface. Grounding makes the outer surface at zero potential, and with no external field the outer surface charge is zero, leaving the shell with net charge −Q.

Question 5Physics

A large open tank filled with water to height H has a small hole at height y above the ground. The water jet leaves horizontally. For what value of y is the horizontal range on the ground maximum?

  1. y = 3H/4
  2. y = H
  3. y = H/2 Correct answer.
  4. y = H/4

Explanation

By Torricelli's theorem the efflux speed is v = √(2g(H − y)), and the time of fall is t = √(2y/g). The horizontal range is R = vt = 2√(y(H − y)), which is maximum when y = H/2.

Question 6Physics

A particle executes simple harmonic motion with amplitude A. At displacement x = A/2 from equilibrium, what is the ratio of its speed to its maximum speed?

  1. 1/√2
  2. √3/2 Correct answer.
  3. 3/4
  4. 1/2

Explanation

The speed in simple harmonic motion is v = ω√(A² − x²), and the maximum speed is v_max = ωA. At x = A/2, v = ωA√(1 − 1/4) = (√3/2)ωA, so the ratio is √3/2.

Question 7Physics

A charged particle enters a uniform magnetic field perpendicular to its velocity and moves in a circular path. If the speed is doubled while the field remains unchanged, what happens to the radius and the period?

  1. The radius doubles and the period doubles.
  2. The radius halves and the period is unchanged.
  3. The radius doubles and the period is unchanged. Correct answer.
  4. The radius is unchanged and the period halves.

Explanation

The magnetic force provides the centripetal force, giving r = mv/(qB), so the radius is proportional to speed and doubles. The period is T = 2πm/(qB), which is independent of speed, so it remains unchanged.

Question 8Physics

In a photoelectric experiment, the stopping potential V0 is measured for several frequencies ν of incident light on a given metal. Which statement about the graph of V0 versus ν is correct?

  1. The slope depends on the intensity, and the intercept gives the work function in volts.
  2. The slope is h, and the intercept on the V0-axis gives the work function.
  3. The slope is h/e, and the intercept on the ν-axis gives the threshold frequency. Correct answer.
  4. The graph is a straight line through the origin for every metal.

Explanation

Einstein's photoelectric equation gives eV0 = hν − φ, so V0 = (h/e)ν − φ/e. The graph is a straight line of slope h/e, which is the same for all metals, and its intercept on the frequency axis is the threshold frequency ν0 = φ/h.

Question 9Physics

In a Young double-slit experiment using light of wavelength λ, a thin transparent film of thickness t and refractive index n is placed in front of one slit. The fringe pattern shifts. By how many fringe widths does the central maximum shift?

  1. (n − 1)t/λ Correct answer.
  2. λ/((n − 1)t)
  3. (n + 1)t/λ
  4. (n − 1)λ/t

Explanation

The film introduces an extra optical path difference of (n − 1)t between the two interfering beams. One complete fringe width corresponds to a path difference of λ, so the number of fringes by which the pattern shifts is (n − 1)t/λ.

Question 10Physics

A physical quantity Z is calculated from measured quantities as Z = A²B/C. The fractional errors in A, B and C are 1%, 2% and 3% respectively, all small. What is the maximum fractional error in Z?

  1. 5%
  2. 6%
  3. 4%
  4. 7% Correct answer.

Explanation

For Z = A²B/C, the maximum fractional error is obtained by adding the fractional errors with the magnitudes of the powers: 2(1%) + 1(2%) + 1(3%) = 7%. The exponent 2 on A doubles its contribution, while B and C contribute once each.

Question 11Physics

Which of the following physical quantities has the same dimensional formula as Planck's constant?

  1. Torque
  2. Moment of inertia
  3. Angular momentum Correct answer.
  4. Linear momentum

Explanation

Planck's constant has dimensions [h] = ML²T⁻¹, the same as action. Angular momentum r × p is also ML²T⁻¹, so the two quantities are dimensionally identical even though they describe different physics.

