X2 T06 07 miscellaneous dynamics questions (2011)
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Transcript of X2 T06 07 miscellaneous dynamics questions (2011)
![Page 1: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/1.jpg)
Miscellaneous Dynamics Questions
e.g. (i) (1992) – variable angular velocityThe diagram shows a model train T
that is
moving around a circular track, centre O and radius a
metres.
The train is travelling at a constant speed of u
m/s. The point N
is in the same plane as
the track and is x
metres from the nearest point on the track. The line NO
produced
meets the track at S.diagramin theasand Let TOSTNS
![Page 2: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/2.jpg)
uadtdθ and of in terms Express a)
![Page 3: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/3.jpg)
uadtdθ and of in terms Express a) al
![Page 4: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/4.jpg)
uadtdθ and of in terms Express a) al
dtda
dtdl
![Page 5: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/5.jpg)
uadtdθ and of in terms Express a) al
dtda
dtdl
au
dtd
dtdau
![Page 6: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/6.jpg)
uadtdθ and of in terms Express a) al
dtda
dtdl
au
dtd
dtdau
coscoscos
that;deduceand 0sinsinthat Show b)
aaxu
dtd
ax-a
![Page 7: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/7.jpg)
uadtdθ and of in terms Express a) al
dtda
dtdl
au
dtd
dtdau
coscoscos
that;deduceand 0sinsinthat Show b)
aaxu
dtd
ax-a
),(exterior OTNTOSTNONTO
![Page 8: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/8.jpg)
uadtdθ and of in terms Express a) al
dtda
dtdl
au
dtd
dtdau
coscoscos
that;deduceand 0sinsinthat Show b)
aaxu
dtd
ax-a
),(exterior OTNTOSTNONTO
NTO
NTO
![Page 9: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/9.jpg)
uadtdθ and of in terms Express a) al
dtda
dtdl
au
dtd
dtdau
coscoscos
that;deduceand 0sinsinthat Show b)
aaxu
dtd
ax-a
),(exterior OTNTOSTNONTO
NTO
NTO
sinsin
;In xaaNTO
![Page 10: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/10.jpg)
uadtdθ and of in terms Express a) al
dtda
dtdl
au
dtd
dtdau
coscoscos
that;deduceand 0sinsinthat Show b)
aaxu
dtd
ax-a
),(exterior OTNTOSTNONTO
NTO
NTO
sinsin
;In xaaNTO
0sinsin
sinsin
xaa
xaa
![Page 11: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/11.jpg)
0coscos
respect to with atedifferenti
dtdxa
dtd
dtda
t
![Page 12: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/12.jpg)
0coscos
respect to with atedifferenti
dtdxa
dtd
dtda
t
0coscoscos dtdxaa
dtda
![Page 13: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/13.jpg)
0coscos
respect to with atedifferenti
dtdxa
dtd
dtda
t
0coscoscos dtdxaa
dtda
aua
dtdxaa coscoscos
![Page 14: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/14.jpg)
0coscos
respect to with atedifferenti
dtdxa
dtd
dtda
t
0coscoscos dtdxaa
dtda
aua
dtdxaa coscoscos
coscos
cosxaa
udtd
![Page 15: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/15.jpg)
0coscos
respect to with atedifferenti
dtdxa
dtd
dtda
t
0coscoscos dtdxaa
dtda
aua
dtdxaa coscoscos
coscos
cosxaa
udtd
track. the tol tangentiais when 0 that Show c) NTdtd
when NT
is a tangent;
![Page 16: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/16.jpg)
0coscos
respect to with atedifferenti
dtdxa
dtd
dtda
t
0coscoscos dtdxaa
dtda
aua
dtdxaa coscoscos
coscos
cosxaa
udtd
track. the tol tangentiais when 0 that Show c) NTdtd
N
T
O
![Page 17: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/17.jpg)
0coscos
respect to with atedifferenti
dtdxa
dtd
dtda
t
0coscoscos dtdxaa
dtda
aua
dtdxaa coscoscos
coscos
cosxaa
udtd
track. the tol tangentiais when 0 that Show c) NTdtd
N
T
O
when NT
is a tangent;
![Page 18: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/18.jpg)
0coscos
respect to with atedifferenti
dtdxa
dtd
dtda
t
0coscoscos dtdxaa
dtda
aua
dtdxaa coscoscos
coscos
cosxaa
udtd
track. the tol tangentiais when 0 that Show c) NTdtd
N
T
O
when NT
is a tangent;radius)(tangent 90 NTO
![Page 19: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/19.jpg)
0coscos
respect to with atedifferenti
dtdxa
dtd
dtda
t
0coscoscos dtdxaa
dtda
aua
dtdxaa coscoscos
coscos
cosxaa
udtd
track. the tol tangentiais when 0 that Show c) NTdtd
N
T
O
when NT
is a tangent;radius)(tangent 90 NTO
90
![Page 20: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/20.jpg)
0coscos
respect to with atedifferenti
dtdxa
dtd
dtda
t
0coscoscos dtdxaa
dtda
aua
dtdxaa coscoscos
coscos
cosxaa
udtd
track. the tol tangentiais when 0 that Show c) NTdtd
N
T
O
when NT
is a tangent;radius)(tangent 90 NTO
90
cos90cos90cos
xaau
dtd
![Page 21: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/21.jpg)
0coscos
respect to with atedifferenti
dtdxa
dtd
dtda
t
0coscoscos dtdxaa
dtda
aua
