Question
(a) Define density. [1]
(b) A smooth pebble, made from uniform rock, has the shape of an elongated sphere as shown
in Fig. 1.1.
The length of the pebble is L. The cross-section of the pebble, in the plane perpendicular to L,
is circular with a maximum radius r.
A student investigating the density of the rock makes measurements to determine the values
of L, r and the mass M of the pebble as follows:
L = (0.1242 ± 0.0001)m
r = (0.0420 ± 0.0004)m
M = (1.072 ± 0.001)kg.
(i) State the name of a measuring instrument suitable for making this measurement of L. [1]
(ii) Determine the percentage uncertainty in the measurement of r.
percentage uncertainty = …………………………………………….. % [1]
(c) The density ρ of the rock from which the pebble in (b) is composed is given by
where n is an integer and k is a constant, with no units, that is equal to 2.094.
(i) Use SI base units to show that n is equal to –2. [2]
(ii) Calculate the percentage uncertainty in ρ.
percentage uncertainty = …………………………………………….. % [3]
(iii) Determine ρ with its absolute uncertainty. Give your values to the appropriate number of
significant figures.
ρ = ( ……………………………….. ± ……………….) kgm–3 [3]
[Total: 11]
Answer/Explanation
Ans
(a) mass / volume
(b) (i) (vernier/digital) calipers
(b) (ii) percentage uncertainty = (0.0004 / 0.0420) × 100
= 1%
(c) (i) kg m–3 = kg × mn / m or kg m–3 = kg × mn × m–1
–3 =n – 1 and (so) n = –2
(c) (ii) (Δρ /ρ) = (ΔM / M) + 2(Δr / r) + (ΔL / L)
percentage uncertainty = [(0.001 / 1.072) + 2 × (0.0004 / 0.0420) + (0.0001 / 0.1242)] ( × 100)
= 0.09% + 2 × 0.95% + 0.08%
= 2%
(c) (iii) ρ = (1.072 × 0.0420–2) / (2.094 × 0.1242)
= 2337 (kg m–3)
∆ρ = 0.021 × 2337
= 49 (kgm–3)
ρ = (2340 ± 50) kgm–3
Question
(a) Define momentum. [1]
(b) Two balls X and Y, of equal diameter but different masses 0.24kg and 0.12kg respectively,
slide towards each other on a frictionless horizontal surface, as shown in Fig. 2.1.
Both balls have initial speed 2.3 m/s before they collide with each other. Fig. 2.2 shows the
variation with time t of the force FY exerted on ball Y by ball X during the collision.
(i) Calculate the kinetic energy of ball X before the collision.
kinetic energy = ……………………………………………… J [3]
(ii) The area enclosed by the lines and the time axis in Fig. 2.2 represents the change in
momentum of ball Y during the collision.
Determine the magnitude of the change in momentum of ball Y.
change in momentum = …………………………………………… Ns [2]
(iii) Calculate the magnitude of the velocity of ball Y after the collision.
velocity = ………………………………………… ms–1 [2]
(c) On Fig. 2.3, sketch the variation with time t of the force \(F_X\) exerted on ball X by ball Y during
the collision in (b)
[3]
[Total: 11]
Answer/Explanation
Ans
(a) mass × velocity
(b) (i) kinetic energy = ½mv2
= ½ × 0.24 × 2.32
= 0.63 J
(b) (ii) change in momentum = ½ × 240 × 5.0 × 10–3
= 0.60N s
(b) (iii) (change in velocity of Y) = 0.60 / 0.12
( = 5.0 ms–1)
final velocity of Y = 5.0 – 2.3
= 2.7 ms–1
or
(final momentum of Y) = 0.60 – 0.12 × 2.3
( = 0.324 Ns)
final velocity of Y = 0.324 / 0.12
= 2.7 ms–1
(c) sloping straight line from (0, 0) to t = 3.0ms and another straight line continuous with the first from t = 3.0ms to (5.0, 0)
lines showing maximum force of magnitude 240 N
lines wholly in the negative F region of the graph
Question
(a) A uniform metal bar, initially unstretched, has sides of length w, x and y, as shown in Fig. 3.1.
The bar is now stretched by a tensile force F applied to the shaded ends. The changes in
the lengths x and y are negligible. The bar now has sides of length x, y and z, as shown in
Fig. 3.2.
Determine expressions, in terms of some or all of F, w, x, y and z, for:
(i) the stress σ applied to the bar by the tensile force
σ = ………………………………………………… [1]
(ii) the strain ε in the bar due to the tensile force
ε = ………………………………………………… [1]
(iii) the Young modulus E of the metal from which the bar is made.
E = ………………………………………………… [2]
(b) A copper wire is stretched by a tensile force that gradually increases from 0 to 280N. The
variation with extension of the tensile force is shown in Fig. 3.3.