Question 12Physics

A body starts from rest and moves along a straight line with uniform acceleration. The ratio of the distance covered in the first second to the distance covered in the third second is

  1. 3 : 5
  2. 1 : 3
  3. 1 : 9
  4. 1 : 5 Correct answer.

Explanation

For motion from rest with constant acceleration, the distances covered in successive equal time intervals are in the ratio 1 : 3 : 5 : ... Therefore the first second and the third second give 1 : 5. This follows from evaluating s = ½at² at t = 1 s and taking the difference between t = 3 s and t = 2 s.

Question 13Physics

A metallic wire of resistance R is stretched uniformly until its length becomes twice the original length. The resistance of the stretched wire is

  1. 2R
  2. 8R
  3. R/2
  4. 4R Correct answer.

Explanation

Stretching conserves the volume of the wire, so the product of length and area stays fixed. Since R = ρL/A, the resistance scales as L², and doubling the length multiplies the resistance by four.

Question 14Physics

A parallel-plate capacitor is charged by a battery and then disconnected. A dielectric slab of dielectric constant K > 1 is inserted so that it completely fills the space between the plates. Which quantity remains unchanged?

  1. The energy stored in the capacitor
  2. The electric field between the plates
  3. The potential difference across the plates
  4. The charge on the plates Correct answer.

Explanation

Once the battery is removed, the free charge on the plates has no path to escape, so it stays fixed. The dielectric then reduces the field to E/K, the potential difference to V/K, and the stored energy to U/K because the capacitance rises to KC.

Question 15Physics

A rectangular current-carrying coil is placed in a uniform magnetic field. The torque on the coil is maximum when the plane of the coil is

  1. inclined at 60° to the magnetic field
  2. inclined at 30° to the magnetic field
  3. parallel to the magnetic field Correct answer.
  4. perpendicular to the magnetic field

Explanation

The torque is τ = NIAB sinθ, where θ is measured between the normal to the coil and the magnetic field. The sine reaches its maximum at θ = 90°, which corresponds to the plane of the coil lying parallel to the field.

Question 16Physics

A Carnot engine operates between a hot reservoir at 500 K and a cold reservoir at 300 K. If it absorbs 1000 J of heat from the hot reservoir per cycle, the work done per cycle is

  1. 400 J Correct answer.
  2. 600 J
  3. 750 J
  4. 200 J

Explanation

The Carnot efficiency depends only on the reservoir temperatures: η = 1 − T_c/T_h = 1 − 300/500 = 0.40. The work per cycle is therefore W = ηQ_h = 0.40 × 1000 J = 400 J.

Question 17Physics

For a transparent medium of refractive index √3, the Brewster angle for light incident from air is

  1. 60° Correct answer.
  2. 45°
  3. 75°
  4. 30°

Explanation

At the Brewster angle the reflected and refracted rays are mutually perpendicular, which gives tanθ_B = n. With n = √3, the Brewster angle satisfies tanθ_B = √3, so θ_B = 60°.

Question 18Physics

The minimum energy required to excite a hydrogen atom from its ground state to the n = 3 state is approximately

  1. 1.51 eV
  2. 12.1 eV Correct answer.
  3. 10.2 eV
  4. 3.40 eV

Explanation

Hydrogen levels are E_n = −13.6/n² eV, so the excitation energy is 13.6(1 − 1/9) = 13.6 × 8/9 ≈ 12.1 eV. This is the gap between the ground state and the n = 3 level.

Question 19Physics

A uniform thin rod of length L is pivoted at one end and held horizontally. It is released from rest. The speed of the free end as the rod passes through the vertical position is

  1. √(3gL) Correct answer.
  2. √(3gL/2)
  3. √(gL)
  4. √(2gL)

Explanation

The centre of mass falls a distance L/2, so energy conservation gives mg(L/2) = ½Iω² with I = mL²/3 about the pivot. This yields ω = √(3g/L), and the free end moves at v = ωL = √(3gL).