dtdxaa coscoscos
coscos
cosxaa
udtd
track. the tol tangentiais when 0 that Show c) NTdtd
N
T
O
when NT
is a tangent;radius)(tangent 90 NTO
90
cos90cos90cos
xaau
dtd
0dtd
![Page 22: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/22.jpg)
d) Suppose that x
= aShow that the train’s angular velocity about N
when is
times the angular velocity about N
when 2
53
0
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d) Suppose that x
= aShow that the train’s angular velocity about N
when is
times the angular velocity about N
when 2
2when
53
0
![Page 24: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/24.jpg)
d) Suppose that x
= aShow that the train’s angular velocity about N
when is
times the angular velocity about N
when 2
2when
53
0
T
ON
![Page 25: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/25.jpg)
d) Suppose that x
= aShow that the train’s angular velocity about N
when is
times the angular velocity about N
when 2
2when
53
0
T
ON
a
![Page 26: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/26.jpg)
d) Suppose that x
= aShow that the train’s angular velocity about N
when is
times the angular velocity about N
when 2
2when
53
0
T
ON
a
2a
![Page 27: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/27.jpg)
d) Suppose that x
= aShow that the train’s angular velocity about N
when is
times the angular velocity about N
when 2
2when
53
0
T
ON
a
2a
a5
![Page 28: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/28.jpg)
d) Suppose that x
= aShow that the train’s angular velocity about N
when is
times the angular velocity about N
when 2
2when
53
0
T
ON
a
2a
a5 52cos
![Page 29: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/29.jpg)
d) Suppose that x
= aShow that the train’s angular velocity about N
when is
times the angular velocity about N
when 2
2when
53
0
T
ON
a
2a
a5 52cos
51
2cos
![Page 30: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/30.jpg)
d) Suppose that x
= aShow that the train’s angular velocity about N
when is
times the angular velocity about N
when 2
2when
53
0
T
ON
a
2a
a5 52cos
51
2cos
cos22
cos
2cos
aa
u
dtd
![Page 31: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/31.jpg)
d) Suppose that x
= aShow that the train’s angular velocity about N
when is
times the angular velocity about N
when 2
2when
53
0
T
ON
a
2a
a5 52cos
51
2cos
cos22
cos
2cos
aa
u
dtd
522
51
51
aa
u
![Page 32: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/32.jpg)
d) Suppose that x
= aShow that the train’s angular velocity about N
when is
times the angular velocity about N
when 2
2when
53
0
T
ON
a
2a
a5 52cos
51
2cos
cos22
cos
2cos
aa
u
dtd
522
51
51
aa
u
au5
![Page 33: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/33.jpg)
d) Suppose that x
= aShow that the train’s angular velocity about N
when is
times the angular velocity about N
when 2
2when
53
0
T
ON
a
2a
a5 52cos
51
2cos
cos22
cos
2cos
aa
u
dtd
522
51
51
aa
u
au5
0when
![Page 34: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/34.jpg)
d) Suppose that x
= aShow that the train’s angular velocity about N
when is
times the angular velocity about N
when 2
2when
53
0
T
ON
a
2a
a5 52cos
51
2cos
cos22
cos
2cos
aa
u
dtd
522
51
51
aa
u
au5
0when
0cos20cos0cosaa
udtd
![Page 35: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/35.jpg)
d) Suppose that x
= aShow that the train’s angular velocity about N
when is
times the angular velocity about N
when 2
2when
53
0
T
ON
a
2a
a5 52cos
51
2cos
cos22
cos
2cos
aa
u
dtd
522
51
51
aa
u
au5
0when
0cos20cos0cosaa
udtd
au3
![Page 36: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/36.jpg)
d) Suppose that x
= aShow that the train’s angular velocity about N
when is
times the angular velocity about N
when 2
2when
53
0
T
ON
a
2a
a5 52cos
51
2cos
cos22
cos
2cos
aa
u
dtd
522
51
51
aa
u
au5
0when
0cos20cos0cosaa
udtd
au3
au53
5
![Page 37: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/37.jpg)
d) Suppose that x
= aShow that the train’s angular velocity about N
when is
times the angular velocity about N
when 2
2when
53
0
T
ON
a
2a
a5 52cos
51
2cos
cos22
cos
2cos
aa
u
dtd
522
51
51
aa
u
au5
0when
0cos20cos0cosaa
udtd
au3
au53
5
0n locity wheangular ve the
times53 is
2n locity wheangular ve theThus
![Page 38: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/38.jpg)
(ii) (2000) A string of length l
is initially vertical and has a mass P
of m
kg attached to it. The mass P
is given a
horizontal velocity of magnitude V and begins to move along the arc of a circle in a counterclockwise direction.Let O
be the centre of this circle and A
the initial
position of P. Let s
denote the arc length AP, , dtdsv
AOP and let the tension in the string be T. The acceleration due to gravity is g
and there are no frictional forces acting on P.
