foundation studies examinations …flai/theory/exams/ft09_1.pdffoundation studies examinations...

1

FOUNDATION STUDIES

EXAMINATIONS

September 2009

PHYSICS

First Paper

July Fast Track

Time allowed 1.5 hour for writing10 minutes for reading

This paper consists of 4 questions printed on 7 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

dtv =

∫a dt r =

∫v dt

v = u + at a = −gjx = ut + 1

2at2 v = u− gtj

v2 = u2 + 2ax r = ut− 12gt2j

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 + C21C

= 1C1

+ 1C2

R = R1 + R21R

= 1R1

+ 1R2

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)

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)

3

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)

N : x = m(λ2) AN : x = (m + 1

2)(λ

2)

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

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

L ≡ r× p = r×mv

L = rmv = n( h2π

)

δE = hf = Ei − Ef

rn = n2( h2

4π2mke2 ) = n2a0

En = −ke2

2a0( 1

n2 ) = −13.6n2 eV

1λ

= ke2

2a0hc( 1

n2f− 1

n2i) = RH( 1

n2f− 1

n2i)

(a0 = Bohr radius = 0.0529 nm)

(RH = 1.09737× 107 m−1)

(n = 1, 2, 3....) (k ≡ 14πε0

)

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 2009 4

k

m

Question 1 ( 15 marks):

The figure above shows a block of mass, m (kilogram) suspended by a spring, of spring

constant, k (Newton/metre), from a fixed beam. When the block is pulled down and

released, it oscillates vertically, with a period of P (second). You are given, that P may

depend on m, k, and the acceleration due to gravity, g.

Use dimensions to derive an expression for P , in terms of m, k, and g.


Question 2 ( (6 + 6 + 3) = 15 marks):

mµ

M

a

T

The Figure above shows two blocks, of masses m and M , connected by a string and

pulley system. The strings and the pulleys are of negligible mass, and the pulleys have

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

on which it slides is µ. The acceleration of block m is a, and the tension in the string

attached to it is T , as labeled. The system is released from rest.

(i) Draw a diagram of each block, labeling all particular forces that act upon it.

Label also, the acceleration of each block.

(ii) Write down Newton’s equation of motion for each block, in both the vertical, and

horizontal directions (three equations).

(iii) Hence derive an expression for the horizontal acceleration, a, of block m, in

terms of the parameters labeled in the Figure, and the acceleration of gravity, g.


wheel

ladder

M

A

Bµ

6 m

8 m

(µ = 0)

wall

floor

ladder about to slip

10 m

Question 3 ( (4 + 9 + 2) = 15 marks):

The above Figure shows a uniform ladder, AB, of length 10 m, and mass M , which

leans between a vertical wall, and the floor. There is a frictionless wheel at end A of

the ladder, so that there is zero friction between the wall and the ladder at that end.

At end B, however, the coefficient of friction between the ladder and the floor is µ. The

ladder is at the point of slipping, when end A of the ladder is 8 m above the floor, and

end B of the ladder is 6 m horizontally from the wall.

(i) Draw a diagram of the ladder, and label all the forces that act upon it.

(ii) Write down the equations for equilibrium of the ladder (three equations).

(iii) Hence find the value of the coefficient of friction, µ, between the ladder and the

floor.


r

x

y

z

3 m

4 m

3 m

5 N10 N

8 N

P

Q

Question 4 ( 6 + 2 + 7 = 15 marks):

The figure above shows a rectangular box of labeled dimensions, aligned along the x−,y−, and z− axes. Forces of 5 N , 8 N and 10 N , act at corner P of the box, in thedirections indicated.

(i) Write expressions for each of the three forces, in terms of the ijk unit vectors, andhence find their vector sum, F.

(ii) Express the position vector, r, of point P, in terms of the ijk unit vectors.

(iii) Calculate the torque, τ of total force F, about the origin, O, given that -

τ = r× F (cross product)

END OF EXAM

ANSWERS:

Q1. P = C√

mk, where k ≡ dimensionless const.

Q2. (ii) R−mg = 0 , T − µR = ma , Mg − 4T = M a4

; (iii) a = 4g(M−4µmM+16m

) .

Q3. (ii) N − µR = 0 , R−mg = 0 , 3mg − 8N = 0; (iii) µ = 38.

Q4. (i) F = −8i− 8j− 11k N ; (ii) r = 4i + 3j + 3k m; (iii) τ = −9i + 20j− 8k Nm .

1

FOUNDATION STUDIES

EXAMINATIONS

December 2009

PHYSICS

Second Paper

July Fast Track

Time allowed 1.5 hour for writing10 minutes for reading





PHYSICS: Second Paper. July Fast Track 2009 4

m M

M

m

2R

Ox x

y y

v

v

rest rest(a) (b)

O

Figure 1:

Question 1 ( 15 marks):

Figure 1(a) shows two balls, of masses, m and M (M > m), connected at the ends of a

straight rod, of negligible mass, and length, 2R. Initially, the rod is horizontal, aligned

with the x-axis, and pivoted at its centre, O. When the rod is released from rest, M

falls, while m rises, until the rod is vertical, and aligned along the y-axis, as depicted in

Figure 1(b). At this stage, each of the two balls is moving with a velocity, v. There is

negligible friction at the pivot.

Use energy principles to derive an expression for the velocity, v, in terms of M , m, R,

and the acceleration due to gravity, g.

PHYSICS: Second Paper. July Fast Track 2009 5

rest

restx

4 m/s

y

2MM

2M x

1 m/s

y

2M

M

2M

1 m/s

v

θ

Before After

α3 m

4 m

Figure 2:

Question 2 ( (11 + 4) = 15 marks):

One ball, of mass, M , moving with a velocity of 4 m/s in the +x-direction, strikes two

stationary balls, both of mass 2M , as shown in Figure 2. The x-y plane is horizontal.