Question 20Physics

A conducting rod of mass m and length L slides without friction down two long vertical parallel rails separated by L and joined at the top by a resistor R. A uniform horizontal magnetic field B is perpendicular to the plane of the rails. The terminal speed acquired by the rod is

  1. mgR/(BL)
  2. mgR/(BL²)
  3. mgR/(B²L²) Correct answer.
  4. B²L²/(mgR)

Explanation

At terminal speed the upward magnetic force balances the weight, so mg = BIL, with I = BLv/R from the motional emf. Substituting gives mg = B²L²v/R, hence v = mgR/(B²L²).

Question 21Physics

The dimensional formula for the coefficient of viscosity η is

  1. [M L^-1 T^-2]
  2. [M L T^-1]
  3. [M L^-2 T^-2]
  4. [M L^-1 T^-1] Correct answer.

Explanation

For a Newtonian fluid, shear stress equals η times the velocity gradient, so η has dimensions of force divided by area and by velocity per unit length. Simplifying [M L T^-2]/(L^2 · T^-1) gives [M L^-1 T^-1].

Question 22Physics

A point charge q is placed at the centre of an imaginary spherical surface of radius R. The total electric flux through the surface is Φ. If the radius of the spherical surface is doubled while q remains at the centre, the flux becomes

  1. 4Φ
  2. Φ/2
  3. Φ/4
  4. Φ Correct answer.

Explanation

Gauss law states that the total electric flux through a closed surface depends only on the charge enclosed and equals q/ε0. Changing the radius of the imaginary sphere does not change the enclosed charge, so the flux remains Φ.

Question 23Physics

An ideal gas undergoes an isothermal expansion. The change in its internal energy during the process is

  1. zero Correct answer.
  2. negative
  3. equal to the work done by the gas
  4. positive

Explanation

For an ideal gas, internal energy depends only on temperature. Since the process is isothermal, the temperature remains constant and therefore the change in internal energy is zero.

Question 24Physics

An ideal inductor is connected to an AC source of voltage V = V0 sin(ωt). The average power consumed by the inductor over one complete cycle is

  1. V0^2/(2ωL)
  2. zero Correct answer.
  3. V0^2/(ωL)
  4. V0^2/2

Explanation

In an ideal inductor the current lags the voltage by π/2, so the instantaneous power alternates in sign over a cycle. Its average over a full cycle is zero, and no net energy is dissipated.

Question 25Physics

A convex lens of focal length f forms a real image whose size equals the size of the object. The object distance from the lens is

  1. 4f
  2. √2 f
  3. f
  4. 2f Correct answer.

Explanation

Equal image and object sizes for a real image require magnification magnitude 1, so the image distance equals the object distance. The thin-lens equation then gives 1/f = 1/u + 1/u, hence u = 2f.

Question 26Physics

An electron is accelerated from rest through a potential difference V. Its de Broglie wavelength is proportional to

  1. 1/√V Correct answer.
  2. V
  3. √V
  4. 1/V

Explanation

The electron gains kinetic energy eV, so its momentum is p = √(2meV). The de Broglie wavelength λ = h/p is therefore proportional to 1/√V.

Question 27Physics

A galvanometer of resistance G gives full-scale deflection for a current Ig. To convert it into a voltmeter of range V, the resistance connected in series with the galvanometer must be

  1. V/Ig + G
  2. V Ig - G
  3. G - V/Ig
  4. V/Ig - G Correct answer.

Explanation

At full-scale deflection, the current through the voltmeter is Ig and the voltage across it is V. The total series resistance required is V/Ig, so the additional series resistance is V/Ig - G.

Question 28Physics

A projectile is launched from level ground with speed u at angle θ to the horizontal and has range R. Ignoring air resistance, another launch angle that gives the same range with the same speed is

  1. 45° + θ
  2. 90° − θ Correct answer.
  3. θ/2
  4. 180° − θ

Explanation

The horizontal range is R = u^2 sin(2θ)/g. Since sin(2θ) = sin(180° − 2θ), replacing θ by 90° − θ gives the same value of sin(2θ) and hence the same range.