For parts a) to d), assume the mass is moving along the circle.
![Page 39: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/39.jpg)
(ii) (2000) A string of length l
is initially vertical and has a mass P
of m
kg attached to it. The mass P
is given a
horizontal velocity of magnitude V and begins to move along the arc of a circle in a counterclockwise direction.Let O
be the centre of this circle and A
the initial
position of P. Let s
denote the arc length AP, , dtdsv
AOP and let the tension in the string be T. The acceleration due to gravity is g
and there are no frictional forces acting on P.
For parts a) to d), assume the mass is moving along the circle.a) Show that the tangential acceleration of P
is given by
2
2
2
211 v
dd
ldtsd
![Page 40: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/40.jpg)
(ii) (2000) A string of length l
is initially vertical and has a mass P
of m
kg attached to it. The mass P
is given a
horizontal velocity of magnitude V and begins to move along the arc of a circle in a counterclockwise direction.Let O
be the centre of this circle and A
the initial
position of P. Let s
denote the arc length AP, , dtdsv
AOP and let the tension in the string be T. The acceleration due to gravity is g
and there are no frictional forces acting on P.
For parts a) to d), assume the mass is moving along the circle.a) Show that the tangential acceleration of P
is given by
2
2
2
211 v
dd
ldtsd
ls
![Page 41: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/41.jpg)
(ii) (2000) A string of length l
is initially vertical and has a mass P
of m
kg attached to it. The mass P
is given a
horizontal velocity of magnitude V and begins to move along the arc of a circle in a counterclockwise direction.Let O
be the centre of this circle and A
the initial
position of P. Let s
denote the arc length AP, , dtdsv
AOP and let the tension in the string be T. The acceleration due to gravity is g
and there are no frictional forces acting on P.
For parts a) to d), assume the mass is moving along the circle.a) Show that the tangential acceleration of P
is given by
2
2
2
211 v
dd
ldtsd
ls
dtdl
dtdsv
![Page 42: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/42.jpg)
(ii) (2000) A string of length l
is initially vertical and has a mass P
of m
kg attached to it. The mass P
is given a
horizontal velocity of magnitude V and begins to move along the arc of a circle in a counterclockwise direction.Let O
be the centre of this circle and A
the initial
position of P. Let s
denote the arc length AP, , dtdsv
AOP and let the tension in the string be T. The acceleration due to gravity is g
and there are no frictional forces acting on P.
For parts a) to d), assume the mass is moving along the circle.a) Show that the tangential acceleration of P
is given by
2
2
2
211 v
dd
ldtsd
dtdv
dtsd2
2ls
dtdl
dtdsv
![Page 43: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/43.jpg)
(ii) (2000) A string of length l
is initially vertical and has a mass P
of m
kg attached to it. The mass P
is given a
horizontal velocity of magnitude V and begins to move along the arc of a circle in a counterclockwise direction.Let O
be the centre of this circle and A
the initial
position of P. Let s
denote the arc length AP, , dtdsv
AOP and let the tension in the string be T. The acceleration due to gravity is g
and there are no frictional forces acting on P.
