1

FOUNDATION STUDIES

EXAMINATIONS

September 2007

PHYSICS

First Paper

July Fast Track

Time allowed 1.5 hour for writing10 minutes for reading

This paper consists of 4 questions printed on 6 pages.PLEASE CHECK BEFORE COMMENCING.

Candidates should submit answers to ALL QUESTIONS.

Marks on this paper total 60 marks, and count as 15% of the subject.

Start each question at the top of a new page.

2

INFORMATION

a · b = ab cos θ

a× b = ab sin θ c =

∣∣∣∣∣∣i j kax ay az

bx by bz

∣∣∣∣∣∣v ≡ dr

dta ≡ dv

dt

v =∫

a dt r =∫

v dt

v = u + atx = ut + 1

2at2

v2 = u2 + 2ax

a = −gjv = u− gtjr = ut− 1

2gt2j

s = rθ v = rω a = ω2r = v2

r

p = mv

N1 : if∑

F = 0 then δp = 0N2 :

∑F = ma

N3 : FAB = −FBA

W = mg Fr = µR

g =acceleration due to gravity=10m s−2

τ ≡ r× F∑Fx = 0

∑Fy = 0

∑τP = 0

W ≡∫ r2r1

F dr W = F · s

KE = 12mv2 PE = mgh

P ≡ dWdt

= F · v

F = kx PE = 12kx2

dvve

= −dmm

vf − vi = ve ln( mi

mf)

F = |vedmdt|

F = k q1q2

r2 k = 14πε0

≈ 9× 109 Nm2C−2

ε0 = 8.854× 10−12 N−1m−2C 2

E ≡limδq→0

(δFδq

)E = k q

r2 r

V ≡ Wq

E = −dVdx

V = k qr

Φ =∮

E · dA =∑

qε0

C ≡ qV

C = Aεd

E = 12

q2

C= 1

2qV = 1

2CV 2

C = C1 + C2 + C3

1C

= 1C1

+ 1C2

+ 1C3

R = R1 + R2 + R3

1R

= 1R1

+ 1R2

+ 1R3

V = IR V = E − IR

P = V I = V 2

R= I2R

K1 :∑

In = 0K2 :

∑(IR′s) =

∑(EMF ′s)

F = q v ×B dF = i dl×B

F = i l×B τ = niA×B

v = EB

r = mq

EBB0

r = mvqB

T = 2πmBq

KEmax = R2B2q2

2m

dB = µ0

4πidl×r

r2∮B · ds = µ0

∑I µ0 = 4π×10−7 NA−2

φ =∫

areaB · dA φ = B ·A

ε = −N dφdt

ε = NABω sin(ωt)

3

f = 1T

k ≡ 2πλ

ω ≡ 2πf v = fλ

y = f(x∓ vt)

y = a sin k(x− vt) = a sin(kx− ωt)= a sin 2π(x

λ− t

T)

P = 12µvω2a2 v =

√Fµ

s = sm sin(kx− ωt)

∆p = ∆pm cos(kx− ωt)

I = 12ρvω2s2

m

n(db′s) ≡ 10 log I1I2

= 10 log II0

where I0 = 10−12 W m−2

fr = fs

(v±vr

v∓vs

)where v ≡ speed of sound = 340 m s−1

y = y1 + y2

y = [2a sin(kx)] cos(ωt)

(m = 0, 1, 2, 3, 4, ....)

N : x = m(λ2) AN : x = (m + 1

2)(λ

2)

y = [2a cos(ω1−ω2

2)t] sin(ω1+ω2

2)t

fB = |f1 − f2|

y = [2a cos(k∆2

)] sin(kx− ωt + k∆2

)

∆ = d sin θ

Max : ∆ = mλ Min : ∆ = (m + 12)λ

I = I0 cos2(k∆2

)

E = hf c = fλ

KEmax = eV0 = hf − φ

E2 = p2c2 + (m0c2)2

E = m0c2 E = pc

λ = hp

(p = m0v (nonrelativistic))