(i) State the maximum extension of the wire for which it obeys Hooke’s law.
extension = ………………………………………….. mm [1]
(ii) Use Fig. 3.3 to determine the strain energy in the wire when the tensile force is 120N.
strain energy = ……………………………………………… J [3]
(iii) Explain why the work done in stretching the wire to an extension of 12mm is not equal to
the energy recovered when the tensile force is removed. [2]
[Total: 10]
Answer/Explanation
(a) (i) σ = F / xy
(a) (ii) ε = (z –w) / w
(a) (iii) E =σ / ε
= Fw / xy(z–w)
(b) (i) extension = 2.2 mm (allow 2.0 –2.4 mm)
(b) (ii) strain energy = area under graph/line or ½Fx or ½kx2
= ½ × 120 × 1.4 × 10–3 or ½ × 8.6 × 104 × (1.4 × 10–3)2
= 0.084 J
(b) (iii) (some of the) deformation of the wire is plastic/permanent/not elastic
or
wire goes past the elastic limit/enters plastic region
energy (that cannot be recovered) is dissipated as thermal energy/becomes internal energy
Question
(a) By reference to the direction of transfer of energy, state what is meant by a longitudinal wave. [1]
(b) A vehicle travels at constant speed around a wide circular track. It continuously sounds its
horn, which emits a single note of frequency 1.2kHz. An observer is a large distance away
from the track, as shown in the view from above in Fig. 4.1.
Fig. 4.2 shows the variation with time of the frequency f of the sound of the horn that is
detected by the observer. The time taken for the vehicle to travel once around the track is T.
(i) Explain why the frequency of the sound detected by the observer is sometimes above
and sometimes below 1.2kHz. [2]
(ii) State the name of the phenomenon in (b)(i). [1]
(iii) On Fig. 4.1, mark with a letter X the position of the vehicle when it emitted the sound that
is detected at time T. [1]
(iv) On Fig. 4.1, mark with a letter Y the position of the vehicle when it emitted the sound that
is detected at time \( \frac{9T}{4}\) [1]
(c) The speed of the sound in the air is 320ms–1.
Use Fig. 4.2 to determine the speed of the vehicle in (b).
speed = ………………………………………… ms–1 [3]
[Total: 9]
Answer/Explanation
Ans
(a) oscillations (of particles) are parallel to (the direction of) energy transfer
(b) (i) (frequency varies as) vehicle moves relative to (stationary) observer
(vehicle) moving towards (observer) gives higher (observed) frequency (than 1.2 kHz) and (vehicle) moving away (from
observer) gives lower (observed) frequency (than 1.2 kHz)
(b) (ii) Doppler effect
(b) (iii) position of vehicle labelled ‘X’ at top (12 o’clock) position on track
(b) (iv) position of vehicle labelled ‘Y’ at right-hand edge (3 o’clock) position on track B1
(c) maximum frequency = 1.40 (kHz) or 1.40 × 103 (Hz)
1.40 = (1.2 × 320) / (320 –v)
v = 46ms–1
or
minimum frequency = 1.05 (kHz) or 1.05 × 103 (Hz)
1.05 = (1.2 × 320) / (320 + v)
v = 46ms–1
Question
(a) State Kirchhoff’s first law. [2]
(b) The circuit shown in Fig. 5.1 contains a battery of electromotive force (e.m.f.) E and negligible
internal resistance connected to four resistors R1, R2, R3 and R4, each of resistance R.
The current in R3 is 0.30A and the potential difference (p.d.) across R4 is 2.4V.
(i) Show that R is equal to 4.0Ω. [2]
(ii) Determine the e.m.f. E of the battery.
E = ……………………………………………… V [2]
(c) The battery in (b) is replaced with another battery of the same e.m.f. E but with an internal
resistance that is not negligible.
State and explain the change, if any, in the total power produced by the battery. [2]
(d) The resistors in the circuit of Fig. 5.1 are made from nichrome wire of uniform radius 240μm.
The length of this wire needed to make each resistor is 0.67m.
Calculate the resistivity of nichrome.
resistivity = ………………………………………….. Ωm [3]
[Total: 11]
Answer/Explanation
(a) sum of current(s) in = sum of current(s) out
or
(algebraic) sum of current(s) is zero
at a junction (in a circuit) A1
(b) (i) (current in R4 or R1 =) 0.30 + 0.30
(= 0.60A)
(R =) 2.4 / 0.60 = 4.0 (Ω) A1
or
(p.d. across R3 or R2 =) 2.4 / 2
(= 1.2 V)
(R =) 1.2 / 0.30 = 4.0 (Ω)
(b) (ii) E = 2.4 + 2.4 + 1.2 C1
= 6.0 V A1
or
total resistance = 10 (Ω)
E = 10 × 0.60 = 6.0 V
(c) total resistance increases
current decreases (in battery) so total power decreases
(d) resistivity = RA / L
= 4.0 × π × (240 × 10–6)2 / 0.67
= 1.1 × 10–6 Ω m
Question
(a) Complete Table 6.1 to show the masses (in terms of the unified atomic mass unit u) and
charges (in terms of the elementary charge e) of α, β+ and β– particles.
(b) Carbon-14 is radioactive and decays by emission of β– particles.
(i) Nuclei do not contain β– particles.
Explain the origin of the β– particle that is emitted from the nucleus during β– decay. [1]
(ii) State the change in the quark composition of a carbon-14 nucleus when it emits a
β– particle. [1]
(iii) Suggest why the β– particles are emitted with a range of different energies. [2]
[Total: 8]
Answer/Explanation
Ans
6 (a) α-particle mass given as 4u
α-particle charge given as (+)2e
both β-particles mass given as 0.0005u B1
β+ charge given as (+)e and β– charge given as –e
6 (b) (i) neutron decays into proton and an electron / β– particle
6 (b) (ii) down to up
6 (b) (iii) (electron) antineutrino(s) emitted
energy (released in decay)/momentum shared between antineutrino and β– particle