For parts a) to d), assume the mass is moving along the circle.a) Show that the tangential acceleration of P
is given by
2
2
2
211 v
dd
ldtsd
dtdv
dtsd2
2ls
dtdl
dtdsv
dtd
ddv
![Page 44: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/44.jpg)
(ii) (2000) A string of length l
is initially vertical and has a mass P
of m
kg attached to it. The mass P
is given a
horizontal velocity of magnitude V and begins to move along the arc of a circle in a counterclockwise direction.Let O
be the centre of this circle and A
the initial
position of P. Let s
denote the arc length AP, , dtdsv
AOP and let the tension in the string be T. The acceleration due to gravity is g
and there are no frictional forces acting on P.
For parts a) to d), assume the mass is moving along the circle.a) Show that the tangential acceleration of P
is given by
2
2
2
211 v
dd
ldtsd
dtdv
dtsd2
2ls
dtdl
dtdsv
dtd
ddv
lv
ddv
![Page 45: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/45.jpg)
(ii) (2000) A string of length l
is initially vertical and has a mass P
of m
kg attached to it. The mass P
is given a
horizontal velocity of magnitude V and begins to move along the arc of a circle in a counterclockwise direction.Let O
be the centre of this circle and A
the initial
position of P. Let s
denote the arc length AP, , dtdsv
AOP and let the tension in the string be T. The acceleration due to gravity is g
and there are no frictional forces acting on P.
For parts a) to d), assume the mass is moving along the circle.a) Show that the tangential acceleration of P
is given by
2
2
2
211 v
dd
ldtsd
dtdv
dtsd2
2ls
dtdl
dtdsv
dtd
ddv
lv
ddv
vddv
l
1
![Page 46: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/46.jpg)
(ii) (2000) A string of length l
is initially vertical and has a mass P
of m
kg attached to it. The mass P
is given a
horizontal velocity of magnitude V and begins to move along the arc of a circle in a counterclockwise direction.Let O
be the centre of this circle and A
the initial
position of P. Let s
denote the arc length AP, , dtdsv
AOP and let the tension in the string be T. The acceleration due to gravity is g
and there are no frictional forces acting on P.
For parts a) to d), assume the mass is moving along the circle.a) Show that the tangential acceleration of P
is given by
2
2
2
211 v
dd
ldtsd
dtdv
dtsd2
2ls
dtdl
dtdsv
dtd
ddv
lv
ddv
vddv
l
1
2
211 v
dvd
ddv
l
![Page 47: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/47.jpg)
(ii) (2000) A string of length l
is initially vertical and has a mass P
of m
kg attached to it. The mass P
is given a
horizontal velocity of magnitude V and begins to move along the arc of a circle in a counterclockwise direction.Let O
be the centre of this circle and A
the initial
position of P. Let s
denote the arc length AP, , dtdsv
AOP and let the tension in the string be T. The acceleration due to gravity is g
and there are no frictional forces acting on P.
For parts a) to d), assume the mass is moving along the circle.a) Show that the tangential acceleration of P
is given by
2
2
2
211 v
dd
ldtsd
dtdv
dtsd2
2ls
dtdl
dtdsv
dtd
ddv
lv
ddv
vddv
l
1
2
211 v
dvd
ddv
l
2
211 v
dd
l
![Page 48: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/48.jpg)
b) Show that the equation of motion of P
is
sin211 2 gv
dd
l
![Page 49: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/49.jpg)
b) Show that the equation of motion of P
is
sin211 2 gv
dd
l
![Page 50: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/50.jpg)
b) Show that the equation of motion of P
is
sin211 2 gv
dd
l
T
![Page 51: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/51.jpg)
b) Show that the equation of motion of P
is
sin211 2 gv
dd
l
T
mg
![Page 52: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/52.jpg)
b) Show that the equation of motion of P
is
sin211 2 gv
dd
l
T
mg
![Page 53: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/53.jpg)
b) Show that the equation of motion of P
is
sin211 2 gv
dd
l
T
mg
sinmg
![Page 54: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/54.jpg)
b) Show that the equation of motion of P
is
sin211 2 gv
dd
l
T
mg
sm
sinmg
![Page 55: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/55.jpg)