∆x∆px ≥ hπ

∆E∆t ≥ hπ

dNdt

= −λN N = N0 e−λt

R ≡ |dNdt| T 1

2= ln 2

λ= 0.693

λ

MATH:

ax2 + bx + c = 0 → x = −b±√

b2−4ac2a

y dy/dx∫

ydx

xn nx(n−1) 1n+1

xn+1

ekx kekx 1kekx

sin(kx) k cos(kx) − 1k

cos kxcos(kx) −k sin(kx) 1

ksin kx

where k = constant

Sphere: A = 4πr2 V = 43πr3

CONSTANTS:

1u = 1.660× 10−27 kg = 931.50 MeV1eV = 1.602× 10−19 Jc = 3.00× 108m s−1

h = 6.626× 10−34 Jse ≡ electron charge = 1.602× 10−19 C

particle mass(u) mass(kg)

e 5.485 799 031× 10−4 9.109 390× 10−31

p 1.007 276 470 1.672 623× 10−27

n 1.008 664 904 1.674 928× 10−27

PHYSICS: First Paper. July Fast Track 2007 4

Question 1 ( 15 marks):

The total power output, P (W = kgm2s−3), of a wind generator depends on the

total area, A (m2), swept out by the blades, and the velocity, v (ms−1), and density,

ρ (kg m−3), of the wind, at the blades.

Use dimensions to derive an expression for power, P , in terms of A, v, and ρ.

Question 2 ( 4 + 11 = 15 marks):

M

mµ

aT

Figure 1:

A block, of mass m, is coupled to a second block, of mass M , by means of a pulley

system, as depicted in Figure 1. When released from rest, M falls vertically, pulling

m along the horizontal surface. The coefficient of friction between block m and the

horizontal surface is µ. Take the acceleration of block m, to be a. The pulley system

has negligible mass and friction.

(i) Draw a diagram of each of the blocks, of masses, m and M . Label each block

with all of the forces that act individually upon it. Label each block with its acceleration.

(ii) Use Newton’s laws of motion to derive an expression, in terms of m, M , µ, and

the acceleration of gravity, g, for the horizontal acceleration, a, of the block of mass m.

PHYSICS: First Paper. July Fast Track 2007 5

H

8 kN

E

T

5 m

5 m

C8 m

horizontal

6 m

2 kN

Figure 2:

Question 3 ( 4 + 11 = 15 marks):

Figure 2 shows a crane, lifting a body of weight 8 kN. The main beam, HE, of the crane

has a length of 10 m, a total weight of 2 kN, and is hinged at end H. It is held at the

angle illustrated, by a horizontal cable, that attaches to centre point, C, of the beam.

Other relevant dimensions of the system are labeled.

(i) Draw a diagram to show all the forces that act on the beam, HE.

(ii) Use the conditions for equilibrium of beam, HE, to calculate the horizontal

tension force, T, in the cable, attached to point C. Calculate also, the horizontal and

vertical components (Rx and Ry) of the reaction of the hinge at H, on the end of the beam.

PHYSICS: First Paper. July Fast Track 2007 6

CCrest

kRunstretched

(a) (b)

M

M

vB B

RR

T

Figure 3:

Question 4 ( 2 + 13 = 15 marks):

Figure 3 shows a massless rod of length, 2R, attached, at its centre point, to a fixed

pivot, C. A ball of mass, M , is attached at one end of this rod, while the other end is

connected to a massless spring, of spring constant, k. This spring’s other end is attached

to fixed base, at point B.

Initially the rod is held horizontally at rest, as shown in Figure 3 (a). In this position,

the spring is unstretched. When the rod is released, the mass, M , falls, rotating the rod

anticlockwise, about pivot, C, and stretching the spring. Figure 3 (b) shows the rod in

the vertical position.

(i) Prove that the extended length of the spring from B to T in Figure 3 (b)

= BT =√

5R.

(ii) Use energy principles, and the result BT =√

5R, to derive an expression for the

speed, v, of mass, M , in terms of M , k, R, and the acceleration of gravity, g, at the

moment when the rod becomes vertical (Figure 3 (b)).

END OF EXAM

1

FOUNDATION STUDIES

EXAMINATIONS

January 2008

PHYSICS

Final Paper

July Fast Track

Time allowed 3 hours for writing10 minutes for reading

This paper consists of 6 questions printed on 13 pages.PLEASE CHECK BEFORE COMMENCING.