Question 29Physics

A uniform flexible chain of length L lies on a frictionless horizontal table with a length x hanging vertically over the edge. The chain is released from rest. What is its speed at the instant the last link leaves the table?

  1. √(g(L^2 − x^2)/L) Correct answer.
  2. √(g(L − x))
  3. √(2g(L − x))
  4. √(2g(L^2 − x^2)/L)

Explanation

Taking the table top as zero potential energy, the initial potential energy is −(M/L)g x^2/2. When the last link leaves the table, the whole chain hangs vertically with centre of mass at −L/2, so energy conservation gives v^2 = g(L^2 − x^2)/L.

Question 30Physics

A square loop of side a and resistance R rotates with constant angular speed ω about an axis through its centre and perpendicular to a uniform magnetic field B. The self-inductance of the loop is negligible. The average electrical power dissipated in the loop over one full rotation is

  1. B^2 a^4 ω/(2R)
  2. B^2 a^4 ω^2/R
  3. B^2 a^2 ω^2/(2R)
  4. B^2 a^4 ω^2/(2R) Correct answer.

Explanation

The magnetic flux is BA cos(ωt) with A = a^2, so the induced emf has peak value B a^2 ω. The average power dissipated in resistance R is half the peak-emf squared divided by R, giving B^2 a^4 ω^2/(2R).

Question 31Physics

A uniform circular disc of mass M and radius R lies in a plane. Its moment of inertia about any one of its diameters is

  1. MR²/2
  2. MR²/4 Correct answer.
  3. 2MR²/5
  4. MR²

Explanation

The perpendicular-axis theorem gives I_z = I_x + I_y = MR²/2 for a disc about the axis through its centre normal to its plane. By symmetry the two perpendicular diameters are equivalent, so each diameter has moment of inertia MR²/4.

Question 32Physics

The escape speed from the surface of a certain planet of mass M and radius R is v. A projectile of mass 2m is fired from the surface of the same planet. The minimum launch speed needed for it to escape to infinity is

  1. 2v
  2. v Correct answer.
  3. v/2
  4. √2 v

Explanation

Escape speed follows from (1/2)mv² = GMm/R, where the projectile mass m cancels from both sides. Hence v = √(2GM/R) depends only on the planet's mass and radius, so doubling the projectile mass leaves the required launch speed equal to v.

Question 33Physics

A string of length L is fixed at both ends and vibrates in its fundamental mode. The wavelength of the resulting standing wave is

  1. 2L Correct answer.
  2. L
  3. 4L
  4. L/2

Explanation

In the fundamental mode of a fixed-fixed string, the two ends are nodes and the midpoint is an antinode, so exactly half a wavelength fits on the string. Therefore L = λ/2, giving λ = 2L.

Question 34Physics

A wire of total resistance 12 Ω is bent into a closed equilateral triangle. The resistance measured between any two vertices of the triangle is

  1. 8/3 Ω Correct answer.
  2. 6 Ω
  3. 3 Ω
  4. 4 Ω

Explanation

Each side of the equilateral triangle has resistance 12/3 = 4 Ω. Between two chosen vertices, the direct side of 4 Ω is in parallel with the other two sides in series (8 Ω), giving (4 × 8)/(4 + 8) = 8/3 Ω.

Question 35Physics

A source emitting sound of frequency 500 Hz moves directly towards a stationary observer with a speed of 40 m/s. Taking the speed of sound in air as 340 m/s, the frequency heard by the observer is closest to

  1. 500 Hz
  2. 625 Hz
  3. 567 Hz Correct answer.
  4. 447 Hz

Explanation

For a source approaching a stationary observer, the observed frequency is f' = f v/(v − u) with v the speed of sound and u the source speed. Substituting, f' = 500 × 340/300 ≈ 567 Hz, higher than the emitted frequency as expected.