b) Show that the equation of motion of P
is
sin211 2 gv
dd
l
T
mg
sm
sinmg
sinmgsm
![Page 56: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/56.jpg)
b) Show that the equation of motion of P
is
sin211 2 gv
dd
l
T
mg
sm
sinmg
sinmgsm sings
![Page 57: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/57.jpg)
b) Show that the equation of motion of P
is
sin211 2 gv
dd
l
T
mg
sm
sinmg
sinmgsm sings
sin211 2 gv
dd
l
![Page 58: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/58.jpg)
b) Show that the equation of motion of P
is
sin211 2 gv
dd
l
T
mg
sm
sinmg
sinmgsm sings
sin211 2 gv
dd
l
cos12 that Deduce c) 22 glvV
![Page 59: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/59.jpg)
b) Show that the equation of motion of P
is
sin211 2 gv
dd
l
T
mg
sm
sinmg
sinmgsm sings
sin211 2 gv
dd
l
cos12 that Deduce c) 22 glvV
sin211 2 gv
dd
l
![Page 60: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/60.jpg)
b) Show that the equation of motion of P
is
sin211 2 gv
dd
l
T
mg
sm
sinmg
sinmgsm sings
sin211 2 gv
dd
l
cos12 that Deduce c) 22 glvV
sin211 2 gv
dd
l
sin21 2 glv
dd
![Page 61: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/61.jpg)
b) Show that the equation of motion of P
is
sin211 2 gv
dd
l
T
mg
sm
sinmg
sinmgsm sings
sin211 2 gv
dd
l
cos12 that Deduce c) 22 glvV
sin211 2 gv
dd
l
sin21 2 glv
dd
cglv
cglv
cos2
cos21
2
2
![Page 62: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/62.jpg)
b) Show that the equation of motion of P
is
sin211 2 gv
dd
l
T
mg
sm
sinmg
sinmgsm sings
sin211 2 gv
dd
l
cos12 that Deduce c) 22 glvV
sin211 2 gv
dd
l
sin21 2 glv
dd
cglv
cglv
cos2
cos21
2
2
glVccglV
Vv
22
0,when
2
2
![Page 63: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/63.jpg)
b) Show that the equation of motion of P
is
sin211 2 gv
dd
l
T
mg
sm
sinmg
sinmgsm sings
sin211 2 gv
dd
l
cos12 that Deduce c) 22 glvV
sin211 2 gv
dd
l
sin21 2 glv
dd
cglv
cglv
cos2
cos21
2
2
glVccglV
Vv
22
0,when
2
2
cos12cos22
2cos2
22
22
22
glvVglglvV
glVglv
![Page 64: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/64.jpg)
21cosy Explain wh d) mvl
θT-mg
![Page 65: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/65.jpg)
21cosy Explain wh d) mvl
θT-mg
![Page 66: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/66.jpg)
21cosy Explain wh d) mvl
θT-mg T
![Page 67: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/67.jpg)
21cosy Explain wh d) mvl
θT-mg T cosmg
![Page 68: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/68.jpg)
21cosy Explain wh d) mvl
θT-mg T cosmgxm
![Page 69: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/69.jpg)
21cosy Explain wh d) mvl
θT-mg T cosmgxm
cosmgTxm
![Page 70: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/70.jpg)
21cosy Explain wh d) mvl
θT-mg T cosmgxm
cosmgTxm But, the resultant force towards the centre is centripetal force.
![Page 71: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/71.jpg)
21cosy Explain wh d) mvl
θT-mg T cosmgxm
cosmgTxm
2
2
1cos
cos
mvl
mgT
mgTl
mv
But, the resultant force towards the centre is centripetal force.
![Page 72: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/72.jpg)
21cosy Explain wh d) mvl
θT-mg T cosmgxm
cosmgTxm
2
2
1cos
cos
mvl
mgT
mgTl
mv
But, the resultant force towards the centre is centripetal force.
0at which of value theFind .3 that Suppose e) 2 TglV
![Page 73: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/73.jpg)
21cosy Explain wh d) mvl
θT-mg T cosmgxm
cosmgTxm
2
2
1cos
cos
mvl
mgT
mgTl
mv
But, the resultant force towards the centre is centripetal force.
0at which of value theFind .3 that Suppose e) 2 TglV
cos121cos 2 glVml
mgT
![Page 74: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/74.jpg)
21cosy Explain wh d) mvl
θT-mg T cosmgxm
cosmgTxm
2
2
1cos
cos
mvl
mgT
mgTl
mv
But, the resultant force towards the centre is centripetal force.
0at which of value theFind .3 that Suppose e) 2 TglV
cos121cos 2 glVml
mgT
cos1231cos0 glglml
mg
![Page 75: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/75.jpg)
21cosy Explain wh d) mvl
θT-mg T cosmgxm
cosmgTxm
2
2
1cos
cos
mvl
mgT
mgTl
mv
But, the resultant force towards the centre is centripetal force.
