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Question 1
Vect°'s v1 , ~ . , v. are def1ned as. follows:
V ¡= k X j, V2 = ( i + j ) X (j + k), V3 = - k · (j + k) i. v, = i X ( j X k).
Of t hese vectOfs, the ones that are equal to - i are
(bl v,, v,
(e) v,, v,
(d) v3 , v,
Question 2
The accelerat1no of a vehicle is given by a = rl:,i rns- 2. lf rt starts with velocity 5 j rns-1.
after 4 seconds 1ts velocity in ms- 1 will be
(a) (5 - In 6) i
(b) ~ ¡
(e) 5j
(d) i i+ 5j
(e) ln3 i + Sj
Quest ion 3
A part,cle moves ,n the xy- plane lt starts from (O, O) and. alter a bme T, reaches the point
(O. y) . lts average velocity over the period t E [O. T ) is
(a) not possible to calculate from the infOfmation g,ven
(b) (y /T ) ms·'
(e) oí unknown direction, bul has magnitude y / T ms· •
(d) on the j -directoon, but the magnitu de depends on the path taken
(e) (y/T) j ms·•
Quest ion 4
The path, r (t), of a part1de of mass mis gIven by r( t ) = t3 i + t2 j - t k. The fOfce F actmg
oo th e partIcle is
[a) 3l 2 i +2l j - t k
(b) m,19t>+1
(e) 2m(3t i + j )
(d) m(6 i +2 j - k)
(e) m( t' i + t2 j - tk)/i'
Question 5
The drag, fd,,., on an obJect of mass m fall1ng through mr wrrh velocity v Is gIven by
r.,,,9 = - -rv. where 1 is a constant. Taking g = 9.8 m s· 2 , -r = 0.14 kg s· 1 and m = 0.2
kg. t l1e terminal speed of the obJect. 1.e.. maximum speed. v,, lt w1II attaIn when in vertical
free-fall. ,s
(a) v, = 14.0 m s·•
(b) v, = 15.8 m s 1
(e) v, = 17.1 m ,- 1
(d) v, = 18.0 m s·'
(e) v, = 24.4 m ,- 1
Question ti
A h0f 1zonta l force of magn11.udeF IS apphed to a box oí mass m at rest on a h0f1zontal rough
floor . The coefficienrs of stat Ic and k1net1c fr1ction are μ, and μ , respect1vely In Ofder to
start to move the box hOfizontally, we must have that
(a) F > μ•mg
(b) F $ μ,mg
(e) F > ¡,,g
(d) F > μ,mg
(e) F > mg
Question 7
At 1 = O, a pr0Ject1le Is f,red overa cfiff w,t h 1n1t1al velocIty u = u1 i + 1.1j:., Here. i , j are unlt
vectors point1ng, respectw ely, honzontally ªWil'I from the cliff edge, and vert ically upwards
T he pos1tion of the project:ile 15 gIven by the vectOI' r(t), with the di ff edge be1ng at r = O.
Neglectmg drag due to a1r res.istancet, he posmon of the pr0Jectl1e while 1t 1sm fl1ght 1s
(a) r( I) = u1 t i - (u,+ ½gt)t j
(b) r( c) =(u, + u,) ci - ½gr' j
(e) r(t) = u1t i
(d) r(t) = u1 t i - }gt 2 j
(e) r(t)= u1 t i +(u 2 -½gt)t j
Question 8
T he d,mensions of the SI unit of fo,ce. the Newton, ,n te<ms of mass (M), length (L) and
t11ne (T) are
(a) ML'T- 2
(b) ML T- 2
(e) (MLT) 2
(d) L'(M T) - 2
(e) ML T- 1
Question 9
lf f (x, y, z) = xy + yz + zx, then the total different ial, df, off is
(a) df = xdx + ydy + zdz
(b) df = (xy + yz + zx)dx dy dz
(e) df = xydx + yzdy + zxdz
(d) df =(y+z)dx+(x+z)dy+(Y+x)dz
Question 10
A constant ÍOl'ce of S N is applied to an object of mass 2 kg. initially at rest on a smooth,
homontal surface. After the ob¡ect has movec a d1stance of 3 m, 1ts kmetic energy, K E, and
speed, V, a,e
(a) KE = 30 J. v = ,/30 ms- 1
(b) KE = 15 J, v = ,/30 ms- I
(e) KE = 30 J. v = 6 m, - >
(d) KE = 15 J, V = Jis ms- I
(e) KE = S N, v = 10 ms- 1
Question 11
A block sl,des on a srnooth cul\/ed tra ck whose he,ght, y, depends only on the horizontal
d1splacement, x, v1a y= x + 2x 3. Take the accek:rat10n dueto grav1ty g = 9.8 ms- 2. lf the
block starts from rest at x = 3. ns speed. v. when x = 2 ,s apprmomately
(a) v = 21.7 ms- 1
(b) v = 14.6 ms-•
(e) v = 27.6 ms- 1
(d) v = 8.0 rns- 1
(e) v = 12.7 ms- 1
Question 12
In a 1-dimens,onal colhs,on, obJect A of mass mA = 2 kg mov,ng w1th speed uA = 4 ms- I
colhdes w,th obJect B of mass m8 = 4 kg which ,s ,n111ally at res!. The speeds of ob1ects A
and B after colhs1on al'e v,. and v8 respecnvely. T he equat,on that relates vA and v8 ,s