Question 36Physics

Two thin concentric spherical shells of radii R and 2R carry charges +Q and −Q respectively, uniformly distributed. The electric potential at their common centre is

  1. kQ/(2R) Correct answer.
  2. kQ/R
  3. −kQ/(2R)
  4. Zero

Explanation

A uniformly charged shell of radius a contributes kQ/a to the potential at its own centre. The inner shell therefore contributes kQ/R and the outer shell contributes −kQ/(2R), whose sum is kQ/R − kQ/(2R) = kQ/(2R).

Question 37Physics

A light ray travelling in glass of refractive index 5/3 strikes a plane glass–water interface, the water having refractive index 4/3. The critical angle for total internal reflection inside the glass is

  1. 60°
  2. 37°
  3. 45°
  4. 53° Correct answer.

Explanation

Total internal reflection begins when the refracted ray grazes the interface, so sin C = n_water/n_glass = (4/3)/(5/3) = 4/5. Hence C = sin⁻¹(0.8) ≈ 53°.

Question 38Physics

The activity of a radioactive sample is 8000 Bq at t = 0 and 1000 Bq at t = 90 min. The half-life of the nuclide is

  1. 20 min
  2. 45 min
  3. 30 min Correct answer.
  4. 15 min

Explanation

The activity falls by a factor of 8000/1000 = 8 = 2³ in 90 minutes, so exactly three half-lives have elapsed. Therefore the half-life is 90/3 = 30 min.

Question 39Physics

One mole of an ideal monatomic gas is taken quasistatically through a process in which P = kV², where k is a positive constant and V is the volume. The molar heat capacity of the gas in this process is

  1. 13R/6
  2. 11R/6 Correct answer.
  3. 3R/2
  4. 2R

Explanation

With P = kV², the ideal-gas law gives kV³ = RT, so V ∝ T^(1/3) and dV/dT = V/(3T). Then C = C_V + P dV/dT = 3R/2 + (RT/V)(V/3T) = 3R/2 + R/3 = 11R/6.

Question 40Physics

A solid sphere is released from rest on a rough inclined plane of inclination θ and rolls down without slipping. The smallest coefficient of static friction for which rolling without slipping is maintained is

  1. (5/7)tanθ
  2. tanθ
  3. (2/7)tanθ Correct answer.
  4. (2/5)tanθ

Explanation

For a solid sphere rolling down an incline, the required friction is f = (I/(I + mR²)) mg sinθ = (2/7) mg sinθ, while the normal force is N = mg cosθ. Rolling is sustained provided μ ≥ f/N = (2/7)tanθ.

Question 41Physics

A parallel-plate capacitor is charged by a battery, the battery is then disconnected, and a dielectric slab of relative permittivity εr > 1 is inserted so that it fills the space between the plates. Which quantity remains unchanged?

  1. The charge on the plates Correct answer.
  2. The electric field between the plates
  3. The potential difference between the plates
  4. The energy stored in the capacitor

Explanation

With the battery disconnected, there is no conducting path by which free charge can leave or enter the plates, so the plate charge Q stays fixed. The dielectric increases the capacitance C, so V = Q/C decreases, the field E decreases, and the stored energy U = Q²/(2C) decreases.

Question 42Physics

A proton moves with velocity v parallel to a uniform magnetic field B. The magnetic force on the proton is

  1. zero, since v is parallel to B Correct answer.
  2. equal to qvB and perpendicular to the velocity
  3. equal to qvB and along the velocity
  4. equal to qvB and opposite to the velocity

Explanation

The magnetic force is F = qv × B, whose magnitude is qvB sinθ, where θ is the angle between v and B. Here θ = 0°, so sinθ = 0 and the force vanishes. A nonzero magnetic force would also have to be perpendicular to both v and B.

Question 43Physics

An ideal gas enclosed in an insulated vessel undergoes free expansion into an evacuated chamber. For the gas, which statement is correct?

  1. Its temperature and internal energy remain constant. Correct answer.
  2. Its internal energy increases while its temperature stays constant.
  3. Its temperature rises because the gas does negative work.
  4. Its temperature falls because the gas does positive work.