0at which of value theFind .3 that Suppose e) 2 TglV
cos121cos 2 glVml
mgT
cos1231cos0 glglml
mg
cos2cos ggmmg
![Page 76: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/76.jpg)
21cosy Explain wh d) mvl
θT-mg T cosmgxm
cosmgTxm
2
2
1cos
cos
mvl
mgT
mgTl
mv
But, the resultant force towards the centre is centripetal force.
0at which of value theFind .3 that Suppose e) 2 TglV
cos121cos 2 glVml
mgT
cos1231cos0 glglml
mg
cos2cos ggmmg
mgmg cos3
![Page 77: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/77.jpg)
21cosy Explain wh d) mvl
θT-mg T cosmgxm
cosmgTxm
2
2
1cos
cos
mvl
mgT
mgTl
mv
But, the resultant force towards the centre is centripetal force.
0at which of value theFind .3 that Suppose e) 2 TglV
cos121cos 2 glVml
mgT
cos1231cos0 glglml
mg
cos2cos ggmmg
mgmg cos3
31cos
![Page 78: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/78.jpg)
21cosy Explain wh d) mvl
θT-mg T cosmgxm
cosmgTxm
2
2
1cos
cos
mvl
mgT
mgTl
mv
But, the resultant force towards the centre is centripetal force.
0at which of value theFind .3 that Suppose e) 2 TglV
cos121cos 2 glVml
mgT
cos1231cos0 glglml
mg
cos2cos ggmmg
mgmg cos3
31cos
radians911.1
![Page 79: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/79.jpg)
f) Consider the situation in part e). Briefly describe, in words, the path of P
after the tension T
becomes zero.
![Page 80: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/80.jpg)
f) Consider the situation in part e). Briefly describe, in words, the path of P
after the tension T
becomes zero.
When T = 0, the particle would undergo projectile motion, i.e. it would follow a parabolic arc.Its initial velocity would be tangential to the circle with magnitude;
![Page 81: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/81.jpg)
f) Consider the situation in part e). Briefly describe, in words, the path of P
after the tension T
becomes zero.
When T = 0, the particle would undergo projectile motion, i.e. it would follow a parabolic arc.Its initial velocity would be tangential to the circle with magnitude;
21cos mvl
mgT
![Page 82: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/82.jpg)
f) Consider the situation in part e). Briefly describe, in words, the path of P
after the tension T
becomes zero.
When T = 0, the particle would undergo projectile motion, i.e. it would follow a parabolic arc.Its initial velocity would be tangential to the circle with magnitude;
21cos mvl
mgT
2131 mv
lmg
![Page 83: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/83.jpg)
f) Consider the situation in part e). Briefly describe, in words, the path of P
after the tension T
becomes zero.
When T = 0, the particle would undergo projectile motion, i.e. it would follow a parabolic arc.Its initial velocity would be tangential to the circle with magnitude;
21cos mvl
mgT
2131 mv
lmg
3
32
glv
glv
![Page 84: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/84.jpg)
f) Consider the situation in part e). Briefly describe, in words, the path of P
after the tension T
becomes zero.
When T = 0, the particle would undergo projectile motion, i.e. it would follow a parabolic arc.Its initial velocity would be tangential to the circle with magnitude;
21cos mvl
mgT
2131 mv
lmg
3
32
glv
glv
![Page 85: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/85.jpg)
f) Consider the situation in part e). Briefly describe, in words, the path of P
after the tension T
becomes zero.
When T = 0, the particle would undergo projectile motion, i.e. it would follow a parabolic arc.Its initial velocity would be tangential to the circle with magnitude;
21cos mvl
mgT
2131 mv
lmg
3
32
glv
glv
![Page 86: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/86.jpg)
f) Consider the situation in part e). Briefly describe, in words, the path of P
after the tension T
becomes zero.
When T = 0, the particle would undergo projectile motion, i.e. it would follow a parabolic arc.Its initial velocity would be tangential to the circle with magnitude;
21cos mvl
mgT
2131 mv
lmg
3
32
glv
glv
![Page 87: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/87.jpg)
f) Consider the situation in part e). Briefly describe, in words, the path of P
after the tension T
becomes zero.
When T = 0, the particle would undergo projectile motion, i.e. it would follow a parabolic arc.Its initial velocity would be tangential to the circle with magnitude;
21cos mvl
mgT
2131 mv
lmg
3
32
glv
glv
![Page 88: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/88.jpg)
f) Consider the situation in part e). Briefly describe, in words, the path of P
after the tension T
becomes zero.