After the collision. the M ball rebounds back along the x-axis with a velocity of 1 m/s,

while one of the 2M balls moves with a velocity of 1 m/s at an angle of α (where

tan α = 34) above the x-axis, as illustrated.

(i) Using momentum principles, find the velocity of the other ball, of mass 2M , after

the collision. Find both magnitude (v), and direction (θ).

(ii) Is this collision elastic? Show your reasoning.

1

FOUNDATION STUDIES

EXAMINATIONS

January 2010

PHYSICS

Final Paper

July Fast Track

Time allowed 3 hours for writing10 minutes for reading





PHYSICS: Final Paper. July Fast Track 2009 4

x

y

z

O

A

B

F = 10 N

3 m

2 m

4 m

r

Figure 1:

Question 1 ( (2 + 3 + 5) + (6 + 2 + 2) = 20 marks):

Part (a):

Figure 1 shows a rectangular box, with a corner at the origin, O, and with its sidesaligned along the x-,y- and z-axes. Dimensions of the box are labeled. A force of 10 Nacts at corner, A, of the box, in the direction of the diagonal, AB.

(i) Write down an expression for the position vector, r, of point A, in terms of unitvectors ijk.

(ii) Express force, F, in terms of unit vectors ijk.

(iii) Find the torque, τ , of force, F, on the box, about the origin, O as pivot. Giveyour answer in terms of ijk unit vectors. You are given that -

τ = r × F (cross product)


M

8 m

6 m

20 kg

10 m

C

E

H

F50 kg

Figure 3:

Question 2 ( (2 + 8) + (3 + 5 + 2) = 20 marks):

Part (a):

Figure 3 shows a lever, HE, of mass 20 kg, and length, 10 m, being used to lift a block

of mass, M = 50 kg. A massless cable runs horizontally from the centre, C, of the lever,

over a frictionless pulley, and vertically down to the block. The lever is hinged at end

H. A force, F , is exerted at end E, of the lever, via a vertical cable. Dimensions are

labeled. Take the acceleration due to gravity g = 10 ms−2.

(i) Draw a labeled diagram of the lever, showing all forces that act upon it, as the

block is just lifted.

(ii) Use conditions for equilibrium, to determine the value of the force, F , required to

just lift the block. Determine also, the reaction of the hinge at H on the lever, at this

stage.


y

disc

ω

3M

2M

µ

r

rotating

M

r

µ

Figure 4:

Part (b):

Figure 4 shows three blocks, of masses 3M , M and 2M , connected by a string, of

negligible mass, over a pulley, of negligible mass and friction. Blocks M and 2M rest

on the horizontal surface of a disc, at distances r and 2r along the same radius from its

centre. Block 3M rests on a horizontal surface, at the bottom of a hole in the centre of

the disc. The coefficient of static friction between each of blocks M and 2M , and the

surface of the disc is µ. The disc is rotated with a slowly increasing angular velocity, ω,

with the vertical y-axis as its spin axis. If ω continues to increase, blocks M and 2M

will eventually slip, on the surface of the disc.

(i) Draw a diagram of each block, labeling all particular forces that act on each, just

before blocks M and 2M slip. Label also, any acceleration of each block.

(ii) Write down Newton’s equation of motion for each block, in both the vertical, and

horizontal directions, just before blocks M and 2M slip (five equations).

(iii) Hence find the maximum angular velocity, ωm, at which the disc can spin, before

the blocks slip on the disc surface. Express ωm in terms of µ, r, and the acceleration of

gravity, g.


d

k rest

rest

m

M

rest

µ

Figure 5:

Question 3 ( (10) + (10) = 20 marks):

Part (a):

Figure 5 shows two blocks, of masses m and M , and a spring, of spring constant, k,

connected by a massless string, which passes over a massless, frictionless pulley between

the two blocks. The system is released from rest. At this stage the spring is unstretched.

Block, M , falls a vertical distance, d, before coming momentarily to rest, dragging block

m along the horizontal surface, with which it has a kinetic friction coefficient, µ, and

extending the spring.

Use energy principles to derive an expression for d, in terms of the parameters labeled

in Figure 5, and the acceleration due to gravity, g.


M 2M

2 m/s rest

Figure 6:

Part (b):

Figure 6 shows two balls, of masses M and 2M , that hang side-by-side just touching.

Ball of mass M is pulled aside and then released, so that it has a velocity of 2 m/s just

before it makes an inline elastic collision with the other stationary ball.

Use momentum and energy principles to determine the velocities of both balls, immedi-

ately after the collision.


Question 5 ( (2 + 2 + 3 + 3) + (4 + 3 + 3) = 20 marks):

Part (a):

A vibrator source, set to a frequency, f = 500 Hz, and an amplitude of a = 1.00 mm,

sends a transverse wave along a string, of mass per unit length, µ = 2 × 10−3 kg/m,

which is stretched to a tension of T = 10 N .

(i) Calculate the speed, v, with which the wave travels along the string.

(ii) Calculate the wavelength of the wave.

(iii) Write down a possible particular wave function for the wave.

(iv) What is the power, P , of the source of this wave?

Part (b):

The Paschen series of spectral lines for atomic hydrogen, are formed by electron

transitions terminating on the n = 3 energy level. Use the Bohr theory for the hydrogen

atom to answer the following questions.

(i) Calculate the longest and shortest wavelengths in this spectral series.

(ii) What is the total energy of an electron in the n = 3 level?

(iii) What is the radius of the orbit of an electron in the n = 3 level?

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