Explanation

The vessel is insulated, so the heat exchange is zero, and expansion into vacuum means the gas does no work against external pressure. Thus ΔU = Q − W = 0, so the internal energy of the ideal gas remains constant. For an ideal gas, internal energy depends only on temperature, so the temperature also remains constant.

Question 44Physics

A uniform rod of mass m and length L is pivoted at one end and released from rest in the horizontal position. What is its angular speed when it reaches the vertical position?

  1. √(3g/2L)
  2. √(g/L)
  3. √(2g/L)
  4. √(3g/L) Correct answer.

Explanation

Mechanical energy is conserved about the pivot. The centre of mass descends by L/2, so mgL/2 = 1/2 Iω², with I = mL²/3 for a rod about one end. Solving gives ω² = 3g/L.

Question 45Physics

An electric dipole of moment p is oriented parallel to a uniform electric field E. The work done by an external agent to rotate it to the antiparallel orientation is

  1. pE
  2. zero
  3. 2pE Correct answer.
  4. −2pE

Explanation

The dipole potential energy in a uniform field is U = −p·E. Initially U = −pE, and finally U = +pE when the dipole is antiparallel to the field. The external work is the change in potential energy, so W = 2pE.

Question 46Physics

A prism has refracting angle 60°. At minimum deviation, the angle of deviation is 30°. The refractive index of the prism material is

  1. √(3/2)
  2. √2 Correct answer.
  3. 2/√3
  4. √3

Explanation

At minimum deviation, the prism formula is n = sin[(A + δm)/2] / sin(A/2). Substituting A = 60° and δm = 30° gives n = sin45°/sin30° = (√2/2)/(1/2) = √2.

Question 47Physics

In a photoelectric-effect experiment, the stopping potential V0 is plotted against the frequency ν of incident light for a given metal. The slope of the straight-line graph is

  1. e/h
  2. φ/e
  3. h
  4. h/e Correct answer.

Explanation

Einstein's photoelectric equation gives eV0 = hν − φ, where φ is the work function. Dividing by e gives V0 = (h/e)ν − φ/e, so the graph of V0 versus ν is a straight line of slope h/e. The intercept is related to the work function, not the slope.

Question 48Physics

A series LCR circuit is driven at its resonant angular frequency. The power factor of the circuit is

  1. zero
  2. 1 Correct answer.
  3. 1/√2
  4. 1/2

Explanation

At resonance, the inductive and capacitive reactances cancel, so the impedance is purely resistive and Z = R. The phase angle between voltage and current is therefore zero, and the power factor cosφ equals 1.

Question 49Physics

Two blocks of masses m and 2m hang from a light string passing over a pulley modelled as a uniform solid cylinder of mass m and radius R. The string does not slip on the pulley. If the blocks are released from rest, the magnitude of their acceleration is

  1. 2g/7 Correct answer.
  2. 2g/5
  3. g/2
  4. g/3

Explanation

For the 2m block, 2mg − T2 = 2ma; for the m block, T1 − mg = ma. The pulley obeys (T2 − T1)R = Iα, with I = mR²/2 and α = a/R, so T2 − T1 = ma/2. Adding the equations gives mg = (7/2)ma, hence a = 2g/7.

Question 50Physics

A horizontal conducting rod of length L and mass m slides without friction on two parallel conducting rails separated by L. The rails are connected through a resistor R, and a uniform vertical magnetic field B points upward. The rod is given an initial horizontal speed v0. What is the total distance travelled before it stops?

  1. B² L² v0/(m R)
  2. m v0 R/(B² L²) Correct answer.
  3. m v0/(B² L² R)
  4. B² L²/(m v0 R)

Explanation

The motional emf is BLv, so the current is BLv/R and the retarding magnetic force is B²L²v/R. Thus m dv/dt = −(B²L²/R)v, giving an exponentially decaying speed. The total distance is ∫v dt = m v0 R/(B²L²).