When T = 0, the particle would undergo projectile motion, i.e. it would follow a parabolic arc.Its initial velocity would be tangential to the circle with magnitude;
21cos mvl
mgT
2131 mv
lmg
3
32
glv
glv
![Page 89: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/89.jpg)
(iii) (2003)A particle of mass m
is thrown from the top, O, of a very tall building
with an initial velocity u
at an angle of to the horizontal. The particle experiences the effect of gravity, and a resistance proportional to its velocity in both directions.
The equations of motion in the horizontal and vertical directions are given respectively by
gykyxkx and where k is a constant and the acceleration due to gravity is g.
(You are NOT required to show these)
![Page 90: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/90.jpg)
cosresult theDerive a) ktuex
![Page 91: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/91.jpg)
cosresult theDerive a) ktuex
xkdtxd
![Page 92: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/92.jpg)
cosresult theDerive a) ktuex
xkdtxd
x
u xxd
kt
cos
1
![Page 93: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/93.jpg)
cosresult theDerive a) ktuex
xkdtxd
x
u xxd
kt
cos
1
xuxk
t coslog1
![Page 94: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/94.jpg)
cosresult theDerive a) ktuex
xkdtxd
x
u xxd
kt
cos
1
xuxk
t coslog1
cosloglog1 uxk
t
![Page 95: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/95.jpg)
cosresult theDerive a) ktuex
xkdtxd
x
u xxd
kt
cos
1
xuxk
t coslog1
cosloglog1 uxk
t
coslog1
ux
kt
![Page 96: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/96.jpg)
cosresult theDerive a) ktuex
xkdtxd
x
u xxd
kt
cos
1
xuxk
t coslog1
cosloglog1 uxk
t
coslog1
ux
kt
coscos
coslog
kt
kt
uex
eu
xu
xkt
![Page 97: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/97.jpg)
cosresult theDerive a) ktuex
xkdtxd
x
u xxd
kt
cos
1
xuxk
t coslog1
cosloglog1 uxk
t
coslog1
ux
kt
coscos
coslog
kt
kt
uex
eu
xu
xkt
condition initial andmotion ofequation
eappropriat thesatisfies sin1t Verify tha b) gegkuk
y kt
![Page 98: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/98.jpg)
cosresult theDerive a) ktuex
xkdtxd
x
u xxd
kt
cos
1
xuxk
t coslog1
cosloglog1 uxk
t
coslog1
ux
kt
coscos
coslog
kt
kt
uex
eu
xu
xkt
condition initial andmotion ofequation
eappropriat thesatisfies sin1t Verify tha b) gegkuk
y kt
gykdtyd
![Page 99: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/99.jpg)
cosresult theDerive a) ktuex
xkdtxd
x
u xxd
kt
cos
1
xuxk
t coslog1
cosloglog1 uxk
t
coslog1
ux
kt
coscos
coslog
kt
kt
uex
eu
xu
xkt
condition initial andmotion ofequation
eappropriat thesatisfies sin1t Verify tha b) gegkuk
y kt
gykdtyd
y
u gykydt
sin
![Page 100: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/100.jpg)
cosresult theDerive a) ktuex
xkdtxd
x
u xxd
kt
cos
1
xuxk
t coslog1
cosloglog1 uxk
t
coslog1
ux
kt
coscos
coslog
kt
kt
uex
eu
xu
xkt
condition initial andmotion ofequation
eappropriat thesatisfies sin1t Verify tha b) gegkuk
y kt
gykdtyd
y
u gykydt
sin
yugyk
kt sinlog1
![Page 101: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/101.jpg)
cosresult theDerive a) ktuex
xkdtxd
x
u xxd
kt
cos
1
xuxk
t coslog1
cosloglog1 uxk
t
coslog1
ux
kt
coscos
coslog
kt
kt
uex
eu
xu
xkt
condition initial andmotion ofequation
eappropriat thesatisfies sin1t Verify tha b) gegkuk
y kt
gykdtyd
y
u gykydt
sin
yugyk
kt sinlog1
gkugykkt sinloglog
![Page 102: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/102.jpg)
cosresult theDerive a) ktuex
xkdtxd
x
u xxd
kt
cos
1
xuxk
t coslog1
cosloglog1 uxk
t
coslog1
ux
kt
coscos
coslog
kt
kt
uex
eu
xu
xkt
condition initial andmotion ofequation
eappropriat thesatisfies sin1t Verify tha b) gegkuk
y kt
gykdtyd
y
u gykydt
sin
yugyk
kt sinlog1
gkugykkt sinloglog
gkugykkt
sinlog
![Page 103: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/103.jpg)
cosresult theDerive a) ktuex
xkdtxd
x
u xxd
kt
cos
1
xuxk
t coslog1
cosloglog1 uxk