Candidates should submit answers to ALL QUESTIONS.

Marks on this paper total 120 marks, and count as 45% of the subject.

Start each question at the top of a new page.

PHYSICS: Final Paper. July Fast Track 2007 4

3 m

4 m

5 m P

Q

R

x

z

y

Figure 1:

Question 1 ( (4 + 6) + (10) = 20 marks):

Part (a):

Figure 1 shows a rectangular box with sides of lengths 3 m, 4 m and 5 m, as labeled. A

triangle, PQR, has been drawn within this box. You are given, that the vector area, A,

of the triangle is given by -

A = 12(−→PQ×

−→PR) (cross product)

where−→PQ and

−→PR are displacements along the sides of the triangle.

(i) Express each of the two displacements−→PQ and

−→PR in terms of the ijk unit vectors.

(ii) Find an expression for the vector area A, in terms of the ijk unit vectors.

PHYSICS: Final Paper. July Fast Track 2007 5

M

M

M

m

m

m

(a)

(b)

(c)

vu

x xrest rest

rest restk uncompressed

Figure 2:

Part (b):

Two blocks, of masses, m and M , on a frictionless horizontal surface, are brought

together against a spring, of spring constant, k, and negligible mass, which rests against

their adjacent faces. In Figure 2(a), the blocks first touch the spring. At this stage,

the spring is uncompressed. In Figure 2(b) the blocks has been moved closer together,

thereby compressing the spring a total distance of 2x. The blocks are then released and

are pushed apart by the spring. Figure 2(c) shows the blocks after they have separated

from the spring, and have velocities, u and v, as labeled. There is no energy left in the

spring.

Use momentum and energy principles, to derive expressions for the velocities, u and v,

of the two blocks, in terms of k, m, M , and x.

PHYSICS: Final Paper. July Fast Track 2007 6

R

ω

y

x

mµ

Figure 3:

Question 2 ( (2 + 8) + (3 + 7) = 20 marks):

Part (a):

An experiment is designed to ”stick” a human to a vertical wall, by friction. To do this,

the human, of mass, m, is placed against the inside wall of a hollow cylinder, of inside

radius, R, rotating at an angular velocity of ω, with the y-axis as its spin axis, as shown

in Figure 3.

(i) Draw a diagram showing all of the forces that act on the human. Also label the

human’s acceleration.

(ii) If the cylinder spins too slowly, the human will slip. Use Newton’s laws of

motion, to find an expression, in terms of R, µ, and the acceleration of gravity, g, for

the minimum angular velocity, ω0, to make the human ”stick”.

PHYSICS: Final Paper. July Fast Track 2007 7

Part (b):

Figure 4:

Figure 4 shows A square sign, EFGH, of mass M = 200 kg, hinged to a steep hillside at

H. The sign is held with side HE vertical, by a horizontal cable, attached to the hillside at

D. The dimensions of the figure are illustrated. Take acceleration of gravity g = 10 ms−2.

(i) Draw a carefully labeled diagram, showing all the forces that act on the sign.

(ii) Using the conditions for the equilibrium of the sign, find the tension, T , in the

cable, and the reaction of the hinge, at H, on the sign.

PHYSICS: Final Paper. July Fast Track 2007 12

S

M

µP f

Figure 9:

Question 5 ( (3 + 3 + 4) + (5 + 5) = 20 marks):

Part (a):

Figure 9 shows a string, of mass per unit length, µ = 50 gram m−1, that passes from a

wave source, S, via a pulley, to a suspended mass, M = 8 kg. The source, S, has power,

P = 700 W , and transmits a wave of frequency, f = 60 Hz, to the string. Take the

acceleration due to gravity = g = 10 ms−2.

(i) Calculate the speed, v, of the wave along the string.

(ii) Calculate the amplitude of the wave as it passes along the string.

(iii) Write down a possible numerical wave function for the wave on the string.

Part (b):

Copper has a photoelectric work function, φ = 4.70 eV .

(i) Calculate the threshold wavelength for copper.

(ii) A photoelectric cell has a copper photoelectrode. Calculate the stopping potential

for this cell, when it is illuminated by light whose wavelength is 5 % shorter than the

threshold wavelength for copper.