(a) 2 VA + Vs = O
(b) v,.+2vs= 4
(e) v,.= va
(d) -1 + 2v~ = 4
(e) ~ = 4
Question 13
In plane polar co-ordinates (r . 4'), the unlt vectors are r. $. The first derivatives w,th respect
to t111e1 t of ? and i/,a re
(a) ~= O d$ = 0
dt ' dt
(b) ~ = ~~ d$ = _ dq,,
dt rdt' dt dt tl>
dr l , d$ q,_
(e) dt = t i/>, dt = - t r
(d) ~ = i + J. !~ = - i + j
dr d,t, d$ d4',
(e) dt=--;¡¡ $ , --;¡¡= - dt '
Question 14
A force F = 2 i - 3 j + k ,s applted at a po,nt whose posit,on vector relat,ve to a hxed ongon
IS r = - i + 2j + 2 k. T he torque G 1s
(a) G = i - j + 3 k
(b) G = - i + j - 3 k
(e) G= -2 i - 6j +2 k
(d) G = -8 i - 5j + k
(e) G =B i + Sj - k
Question 15
The pos1t1on vector of a part ide as a funct 1on of time t 1s g,ven by r( t) = t2 i - r k. lts
~ngular velocity w at t = 1 ts
(a) ( i - k)/ ./2
(b) - i+ j + k
(e) - j/2
(d) ;J.( i + j+ k)
(e) i - k
Question 16
Of the follow1ng forces
F,( r) = 3x2y i+(x3+2yz 2 ) j +2y 2z k, F2(r) = 2y i+ 4z j -2x k, F3 (r) = x- 1 i+y- 1 j +z- 1 k,
the conservatwe forces are
(a) F2 only
{b) F 1, Fa
(e) F,. F,
{d) F,, F,, F3
(e) F1 only
Quest ion 17
Let r = x i -r y j + z k be the pos1t1on vector 1n 1131 : and dehne f (r ) = v(x'y + z') . Let
ro=~ i + >1,j + Zo k and r1 = x1 i + y1 j + z1 k, and define
1.
,,
1 = .f (r) • d r.
The11
(a) /=O
(b) / cannot be computed - the path from ro to r1 is roeeded
(e) 1 = lf (r,)1 - ]f (ro)I
(d) l = xfy1 + zi - xfro - zJ
(e) / = l•1I - l•ol
Question 18
Two part1cles , eacho f massm . move m one dimens1oann d colhdem elast1cal.J Tyhe coefficient
of restitution is ,. T he m1tial speeds of the part,des are u1 and O and their final speeds are
Vi and v2 . The relat1onsh1bpe tweenv 2 and Ut 1s
(a) Vz = U¡(l - <)
(b) vi=U¡E
(e) V,= 11¡/2
(d) v, = u 1(1 + <)/2
(e) V:2= U1
Question 19
A one-d,mensional efast1cc ollision takes place beti.veent wo part1des of equal mass. Their
m1t1I~ velocities are u1 i, LJii. the1r fanal velooties are v1i , ½ i :ind u1 =f v1. Wh,ch one of the
following is true 7
(e) u, - "' and v1 = v,
{d) v1 = ui/ 2 and "-' = u,/2
(e) 111- U, = V¡ - "l,
Ouestion 20
Let a, b E R3 be vectors and let e ;,, O be a scalar. G,veo that a x b = O. !al = 1/ ../3 and
a • b = e, lhen we can deduce that
(a) lb!=c
(b) 3b = ca
(e) b is ort hogonal to a
(e) b = 3ca
Question 21
A uniform hOfizontal plank of mass m and length 21, and whose ends are at points A and B,
rests on a p,vot at point P. The d1stance AP is b. where O < b < f. A mass M Is attached
at point A. lf the system 1s ,n equll1briurn, then we have that M/m =
{a) bg/(1 + b)
(b) (/ - b)/(1 + b)
(e) (1- b)/b
(d) (b- /)9
(e) (b- 1)/1
Question 22
One end of a light rod is at the origin, O, and the other end is at postion r relative to O.
T he rod is subJect to forces F, = - F1 i at posit1011j r and F2 = - F2 j at posit1on r . lt is in
equ1llbnum. By f,nd,ng torques about O. we can deduce lhat
(a) F1 = F,
(b) r = c(2F1 i+ 3F, j), 1"'1ere e 1s a scala,
( e) There is no add1t1onal apphed force at O
(d) 2F1 +3F 2 = O
(e) Equ1l1bnum IS ,mpossible in thlS snuat,on
Quest ion 23
T he situatlon is ldentical to that of the previous question. We can also deduce that
(a) F¡ + F2 = O
(b) r - c(3F1 i+ 2F2 j ), where e IS a scalar
(c) There must be an addit1onal applied force at O
(d) 3F1 +2F, = O
(e) Equd1bnum ,s only poss,ble 1f the rod ,s rotaMg
Ouestion 24
A co1nl les at a d1st:ince0 .15 m rroml he centre of a hor1zonLadl rsc, wh1ch 1sr otat1ng
with angular veloaty w k . The coefflcient of static fnctK>n between the com and the d1sc 1s
¡,, = 0.75. Tak1ng g = 9.81 ms 1, the maximum value of w such that the coin does not
shde is approXJmately
(a) w: 4.7 rad , - 1
(b) w = 5.4 rad s- 1
(e) w = 7.0 rad s- 1
( d) w = 7 5 rad s 1
(e) w = 8.2 rad s- 1