t
coslog1
ux
kt
coscos
coslog
kt
kt
uex
eu
xu
xkt
condition initial andmotion ofequation
eappropriat thesatisfies sin1t Verify tha b) gegkuk
y kt
gykdtyd
y
u gykydt
sin
yugyk
kt sinlog1
gkugykkt sinloglog
gkugykkt
sinlog
ktegku
gyk
sin
![Page 104: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/104.jpg)
cosresult theDerive a) ktuex
xkdtxd
x
u xxd
kt
cos
1
xuxk
t coslog1
cosloglog1 uxk
t
coslog1
ux
kt
coscos
coslog
kt
kt
uex
eu
xu
xkt
condition initial andmotion ofequation
eappropriat thesatisfies sin1t Verify tha b) gegkuk
y kt
gykdtyd
y
u gykydt
sin
yugyk
kt sinlog1
gkugykkt sinloglog
gkugykkt
sinlog
ktegku
gyk
sin
gegkuk
y kt sin1
![Page 105: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/105.jpg)
c) Find the value of t
when the particle reaches its maximum height
![Page 106: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/106.jpg)
c) Find the value of t
when the particle reaches its maximum height0when occursheight Maximum y
![Page 107: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/107.jpg)
c) Find the value of t
when the particle reaches its maximum height0when occursheight Maximum y
0sinlog1ugyk
kt
![Page 108: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/108.jpg)
c) Find the value of t
when the particle reaches its maximum height0when occursheight Maximum y
0sinlog1ugyk
kt
gkugk
t sinloglog1
![Page 109: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/109.jpg)
c) Find the value of t
when the particle reaches its maximum height0when occursheight Maximum y
0sinlog1ugyk
kt
gkugk
t sinloglog1
ggku
kt sinlog1
![Page 110: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/110.jpg)
c) Find the value of t
when the particle reaches its maximum height0when occursheight Maximum y
0sinlog1ugyk
kt
gkugk
t sinloglog1
ggku
kt sinlog1
d) What is the limiting value of the horizontal displacement of the particle?
![Page 111: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/111.jpg)
c) Find the value of t
when the particle reaches its maximum height0when occursheight Maximum y
0sinlog1ugyk
kt
gkugk
t sinloglog1
ggku
kt sinlog1
d) What is the limiting value of the horizontal displacement of the particle?
cos
cos
kt
kt
uedtdx
uex
![Page 112: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/112.jpg)
c) Find the value of t
when the particle reaches its maximum height0when occursheight Maximum y
0sinlog1ugyk
kt
gkugk
t sinloglog1
ggku
kt sinlog1
d) What is the limiting value of the horizontal displacement of the particle?
cos
cos
kt
kt
uedtdx
uex
0
coslim dteux kt
t
![Page 113: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/113.jpg)
c) Find the value of t
when the particle reaches its maximum height0when occursheight Maximum y
0sinlog1ugyk
kt
gkugk
t sinloglog1
ggku
kt sinlog1
d) What is the limiting value of the horizontal displacement of the particle?
cos
cos
kt
kt
uedtdx
uex
0
coslim dteux kt
t
tkt
te
kux
0
1coslim
![Page 114: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/114.jpg)
c) Find the value of t
when the particle reaches its maximum height0when occursheight Maximum y
0sinlog1ugyk
kt
gkugk
t sinloglog1
ggku
kt sinlog1
d) What is the limiting value of the horizontal displacement of the particle?
cos
cos
kt
kt
uedtdx
uex
0
coslim dteux kt
t
tkt
te
kux
0
1coslim
1coslim
kt
te
kux
![Page 115: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/115.jpg)
c) Find the value of t
when the particle reaches its maximum height0when occursheight Maximum y
0sinlog1ugyk
kt
gkugk
t sinloglog1
ggku
kt sinlog1
d) What is the limiting value of the horizontal displacement of the particle?
cos
cos
kt
kt
uedtdx
uex
0
coslim dteux kt
t
tkt
te
kux
0
1coslim
1coslim
kt
te
kux
kux cos
![Page 116: X2 T06 07 miscellaneous dynamics questions (2011)](https://reader033.fdocuments.us/reader033/viewer/2022052900/555e0476d8b42a3f618b57ce/html5/thumbnails/116.jpg)
Exercise 9E; 1 to 4, 7
Exercise 9F; 1, 2, 4, 7, 9, 12, 14, 16, 20, 22, 25